C3.qti

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<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
  <objectbank ident="C3">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- C3</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="C3-2287" title="C3 | Homogeneous second-order linear ODE | ver. 2287"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 128 \, {x} = -32 \, {x'}" alt="-2 \, {x''} - 128 \, {x} = -32 \, {x'}" title="-2 \, {x''} - 128 \, {x} = -32 \, {x'}" data-latex="-2 \, {x''} - 128 \, {x} = -32 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y''} - 12 \, {y'} = 24 \, {y}" alt="-3 \, {y''} - 12 \, {y'} = 24 \, {y}" title="-3 \, {y''} - 12 \, {y'} = 24 \, {y}" data-latex="-3 \, {y''} - 12 \, {y'} = 24 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%20128%20%5C,%20%7Bx%7D%20=%20-32%20%5C,%20%7Bx'%7D" alt="-2 \, {x''} - 128 \, {x} = -32 \, {x'}" title="-2 \, {x''} - 128 \, {x} = -32 \, {x'}" data-latex="-2 \, {x''} - 128 \, {x} = -32 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By''%7D%20-%2012%20%5C,%20%7By'%7D%20=%2024%20%5C,%20%7By%7D" alt="-3 \, {y''} - 12 \, {y'} = 24 \, {y}" title="-3 \, {y''} - 12 \, {y'} = 24 \, {y}" data-latex="-3 \, {y''} - 12 \, {y'} = 24 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5044" title="C3 | Homogeneous second-order linear ODE | ver. 5044"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?39 \, {x} + 3 \, {x''} = 18 \, {x'}" alt="39 \, {x} + 3 \, {x''} = 18 \, {x'}" title="39 \, {x} + 3 \, {x''} = 18 \, {x'}" data-latex="39 \, {x} + 3 \, {x''} = 18 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-128 \, {y} = 32 \, {y'} + 2 \, {y''}" alt="-128 \, {y} = 32 \, {y'} + 2 \, {y''}" title="-128 \, {y} = 32 \, {y'} + 2 \, {y''}" data-latex="-128 \, {y} = 32 \, {y'} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?39%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D%20=%2018%20%5C,%20%7Bx'%7D" alt="39 \, {x} + 3 \, {x''} = 18 \, {x'}" title="39 \, {x} + 3 \, {x''} = 18 \, {x'}" data-latex="39 \, {x} + 3 \, {x''} = 18 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-128%20%5C,%20%7By%7D%20=%2032%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-128 \, {y} = 32 \, {y'} + 2 \, {y''}" title="-128 \, {y} = 32 \, {y'} + 2 \, {y''}" data-latex="-128 \, {y} = 32 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7815" title="C3 | Homogeneous second-order linear ODE | ver. 7815"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, {y'} = 27 \, {y} + 3 \, {y''}" alt="18 \, {y'} = 27 \, {y} + 3 \, {y''}" title="18 \, {y'} = 27 \, {y} + 3 \, {y''}" data-latex="18 \, {y'} = 27 \, {y} + 3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0" alt="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0" title="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0" data-latex="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18%20%5C,%20%7By'%7D%20=%2027%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By''%7D" alt="18 \, {y'} = 27 \, {y} + 3 \, {y''}" title="18 \, {y'} = 27 \, {y} + 3 \, {y''}" data-latex="18 \, {y'} = 27 \, {y} + 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-102%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20-%2018%20%5C,%20%7Bx'%7D%20=%200" alt="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0" title="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0" data-latex="-102 \, {x} - 3 \, {x''} - 18 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7223" title="C3 | Homogeneous second-order linear ODE | ver. 7223"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x'} = -10 \, {x} - 2 \, {x''}" alt="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" title="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" data-latex="-4 \, {x'} = -10 \, {x} - 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" alt="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" title="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" data-latex="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx'%7D%20=%20-10%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7Bx''%7D" alt="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" title="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" data-latex="-4 \, {x'} = -10 \, {x} - 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20243%20%5C,%20%7By%7D%20+%2054%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D" alt="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" title="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" data-latex="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9633" title="C3 | Homogeneous second-order linear ODE | ver. 9633"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}" alt="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}" title="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}" data-latex="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = -60 \, {y'} - 300 \, {y}" alt="3 \, {y''} = -60 \, {y'} - 300 \, {y}" title="3 \, {y''} = -60 \, {y'} - 300 \, {y}" data-latex="3 \, {y''} = -60 \, {y'} - 300 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-16%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D%20-%2064%20%5C,%20%7Bx%7D" alt="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}" title="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}" data-latex="0 = -16 \, {x'} - 2 \, {x''} - 64 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%20-60%20%5C,%20%7By'%7D%20-%20300%20%5C,%20%7By%7D" alt="3 \, {y''} = -60 \, {y'} - 300 \, {y}" title="3 \, {y''} = -60 \, {y'} - 300 \, {y}" data-latex="3 \, {y''} = -60 \, {y'} - 300 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3912" title="C3 | Homogeneous second-order linear ODE | ver. 3912"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} = -4 \, {y'} - 2 \, {y}" alt="2 \, {y''} = -4 \, {y'} - 2 \, {y}" title="2 \, {y''} = -4 \, {y'} - 2 \, {y}" data-latex="2 \, {y''} = -4 \, {y'} - 2 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24 \, {x} = -12 \, {x'} - 3 \, {x''}" alt="24 \, {x} = -12 \, {x'} - 3 \, {x''}" title="24 \, {x} = -12 \, {x'} - 3 \, {x''}" data-latex="24 \, {x} = -12 \, {x'} - 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20=%20-4%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By%7D" alt="2 \, {y''} = -4 \, {y'} - 2 \, {y}" title="2 \, {y''} = -4 \, {y'} - 2 \, {y}" data-latex="2 \, {y''} = -4 \, {y'} - 2 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24%20%5C,%20%7Bx%7D%20=%20-12%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx''%7D" alt="24 \, {x} = -12 \, {x'} - 3 \, {x''}" title="24 \, {x} = -12 \, {x'} - 3 \, {x''}" data-latex="24 \, {x} = -12 \, {x'} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9471" title="C3 | Homogeneous second-order linear ODE | ver. 9471"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}" alt="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}" title="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}" data-latex="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, {y'} = 2 \, {y''} + 32 \, {y}" alt="-16 \, {y'} = 2 \, {y''} + 32 \, {y}" title="-16 \, {y'} = 2 \, {y''} + 32 \, {y}" data-latex="-16 \, {y'} = 2 \, {y''} + 32 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7Bx''%7D%20+%2020%20%5C,%20%7Bx'%7D%20+%2058%20%5C,%20%7Bx%7D" alt="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}" title="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}" data-latex="0 = 2 \, {x''} + 20 \, {x'} + 58 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20%7By'%7D%20=%202%20%5C,%20%7By''%7D%20+%2032%20%5C,%20%7By%7D" alt="-16 \, {y'} = 2 \, {y''} + 32 \, {y}" title="-16 \, {y'} = 2 \, {y''} + 32 \, {y}" data-latex="-16 \, {y'} = 2 \, {y''} + 32 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2204" title="C3 | Homogeneous second-order linear ODE | ver. 2204"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} = -36 \, {y} - 2 \, {y''}" alt="-12 \, {y'} = -36 \, {y} - 2 \, {y''}" title="-12 \, {y'} = -36 \, {y} - 2 \, {y''}" data-latex="-12 \, {y'} = -36 \, {y} - 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 128 \, {x} = 32 \, {x'}" alt="-2 \, {x''} - 128 \, {x} = 32 \, {x'}" title="-2 \, {x''} - 128 \, {x} = 32 \, {x'}" data-latex="-2 \, {x''} - 128 \, {x} = 32 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20=%20-36%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D" alt="-12 \, {y'} = -36 \, {y} - 2 \, {y''}" title="-12 \, {y'} = -36 \, {y} - 2 \, {y''}" data-latex="-12 \, {y'} = -36 \, {y} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%20128%20%5C,%20%7Bx%7D%20=%2032%20%5C,%20%7Bx'%7D" alt="-2 \, {x''} - 128 \, {x} = 32 \, {x'}" title="-2 \, {x''} - 128 \, {x} = 32 \, {x'}" data-latex="-2 \, {x''} - 128 \, {x} = 32 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3005" title="C3 | Homogeneous second-order linear ODE | ver. 3005"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, {x} = -2 \, {x''} - 12 \, {x'}" alt="18 \, {x} = -2 \, {x''} - 12 \, {x'}" title="18 \, {x} = -2 \, {x''} - 12 \, {x'}" data-latex="18 \, {x} = -2 \, {x''} - 12 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} = -58 \, {y} - 20 \, {y'}" alt="2 \, {y''} = -58 \, {y} - 20 \, {y'}" title="2 \, {y''} = -58 \, {y} - 20 \, {y'}" data-latex="2 \, {y''} = -58 \, {y} - 20 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7Bx''%7D%20-%2012%20%5C,%20%7Bx'%7D" alt="18 \, {x} = -2 \, {x''} - 12 \, {x'}" title="18 \, {x} = -2 \, {x''} - 12 \, {x'}" data-latex="18 \, {x} = -2 \, {x''} - 12 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20=%20-58%20%5C,%20%7By%7D%20-%2020%20%5C,%20%7By'%7D" alt="2 \, {y''} = -58 \, {y} - 20 \, {y'}" title="2 \, {y''} = -58 \, {y} - 20 \, {y'}" data-latex="2 \, {y''} = -58 \, {y} - 20 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3702" title="C3 | Homogeneous second-order linear ODE | ver. 3702"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0" alt="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0" title="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0" data-latex="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18 \, {y} = 2 \, {y''} + 12 \, {y'}" alt="-18 \, {y} = 2 \, {y''} + 12 \, {y'}" title="-18 \, {y} = 2 \, {y''} + 12 \, {y'}" data-latex="-18 \, {y} = 2 \, {y''} + 12 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20+%2078%20%5C,%20%7Bx%7D%20+%206%20%5C,%20%7Bx'%7D%20=%200" alt="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0" title="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0" data-latex="3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18%20%5C,%20%7By%7D%20=%202%20%5C,%20%7By''%7D%20+%2012%20%5C,%20%7By'%7D" alt="-18 \, {y} = 2 \, {y''} + 12 \, {y'}" title="-18 \, {y} = 2 \, {y''} + 12 \, {y'}" data-latex="-18 \, {y} = 2 \, {y''} + 12 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5060" title="C3 | Homogeneous second-order linear ODE | ver. 5060"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} = 30 \, {x'} - 75 \, {x}" alt="3 \, {x''} = 30 \, {x'} - 75 \, {x}" title="3 \, {x''} = 30 \, {x'} - 75 \, {x}" data-latex="3 \, {x''} = 30 \, {x'} - 75 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} - 6 \, {y'} = -3 \, {y''}" alt="6 \, {y} - 6 \, {y'} = -3 \, {y''}" title="6 \, {y} - 6 \, {y'} = -3 \, {y''}" data-latex="6 \, {y} - 6 \, {y'} = -3 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20=%2030%20%5C,%20%7Bx'%7D%20-%2075%20%5C,%20%7Bx%7D" alt="3 \, {x''} = 30 \, {x'} - 75 \, {x}" title="3 \, {x''} = 30 \, {x'} - 75 \, {x}" data-latex="3 \, {x''} = 30 \, {x'} - 75 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20-%206%20%5C,%20%7By'%7D%20=%20-3%20%5C,%20%7By''%7D" alt="6 \, {y} - 6 \, {y'} = -3 \, {y''}" title="6 \, {y} - 6 \, {y'} = -3 \, {y''}" data-latex="6 \, {y} - 6 \, {y'} = -3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5491" title="C3 | Homogeneous second-order linear ODE | ver. 5491"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-60 \, {y} - 3 \, {y''} = 12 \, {y'}" alt="-60 \, {y} - 3 \, {y''} = 12 \, {y'}" title="-60 \, {y} - 3 \, {y''} = 12 \, {y'}" data-latex="-60 \, {y} - 3 \, {y''} = 12 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?40 \, {x'} + 2 \, {x''} = -200 \, {x}" alt="40 \, {x'} + 2 \, {x''} = -200 \, {x}" title="40 \, {x'} + 2 \, {x''} = -200 \, {x}" data-latex="40 \, {x'} + 2 \, {x''} = -200 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-60%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By''%7D%20=%2012%20%5C,%20%7By'%7D" alt="-60 \, {y} - 3 \, {y''} = 12 \, {y'}" title="-60 \, {y} - 3 \, {y''} = 12 \, {y'}" data-latex="-60 \, {y} - 3 \, {y''} = 12 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?40%20%5C,%20%7Bx'%7D%20+%202%20%5C,%20%7Bx''%7D%20=%20-200%20%5C,%20%7Bx%7D" alt="40 \, {x'} + 2 \, {x''} = -200 \, {x}" title="40 \, {x'} + 2 \, {x''} = -200 \, {x}" data-latex="40 \, {x'} + 2 \, {x''} = -200 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3330" title="C3 | Homogeneous second-order linear ODE | ver. 3330"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} = -8 \, {y'} + 2 \, {y''}" alt="-8 \, {y} = -8 \, {y'} + 2 \, {y''}" title="-8 \, {y} = -8 \, {y'} + 2 \, {y''}" data-latex="-8 \, {y} = -8 \, {y'} + 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30 \, {x'} = 87 \, {x} + 3 \, {x''}" alt="-30 \, {x'} = 87 \, {x} + 3 \, {x''}" title="-30 \, {x'} = 87 \, {x} + 3 \, {x''}" data-latex="-30 \, {x'} = 87 \, {x} + 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20=%20-8%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-8 \, {y} = -8 \, {y'} + 2 \, {y''}" title="-8 \, {y} = -8 \, {y'} + 2 \, {y''}" data-latex="-8 \, {y} = -8 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30%20%5C,%20%7Bx'%7D%20=%2087%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D" alt="-30 \, {x'} = 87 \, {x} + 3 \, {x''}" title="-30 \, {x'} = 87 \, {x} + 3 \, {x''}" data-latex="-30 \, {x'} = 87 \, {x} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5158" title="C3 | Homogeneous second-order linear ODE | ver. 5158"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} = -200 \, {x} - 40 \, {x'}" alt="2 \, {x''} = -200 \, {x} - 40 \, {x'}" title="2 \, {x''} = -200 \, {x} - 40 \, {x'}" data-latex="2 \, {x''} = -200 \, {x} - 40 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0" alt="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0" title="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0" data-latex="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20=%20-200%20%5C,%20%7Bx%7D%20-%2040%20%5C,%20%7Bx'%7D" alt="2 \, {x''} = -200 \, {x} - 40 \, {x'}" title="2 \, {x''} = -200 \, {x} - 40 \, {x'}" data-latex="2 \, {x''} = -200 \, {x} - 40 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By''%7D%20-%2030%20%5C,%20%7By%7D%20=%200" alt="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0" title="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0" data-latex="-18 \, {y'} - 3 \, {y''} - 30 \, {y} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7352" title="C3 | Homogeneous second-order linear ODE | ver. 7352"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} - 243 \, {x} = 54 \, {x'}" alt="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" title="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" data-latex="-3 \, {x''} - 243 \, {x} = 54 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 52 \, {y} = -20 \, {y'}" alt="2 \, {y''} + 52 \, {y} = -20 \, {y'}" title="2 \, {y''} + 52 \, {y} = -20 \, {y'}" data-latex="2 \, {y''} + 52 \, {y} = -20 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20-%20243%20%5C,%20%7Bx%7D%20=%2054%20%5C,%20%7Bx'%7D" alt="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" title="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" data-latex="-3 \, {x''} - 243 \, {x} = 54 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%2052%20%5C,%20%7By%7D%20=%20-20%20%5C,%20%7By'%7D" alt="2 \, {y''} + 52 \, {y} = -20 \, {y'}" title="2 \, {y''} + 52 \, {y} = -20 \, {y'}" data-latex="2 \, {y''} + 52 \, {y} = -20 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2004" title="C3 | Homogeneous second-order linear ODE | ver. 2004"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24 \, {y'} + 2 \, {y''} = -72 \, {y}" alt="-24 \, {y'} + 2 \, {y''} = -72 \, {y}" title="-24 \, {y'} + 2 \, {y''} = -72 \, {y}" data-latex="-24 \, {y'} + 2 \, {y''} = -72 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}" alt="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}" title="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}" data-latex="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D%20=%20-72%20%5C,%20%7By%7D" alt="-24 \, {y'} + 2 \, {y''} = -72 \, {y}" title="-24 \, {y'} + 2 \, {y''} = -72 \, {y}" data-latex="-24 \, {y'} + 2 \, {y''} = -72 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7Bx''%7D%20+%2026%20%5C,%20%7Bx%7D%20+%208%20%5C,%20%7Bx'%7D" alt="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}" title="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}" data-latex="0 = 2 \, {x''} + 26 \, {x} + 8 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" alt="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7041" title="C3 | Homogeneous second-order linear ODE | ver. 7041"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}" alt="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}" title="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}" data-latex="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?108 \, {x} + 3 \, {x''} = 36 \, {x'}" alt="108 \, {x} + 3 \, {x''} = 36 \, {x'}" title="108 \, {x} + 3 \, {x''} = 36 \, {x'}" data-latex="108 \, {x} + 3 \, {x''} = 36 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-12%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D%20+%2020%20%5C,%20%7By%7D" alt="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}" title="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}" data-latex="0 = -12 \, {y'} + 2 \, {y''} + 20 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?108%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D%20=%2036%20%5C,%20%7Bx'%7D" alt="108 \, {x} + 3 \, {x''} = 36 \, {x'}" title="108 \, {x} + 3 \, {x''} = 36 \, {x'}" data-latex="108 \, {x} + 3 \, {x''} = 36 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6490" title="C3 | Homogeneous second-order linear ODE | ver. 6490"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {y'} = -2 \, {y''} - 32 \, {y}" alt="16 \, {y'} = -2 \, {y''} - 32 \, {y}" title="16 \, {y'} = -2 \, {y''} - 32 \, {y}" data-latex="16 \, {y'} = -2 \, {y''} - 32 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {x'} - 50 \, {x} = 2 \, {x''}" alt="12 \, {x'} - 50 \, {x} = 2 \, {x''}" title="12 \, {x'} - 50 \, {x} = 2 \, {x''}" data-latex="12 \, {x'} - 50 \, {x} = 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7By'%7D%20=%20-2%20%5C,%20%7By''%7D%20-%2032%20%5C,%20%7By%7D" alt="16 \, {y'} = -2 \, {y''} - 32 \, {y}" title="16 \, {y'} = -2 \, {y''} - 32 \, {y}" data-latex="16 \, {y'} = -2 \, {y''} - 32 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7Bx'%7D%20-%2050%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D" alt="12 \, {x'} - 50 \, {x} = 2 \, {x''}" title="12 \, {x'} - 50 \, {x} = 2 \, {x''}" data-latex="12 \, {x'} - 50 \, {x} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0401" title="C3 | Homogeneous second-order linear ODE | ver. 0401"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20 \, {x'} - 58 \, {x} = 2 \, {x''}" alt="20 \, {x'} - 58 \, {x} = 2 \, {x''}" title="20 \, {x'} - 58 \, {x} = 2 \, {x''}" data-latex="20 \, {x'} - 58 \, {x} = 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -4 \, {y'} - 2 \, {y''}" alt="2 \, {y} = -4 \, {y'} - 2 \, {y''}" title="2 \, {y} = -4 \, {y'} - 2 \, {y''}" data-latex="2 \, {y} = -4 \, {y'} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20%20%5C,%20%7Bx'%7D%20-%2058%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D" alt="20 \, {x'} - 58 \, {x} = 2 \, {x''}" title="20 \, {x'} - 58 \, {x} = 2 \, {x''}" data-latex="20 \, {x'} - 58 \, {x} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D" alt="2 \, {y} = -4 \, {y'} - 2 \, {y''}" title="2 \, {y} = -4 \, {y'} - 2 \, {y''}" data-latex="2 \, {y} = -4 \, {y'} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7697" title="C3 | Homogeneous second-order linear ODE | ver. 7697"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y'} + 10 \, {y} = -2 \, {y''}" alt="8 \, {y'} + 10 \, {y} = -2 \, {y''}" title="8 \, {y'} + 10 \, {y} = -2 \, {y''}" data-latex="8 \, {y'} + 10 \, {y} = -2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 36 \, {x'} = 162 \, {x}" alt="-2 \, {x''} - 36 \, {x'} = 162 \, {x}" title="-2 \, {x''} - 36 \, {x'} = 162 \, {x}" data-latex="-2 \, {x''} - 36 \, {x'} = 162 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By'%7D%20+%2010%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By''%7D" alt="8 \, {y'} + 10 \, {y} = -2 \, {y''}" title="8 \, {y'} + 10 \, {y} = -2 \, {y''}" data-latex="8 \, {y'} + 10 \, {y} = -2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%2036%20%5C,%20%7Bx'%7D%20=%20162%20%5C,%20%7Bx%7D" alt="-2 \, {x''} - 36 \, {x'} = 162 \, {x}" title="-2 \, {x''} - 36 \, {x'} = 162 \, {x}" data-latex="-2 \, {x''} - 36 \, {x'} = 162 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0288" title="C3 | Homogeneous second-order linear ODE | ver. 0288"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24 \, {x'} = 2 \, {x''} + 72 \, {x}" alt="24 \, {x'} = 2 \, {x''} + 72 \, {x}" title="24 \, {x'} = 2 \, {x''} + 72 \, {x}" data-latex="24 \, {x'} = 2 \, {x''} + 72 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y''} = -4 \, {y'} + 20 \, {y}" alt="-2 \, {y''} = -4 \, {y'} + 20 \, {y}" title="-2 \, {y''} = -4 \, {y'} + 20 \, {y}" data-latex="-2 \, {y''} = -4 \, {y'} + 20 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24%20%5C,%20%7Bx'%7D%20=%202%20%5C,%20%7Bx''%7D%20+%2072%20%5C,%20%7Bx%7D" alt="24 \, {x'} = 2 \, {x''} + 72 \, {x}" title="24 \, {x'} = 2 \, {x''} + 72 \, {x}" data-latex="24 \, {x'} = 2 \, {x''} + 72 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By''%7D%20=%20-4%20%5C,%20%7By'%7D%20+%2020%20%5C,%20%7By%7D" alt="-2 \, {y''} = -4 \, {y'} + 20 \, {y}" title="-2 \, {y''} = -4 \, {y'} + 20 \, {y}" data-latex="-2 \, {y''} = -4 \, {y'} + 20 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5270" title="C3 | Homogeneous second-order linear ODE | ver. 5270"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = -96 \, {y} + 24 \, {y'}" alt="3 \, {y''} = -96 \, {y} + 24 \, {y'}" title="3 \, {y''} = -96 \, {y} + 24 \, {y'}" data-latex="3 \, {y''} = -96 \, {y} + 24 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-147 \, {x} = -42 \, {x'} + 3 \, {x''}" alt="-147 \, {x} = -42 \, {x'} + 3 \, {x''}" title="-147 \, {x} = -42 \, {x'} + 3 \, {x''}" data-latex="-147 \, {x} = -42 \, {x'} + 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%20-96%20%5C,%20%7By%7D%20+%2024%20%5C,%20%7By'%7D" alt="3 \, {y''} = -96 \, {y} + 24 \, {y'}" title="3 \, {y''} = -96 \, {y} + 24 \, {y'}" data-latex="3 \, {y''} = -96 \, {y} + 24 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-147%20%5C,%20%7Bx%7D%20=%20-42%20%5C,%20%7Bx'%7D%20+%203%20%5C,%20%7Bx''%7D" alt="-147 \, {x} = -42 \, {x'} + 3 \, {x''}" title="-147 \, {x} = -42 \, {x'} + 3 \, {x''}" data-latex="-147 \, {x} = -42 \, {x'} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1240" title="C3 | Homogeneous second-order linear ODE | ver. 1240"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y''} = 20 \, {y} + 12 \, {y'}" alt="-2 \, {y''} = 20 \, {y} + 12 \, {y'}" title="-2 \, {y''} = 20 \, {y} + 12 \, {y'}" data-latex="-2 \, {y''} = 20 \, {y} + 12 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?98 \, {x} = -2 \, {x''} + 28 \, {x'}" alt="98 \, {x} = -2 \, {x''} + 28 \, {x'}" title="98 \, {x} = -2 \, {x''} + 28 \, {x'}" data-latex="98 \, {x} = -2 \, {x''} + 28 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By''%7D%20=%2020%20%5C,%20%7By%7D%20+%2012%20%5C,%20%7By'%7D" alt="-2 \, {y''} = 20 \, {y} + 12 \, {y'}" title="-2 \, {y''} = 20 \, {y} + 12 \, {y'}" data-latex="-2 \, {y''} = 20 \, {y} + 12 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?98%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7Bx''%7D%20+%2028%20%5C,%20%7Bx'%7D" alt="98 \, {x} = -2 \, {x''} + 28 \, {x'}" title="98 \, {x} = -2 \, {x''} + 28 \, {x'}" data-latex="98 \, {x} = -2 \, {x''} + 28 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8503" title="C3 | Homogeneous second-order linear ODE | ver. 8503"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}" alt="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}" title="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}" data-latex="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}" alt="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}" title="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}" data-latex="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-8%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D%20-%2026%20%5C,%20%7Bx%7D" alt="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}" title="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}" data-latex="0 = -8 \, {x'} - 2 \, {x''} - 26 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%208%20%5C,%20%7By%7D%20+%208%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}" title="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}" data-latex="0 = 8 \, {y} + 8 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8312" title="C3 | Homogeneous second-order linear ODE | ver. 8312"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y''} = -6 \, {y'} + 51 \, {y}" alt="-3 \, {y''} = -6 \, {y'} + 51 \, {y}" title="-3 \, {y''} = -6 \, {y'} + 51 \, {y}" data-latex="-3 \, {y''} = -6 \, {y'} + 51 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?48 \, {x'} = -3 \, {x''} - 192 \, {x}" alt="48 \, {x'} = -3 \, {x''} - 192 \, {x}" title="48 \, {x'} = -3 \, {x''} - 192 \, {x}" data-latex="48 \, {x'} = -3 \, {x''} - 192 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By''%7D%20=%20-6%20%5C,%20%7By'%7D%20+%2051%20%5C,%20%7By%7D" alt="-3 \, {y''} = -6 \, {y'} + 51 \, {y}" title="-3 \, {y''} = -6 \, {y'} + 51 \, {y}" data-latex="-3 \, {y''} = -6 \, {y'} + 51 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?48%20%5C,%20%7Bx'%7D%20=%20-3%20%5C,%20%7Bx''%7D%20-%20192%20%5C,%20%7Bx%7D" alt="48 \, {x'} = -3 \, {x''} - 192 \, {x}" title="48 \, {x'} = -3 \, {x''} - 192 \, {x}" data-latex="48 \, {x'} = -3 \, {x''} - 192 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5433" title="C3 | Homogeneous second-order linear ODE | ver. 5433"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {y'} = 20 \, {y} + 2 \, {y''}" alt="12 \, {y'} = 20 \, {y} + 2 \, {y''}" title="12 \, {y'} = 20 \, {y} + 2 \, {y''}" data-latex="12 \, {y'} = 20 \, {y} + 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x} = -6 \, {x'} - 3 \, {x''}" alt="3 \, {x} = -6 \, {x'} - 3 \, {x''}" title="3 \, {x} = -6 \, {x'} - 3 \, {x''}" data-latex="3 \, {x} = -6 \, {x'} - 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7By'%7D%20=%2020%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D" alt="12 \, {y'} = 20 \, {y} + 2 \, {y''}" title="12 \, {y'} = 20 \, {y} + 2 \, {y''}" data-latex="12 \, {y'} = 20 \, {y} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx%7D%20=%20-6%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx''%7D" alt="3 \, {x} = -6 \, {x'} - 3 \, {x''}" title="3 \, {x} = -6 \, {x'} - 3 \, {x''}" data-latex="3 \, {x} = -6 \, {x'} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4458" title="C3 | Homogeneous second-order linear ODE | ver. 4458"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}" alt="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}" title="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}" data-latex="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-54 \, {y'} = 3 \, {y''} + 243 \, {y}" alt="-54 \, {y'} = 3 \, {y''} + 243 \, {y}" title="-54 \, {y'} = 3 \, {y''} + 243 \, {y}" data-latex="-54 \, {y'} = 3 \, {y''} + 243 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2012%20%5C,%20%7Bx'%7D%20+%2020%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D" alt="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}" title="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}" data-latex="0 = 12 \, {x'} + 20 \, {x} + 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-54%20%5C,%20%7By'%7D%20=%203%20%5C,%20%7By''%7D%20+%20243%20%5C,%20%7By%7D" alt="-54 \, {y'} = 3 \, {y''} + 243 \, {y}" title="-54 \, {y'} = 3 \, {y''} + 243 \, {y}" data-latex="-54 \, {y'} = 3 \, {y''} + 243 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3294" title="C3 | Homogeneous second-order linear ODE | ver. 3294"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 52 \, {y} = -4 \, {y'}" alt="2 \, {y''} + 52 \, {y} = -4 \, {y'}" title="2 \, {y''} + 52 \, {y} = -4 \, {y'}" data-latex="2 \, {y''} + 52 \, {y} = -4 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-192 \, {x} - 3 \, {x''} = -48 \, {x'}" alt="-192 \, {x} - 3 \, {x''} = -48 \, {x'}" title="-192 \, {x} - 3 \, {x''} = -48 \, {x'}" data-latex="-192 \, {x} - 3 \, {x''} = -48 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%2052%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7By'%7D" alt="2 \, {y''} + 52 \, {y} = -4 \, {y'}" title="2 \, {y''} + 52 \, {y} = -4 \, {y'}" data-latex="2 \, {y''} + 52 \, {y} = -4 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-192%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20=%20-48%20%5C,%20%7Bx'%7D" alt="-192 \, {x} - 3 \, {x''} = -48 \, {x'}" title="-192 \, {x} - 3 \, {x''} = -48 \, {x'}" data-latex="-192 \, {x} - 3 \, {x''} = -48 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4514" title="C3 | Homogeneous second-order linear ODE | ver. 4514"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-60 \, {x'} = 300 \, {x} + 3 \, {x''}" alt="-60 \, {x'} = 300 \, {x} + 3 \, {x''}" title="-60 \, {x'} = 300 \, {x} + 3 \, {x''}" data-latex="-60 \, {x'} = 300 \, {x} + 3 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 100 \, {y} = -20 \, {y'}" alt="2 \, {y''} + 100 \, {y} = -20 \, {y'}" title="2 \, {y''} + 100 \, {y} = -20 \, {y'}" data-latex="2 \, {y''} + 100 \, {y} = -20 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-60%20%5C,%20%7Bx'%7D%20=%20300%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D" alt="-60 \, {x'} = 300 \, {x} + 3 \, {x''}" title="-60 \, {x'} = 300 \, {x} + 3 \, {x''}" data-latex="-60 \, {x'} = 300 \, {x} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%20100%20%5C,%20%7By%7D%20=%20-20%20%5C,%20%7By'%7D" alt="2 \, {y''} + 100 \, {y} = -20 \, {y'}" title="2 \, {y''} + 100 \, {y} = -20 \, {y'}" data-latex="2 \, {y''} + 100 \, {y} = -20 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4302" title="C3 | Homogeneous second-order linear ODE | ver. 4302"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0" alt="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0" title="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0" data-latex="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0" alt="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0" title="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0" data-latex="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?54%20%5C,%20%7Bx'%7D%20+%20243%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D%20=%200" alt="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0" title="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0" data-latex="54 \, {x'} + 243 \, {x} + 3 \, {x''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%2034%20%5C,%20%7By%7D%20+%204%20%5C,%20%7By'%7D%20=%200" alt="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0" title="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0" data-latex="2 \, {y''} + 34 \, {y} + 4 \, {y'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6043" title="C3 | Homogeneous second-order linear ODE | ver. 6043"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y'} - 2 \, {y''} = 40 \, {y}" alt="-8 \, {y'} - 2 \, {y''} = 40 \, {y}" title="-8 \, {y'} - 2 \, {y''} = 40 \, {y}" data-latex="-8 \, {y'} - 2 \, {y''} = 40 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?42 \, {x'} - 3 \, {x''} = 147 \, {x}" alt="42 \, {x'} - 3 \, {x''} = 147 \, {x}" title="42 \, {x'} - 3 \, {x''} = 147 \, {x}" data-latex="42 \, {x'} - 3 \, {x''} = 147 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D%20=%2040%20%5C,%20%7By%7D" alt="-8 \, {y'} - 2 \, {y''} = 40 \, {y}" title="-8 \, {y'} - 2 \, {y''} = 40 \, {y}" data-latex="-8 \, {y'} - 2 \, {y''} = 40 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?42%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx''%7D%20=%20147%20%5C,%20%7Bx%7D" alt="42 \, {x'} - 3 \, {x''} = 147 \, {x}" title="42 \, {x'} - 3 \, {x''} = 147 \, {x}" data-latex="42 \, {x'} - 3 \, {x''} = 147 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3183" title="C3 | Homogeneous second-order linear ODE | ver. 3183"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24 \, {y'} - 51 \, {y} = 3 \, {y''}" alt="-24 \, {y'} - 51 \, {y} = 3 \, {y''}" title="-24 \, {y'} - 51 \, {y} = 3 \, {y''}" data-latex="-24 \, {y'} - 51 \, {y} = 3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} = 24 \, {x'} + 48 \, {x}" alt="-3 \, {x''} = 24 \, {x'} + 48 \, {x}" title="-3 \, {x''} = 24 \, {x'} + 48 \, {x}" data-latex="-3 \, {x''} = 24 \, {x'} + 48 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24%20%5C,%20%7By'%7D%20-%2051%20%5C,%20%7By%7D%20=%203%20%5C,%20%7By''%7D" alt="-24 \, {y'} - 51 \, {y} = 3 \, {y''}" title="-24 \, {y'} - 51 \, {y} = 3 \, {y''}" data-latex="-24 \, {y'} - 51 \, {y} = 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20=%2024%20%5C,%20%7Bx'%7D%20+%2048%20%5C,%20%7Bx%7D" alt="-3 \, {x''} = 24 \, {x'} + 48 \, {x}" title="-3 \, {x''} = 24 \, {x'} + 48 \, {x}" data-latex="-3 \, {x''} = 24 \, {x'} + 48 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(i + 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0852" title="C3 | Homogeneous second-order linear ODE | ver. 0852"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x'} + 26 \, {x} = -2 \, {x''}" alt="-12 \, {x'} + 26 \, {x} = -2 \, {x''}" title="-12 \, {x'} + 26 \, {x} = -2 \, {x''}" data-latex="-12 \, {x'} + 26 \, {x} = -2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-40 \, {y'} = -200 \, {y} - 2 \, {y''}" alt="-40 \, {y'} = -200 \, {y} - 2 \, {y''}" title="-40 \, {y'} = -200 \, {y} - 2 \, {y''}" data-latex="-40 \, {y'} = -200 \, {y} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx'%7D%20+%2026%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7Bx''%7D" alt="-12 \, {x'} + 26 \, {x} = -2 \, {x''}" title="-12 \, {x'} + 26 \, {x} = -2 \, {x''}" data-latex="-12 \, {x'} + 26 \, {x} = -2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-40%20%5C,%20%7By'%7D%20=%20-200%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D" alt="-40 \, {y'} = -200 \, {y} - 2 \, {y''}" title="-40 \, {y'} = -200 \, {y} - 2 \, {y''}" data-latex="-40 \, {y'} = -200 \, {y} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7293" title="C3 | Homogeneous second-order linear ODE | ver. 7293"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, {x'} = 27 \, {x} + 3 \, {x''}" alt="18 \, {x'} = 27 \, {x} + 3 \, {x''}" title="18 \, {x'} = 27 \, {x} + 3 \, {x''}" data-latex="18 \, {x'} = 27 \, {x} + 3 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?68 \, {y} + 2 \, {y''} = -20 \, {y'}" alt="68 \, {y} + 2 \, {y''} = -20 \, {y'}" title="68 \, {y} + 2 \, {y''} = -20 \, {y'}" data-latex="68 \, {y} + 2 \, {y''} = -20 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18%20%5C,%20%7Bx'%7D%20=%2027%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D" alt="18 \, {x'} = 27 \, {x} + 3 \, {x''}" title="18 \, {x'} = 27 \, {x} + 3 \, {x''}" data-latex="18 \, {x'} = 27 \, {x} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?68%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D%20=%20-20%20%5C,%20%7By'%7D" alt="68 \, {y} + 2 \, {y''} = -20 \, {y'}" title="68 \, {y} + 2 \, {y''} = -20 \, {y'}" data-latex="68 \, {y} + 2 \, {y''} = -20 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" alt="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2385" title="C3 | Homogeneous second-order linear ODE | ver. 2385"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, {x} - 2 \, {x''} = 4 \, {x'}" alt="-10 \, {x} - 2 \, {x''} = 4 \, {x'}" title="-10 \, {x} - 2 \, {x''} = 4 \, {x'}" data-latex="-10 \, {x} - 2 \, {x''} = 4 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y''} - 128 \, {y} = -32 \, {y'}" alt="-2 \, {y''} - 128 \, {y} = -32 \, {y'}" title="-2 \, {y''} - 128 \, {y} = -32 \, {y'}" data-latex="-2 \, {y''} - 128 \, {y} = -32 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7Bx''%7D%20=%204%20%5C,%20%7Bx'%7D" alt="-10 \, {x} - 2 \, {x''} = 4 \, {x'}" title="-10 \, {x} - 2 \, {x''} = 4 \, {x'}" data-latex="-10 \, {x} - 2 \, {x''} = 4 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By''%7D%20-%20128%20%5C,%20%7By%7D%20=%20-32%20%5C,%20%7By'%7D" alt="-2 \, {y''} - 128 \, {y} = -32 \, {y'}" title="-2 \, {y''} - 128 \, {y} = -32 \, {y'}" data-latex="-2 \, {y''} - 128 \, {y} = -32 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6273" title="C3 | Homogeneous second-order linear ODE | ver. 6273"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0" alt="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0" title="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0" data-latex="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y'} = 52 \, {y} + 2 \, {y''}" alt="-4 \, {y'} = 52 \, {y} + 2 \, {y''}" title="-4 \, {y'} = 52 \, {y} + 2 \, {y''}" data-latex="-4 \, {y'} = 52 \, {y} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20+%2050%20%5C,%20%7Bx%7D%20+%2020%20%5C,%20%7Bx'%7D%20=%200" alt="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0" title="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0" data-latex="2 \, {x''} + 50 \, {x} + 20 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By'%7D%20=%2052%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D" alt="-4 \, {y'} = 52 \, {y} + 2 \, {y''}" title="-4 \, {y'} = 52 \, {y} + 2 \, {y''}" data-latex="-4 \, {y'} = 52 \, {y} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" alt="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9420" title="C3 | Homogeneous second-order linear ODE | ver. 9420"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, {y} = 2 \, {y''} - 8 \, {y'}" alt="-10 \, {y} = 2 \, {y''} - 8 \, {y'}" title="-10 \, {y} = 2 \, {y''} - 8 \, {y'}" data-latex="-10 \, {y} = 2 \, {y''} - 8 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}" alt="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}" title="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}" data-latex="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20%7By%7D%20=%202%20%5C,%20%7By''%7D%20-%208%20%5C,%20%7By'%7D" alt="-10 \, {y} = 2 \, {y''} - 8 \, {y'}" title="-10 \, {y} = 2 \, {y''} - 8 \, {y'}" data-latex="-10 \, {y} = 2 \, {y''} - 8 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-50%20%5C,%20%7Bx%7D%20+%2020%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D" alt="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}" title="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}" data-latex="0 = -50 \, {x} + 20 \, {x'} - 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0809" title="C3 | Homogeneous second-order linear ODE | ver. 0809"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18 \, {y'} = -3 \, {y''} - 27 \, {y}" alt="-18 \, {y'} = -3 \, {y''} - 27 \, {y}" title="-18 \, {y'} = -3 \, {y''} - 27 \, {y}" data-latex="-18 \, {y'} = -3 \, {y''} - 27 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} = 16 \, {x'} - 64 \, {x}" alt="2 \, {x''} = 16 \, {x'} - 64 \, {x}" title="2 \, {x''} = 16 \, {x'} - 64 \, {x}" data-latex="2 \, {x''} = 16 \, {x'} - 64 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-18%20%5C,%20%7By'%7D%20=%20-3%20%5C,%20%7By''%7D%20-%2027%20%5C,%20%7By%7D" alt="-18 \, {y'} = -3 \, {y''} - 27 \, {y}" title="-18 \, {y'} = -3 \, {y''} - 27 \, {y}" data-latex="-18 \, {y'} = -3 \, {y''} - 27 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20=%2016%20%5C,%20%7Bx'%7D%20-%2064%20%5C,%20%7Bx%7D" alt="2 \, {x''} = 16 \, {x'} - 64 \, {x}" title="2 \, {x''} = 16 \, {x'} - 64 \, {x}" data-latex="2 \, {x''} = 16 \, {x'} - 64 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7301" title="C3 | Homogeneous second-order linear ODE | ver. 7301"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}" alt="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}" title="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}" data-latex="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?42 \, {x'} = -147 \, {x} - 3 \, {x''}" alt="42 \, {x'} = -147 \, {x} - 3 \, {x''}" title="42 \, {x'} = -147 \, {x} - 3 \, {x''}" data-latex="42 \, {x'} = -147 \, {x} - 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%203%20%5C,%20%7By''%7D%20+%2015%20%5C,%20%7By%7D%20+%2012%20%5C,%20%7By'%7D" alt="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}" title="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}" data-latex="0 = 3 \, {y''} + 15 \, {y} + 12 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?42%20%5C,%20%7Bx'%7D%20=%20-147%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D" alt="42 \, {x'} = -147 \, {x} - 3 \, {x''}" title="42 \, {x'} = -147 \, {x} - 3 \, {x''}" data-latex="42 \, {x'} = -147 \, {x} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" alt="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" title="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" title="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1144" title="C3 | Homogeneous second-order linear ODE | ver. 1144"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 18 \, {x} = 12 \, {x'}" alt="-2 \, {x''} - 18 \, {x} = 12 \, {x'}" title="-2 \, {x''} - 18 \, {x} = 12 \, {x'}" data-latex="-2 \, {x''} - 18 \, {x} = 12 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y''} = 102 \, {y} - 18 \, {y'}" alt="-3 \, {y''} = 102 \, {y} - 18 \, {y'}" title="-3 \, {y''} = 102 \, {y} - 18 \, {y'}" data-latex="-3 \, {y''} = 102 \, {y} - 18 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%2018%20%5C,%20%7Bx%7D%20=%2012%20%5C,%20%7Bx'%7D" alt="-2 \, {x''} - 18 \, {x} = 12 \, {x'}" title="-2 \, {x''} - 18 \, {x} = 12 \, {x'}" data-latex="-2 \, {x''} - 18 \, {x} = 12 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By''%7D%20=%20102%20%5C,%20%7By%7D%20-%2018%20%5C,%20%7By'%7D" alt="-3 \, {y''} = 102 \, {y} - 18 \, {y'}" title="-3 \, {y''} = 102 \, {y} - 18 \, {y'}" data-latex="-3 \, {y''} = 102 \, {y} - 18 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2632" title="C3 | Homogeneous second-order linear ODE | ver. 2632"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?50 \, {y} - 16 \, {y'} = -2 \, {y''}" alt="50 \, {y} - 16 \, {y'} = -2 \, {y''}" title="50 \, {y} - 16 \, {y'} = -2 \, {y''}" data-latex="50 \, {y} - 16 \, {y'} = -2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32 \, {x'} - 2 \, {x''} = 128 \, {x}" alt="-32 \, {x'} - 2 \, {x''} = 128 \, {x}" title="-32 \, {x'} - 2 \, {x''} = 128 \, {x}" data-latex="-32 \, {x'} - 2 \, {x''} = 128 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?50%20%5C,%20%7By%7D%20-%2016%20%5C,%20%7By'%7D%20=%20-2%20%5C,%20%7By''%7D" alt="50 \, {y} - 16 \, {y'} = -2 \, {y''}" title="50 \, {y} - 16 \, {y'} = -2 \, {y''}" data-latex="50 \, {y} - 16 \, {y'} = -2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D%20=%20128%20%5C,%20%7Bx%7D" alt="-32 \, {x'} - 2 \, {x''} = 128 \, {x}" title="-32 \, {x'} - 2 \, {x''} = 128 \, {x}" data-latex="-32 \, {x'} - 2 \, {x''} = 128 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7426" title="C3 | Homogeneous second-order linear ODE | ver. 7426"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 20 \, {y'} = -50 \, {y}" alt="2 \, {y''} + 20 \, {y'} = -50 \, {y}" title="2 \, {y''} + 20 \, {y'} = -50 \, {y}" data-latex="2 \, {y''} + 20 \, {y'} = -50 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-50 \, {x} - 12 \, {x'} = 2 \, {x''}" alt="-50 \, {x} - 12 \, {x'} = 2 \, {x''}" title="-50 \, {x} - 12 \, {x'} = 2 \, {x''}" data-latex="-50 \, {x} - 12 \, {x'} = 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%2020%20%5C,%20%7By'%7D%20=%20-50%20%5C,%20%7By%7D" alt="2 \, {y''} + 20 \, {y'} = -50 \, {y}" title="2 \, {y''} + 20 \, {y'} = -50 \, {y}" data-latex="2 \, {y''} + 20 \, {y'} = -50 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-50%20%5C,%20%7Bx%7D%20-%2012%20%5C,%20%7Bx'%7D%20=%202%20%5C,%20%7Bx''%7D" alt="-50 \, {x} - 12 \, {x'} = 2 \, {x''}" title="-50 \, {x} - 12 \, {x'} = 2 \, {x''}" data-latex="-50 \, {x} - 12 \, {x'} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" alt="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6253" title="C3 | Homogeneous second-order linear ODE | ver. 6253"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0" alt="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0" title="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0" data-latex="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, {x} = -2 \, {x''} - 12 \, {x'}" alt="18 \, {x} = -2 \, {x''} - 12 \, {x'}" title="18 \, {x} = -2 \, {x''} - 12 \, {x'}" data-latex="18 \, {x} = -2 \, {x''} - 12 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D%20-%2064%20%5C,%20%7By%7D%20=%200" alt="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0" title="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0" data-latex="16 \, {y'} - 2 \, {y''} - 64 \, {y} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7Bx''%7D%20-%2012%20%5C,%20%7Bx'%7D" alt="18 \, {x} = -2 \, {x''} - 12 \, {x'}" title="18 \, {x} = -2 \, {x''} - 12 \, {x'}" data-latex="18 \, {x} = -2 \, {x''} - 12 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3491" title="C3 | Homogeneous second-order linear ODE | ver. 3491"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-96 \, {x} = -24 \, {x'} + 3 \, {x''}" alt="-96 \, {x} = -24 \, {x'} + 3 \, {x''}" title="-96 \, {x} = -24 \, {x'} + 3 \, {x''}" data-latex="-96 \, {x} = -24 \, {x'} + 3 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0" alt="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0" title="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0" data-latex="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-96%20%5C,%20%7Bx%7D%20=%20-24%20%5C,%20%7Bx'%7D%20+%203%20%5C,%20%7Bx''%7D" alt="-96 \, {x} = -24 \, {x'} + 3 \, {x''}" title="-96 \, {x} = -24 \, {x'} + 3 \, {x''}" data-latex="-96 \, {x} = -24 \, {x'} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?48%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D%20+%20192%20%5C,%20%7By%7D%20=%200" alt="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0" title="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0" data-latex="48 \, {y'} + 3 \, {y''} + 192 \, {y} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6702" title="C3 | Homogeneous second-order linear ODE | ver. 6702"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?87 \, {y} + 3 \, {y''} = -30 \, {y'}" alt="87 \, {y} + 3 \, {y''} = -30 \, {y'}" title="87 \, {y} + 3 \, {y''} = -30 \, {y'}" data-latex="87 \, {y} + 3 \, {y''} = -30 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-108 \, {x} - 3 \, {x''} = -36 \, {x'}" alt="-108 \, {x} - 3 \, {x''} = -36 \, {x'}" title="-108 \, {x} - 3 \, {x''} = -36 \, {x'}" data-latex="-108 \, {x} - 3 \, {x''} = -36 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?87%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By''%7D%20=%20-30%20%5C,%20%7By'%7D" alt="87 \, {y} + 3 \, {y''} = -30 \, {y'}" title="87 \, {y} + 3 \, {y''} = -30 \, {y'}" data-latex="87 \, {y} + 3 \, {y''} = -30 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-108%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20=%20-36%20%5C,%20%7Bx'%7D" alt="-108 \, {x} - 3 \, {x''} = -36 \, {x'}" title="-108 \, {x} - 3 \, {x''} = -36 \, {x'}" data-latex="-108 \, {x} - 3 \, {x''} = -36 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5108" title="C3 | Homogeneous second-order linear ODE | ver. 5108"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36 \, {x} = 2 \, {x''} - 12 \, {x'}" alt="-36 \, {x} = 2 \, {x''} - 12 \, {x'}" title="-36 \, {x} = 2 \, {x''} - 12 \, {x'}" data-latex="-36 \, {x} = 2 \, {x''} - 12 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} + 24 \, {y'} = -72 \, {y}" alt="2 \, {y''} + 24 \, {y'} = -72 \, {y}" title="2 \, {y''} + 24 \, {y'} = -72 \, {y}" data-latex="2 \, {y''} + 24 \, {y'} = -72 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D%20-%2012%20%5C,%20%7Bx'%7D" alt="-36 \, {x} = 2 \, {x''} - 12 \, {x'}" title="-36 \, {x} = 2 \, {x''} - 12 \, {x'}" data-latex="-36 \, {x} = 2 \, {x''} - 12 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20+%2024%20%5C,%20%7By'%7D%20=%20-72%20%5C,%20%7By%7D" alt="2 \, {y''} + 24 \, {y'} = -72 \, {y}" title="2 \, {y''} + 24 \, {y'} = -72 \, {y}" data-latex="2 \, {y''} + 24 \, {y'} = -72 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4611" title="C3 | Homogeneous second-order linear ODE | ver. 4611"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} + 40 \, {x'} = 200 \, {x}" alt="-2 \, {x''} + 40 \, {x'} = 200 \, {x}" title="-2 \, {x''} + 40 \, {x'} = 200 \, {x}" data-latex="-2 \, {x''} + 40 \, {x'} = 200 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36 \, {y} = 12 \, {y'} + 2 \, {y''}" alt="-36 \, {y} = 12 \, {y'} + 2 \, {y''}" title="-36 \, {y} = 12 \, {y'} + 2 \, {y''}" data-latex="-36 \, {y} = 12 \, {y'} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20+%2040%20%5C,%20%7Bx'%7D%20=%20200%20%5C,%20%7Bx%7D" alt="-2 \, {x''} + 40 \, {x'} = 200 \, {x}" title="-2 \, {x''} + 40 \, {x'} = 200 \, {x}" data-latex="-2 \, {x''} + 40 \, {x'} = 200 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36%20%5C,%20%7By%7D%20=%2012%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-36 \, {y} = 12 \, {y'} + 2 \, {y''}" title="-36 \, {y} = 12 \, {y'} + 2 \, {y''}" data-latex="-36 \, {y} = 12 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6497" title="C3 | Homogeneous second-order linear ODE | ver. 6497"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y'} + 15 \, {y} = -3 \, {y''}" alt="6 \, {y'} + 15 \, {y} = -3 \, {y''}" title="6 \, {y'} + 15 \, {y} = -3 \, {y''}" data-latex="6 \, {y'} + 15 \, {y} = -3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-192 \, {x} = 3 \, {x''} + 48 \, {x'}" alt="-192 \, {x} = 3 \, {x''} + 48 \, {x'}" title="-192 \, {x} = 3 \, {x''} + 48 \, {x'}" data-latex="-192 \, {x} = 3 \, {x''} + 48 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By'%7D%20+%2015%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By''%7D" alt="6 \, {y'} + 15 \, {y} = -3 \, {y''}" title="6 \, {y'} + 15 \, {y} = -3 \, {y''}" data-latex="6 \, {y'} + 15 \, {y} = -3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-192%20%5C,%20%7Bx%7D%20=%203%20%5C,%20%7Bx''%7D%20+%2048%20%5C,%20%7Bx'%7D" alt="-192 \, {x} = 3 \, {x''} + 48 \, {x'}" title="-192 \, {x} = 3 \, {x''} + 48 \, {x'}" data-latex="-192 \, {x} = 3 \, {x''} + 48 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" title="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2905" title="C3 | Homogeneous second-order linear ODE | ver. 2905"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300 \, {y} - 60 \, {y'} = -3 \, {y''}" alt="300 \, {y} - 60 \, {y'} = -3 \, {y''}" title="300 \, {y} - 60 \, {y'} = -3 \, {y''}" data-latex="300 \, {y} - 60 \, {y'} = -3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0" alt="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0" title="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0" data-latex="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300%20%5C,%20%7By%7D%20-%2060%20%5C,%20%7By'%7D%20=%20-3%20%5C,%20%7By''%7D" alt="300 \, {y} - 60 \, {y'} = -3 \, {y''}" title="300 \, {y} - 60 \, {y'} = -3 \, {y''}" data-latex="300 \, {y} - 60 \, {y'} = -3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20+%2087%20%5C,%20%7Bx%7D%20-%2012%20%5C,%20%7Bx'%7D%20=%200" alt="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0" title="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0" data-latex="3 \, {x''} + 87 \, {x} - 12 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7974" title="C3 | Homogeneous second-order linear ODE | ver. 7974"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}" alt="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}" title="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}" data-latex="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?32 \, {y'} = 128 \, {y} + 2 \, {y''}" alt="32 \, {y'} = 128 \, {y} + 2 \, {y''}" title="32 \, {y'} = 128 \, {y} + 2 \, {y''}" data-latex="32 \, {y'} = 128 \, {y} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2052%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7Bx'%7D%20+%202%20%5C,%20%7Bx''%7D" alt="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}" title="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}" data-latex="0 = 52 \, {x} + 4 \, {x'} + 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?32%20%5C,%20%7By'%7D%20=%20128%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D" alt="32 \, {y'} = 128 \, {y} + 2 \, {y''}" title="32 \, {y'} = 128 \, {y} + 2 \, {y''}" data-latex="32 \, {y'} = 128 \, {y} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9479" title="C3 | Homogeneous second-order linear ODE | ver. 9479"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y'} = -2 \, {y} - 2 \, {y''}" alt="-4 \, {y'} = -2 \, {y} - 2 \, {y''}" title="-4 \, {y'} = -2 \, {y} - 2 \, {y''}" data-latex="-4 \, {y'} = -2 \, {y} - 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24 \, {x'} = -60 \, {x} - 3 \, {x''}" alt="-24 \, {x'} = -60 \, {x} - 3 \, {x''}" title="-24 \, {x'} = -60 \, {x} - 3 \, {x''}" data-latex="-24 \, {x'} = -60 \, {x} - 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By'%7D%20=%20-2%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D" alt="-4 \, {y'} = -2 \, {y} - 2 \, {y''}" title="-4 \, {y'} = -2 \, {y} - 2 \, {y''}" data-latex="-4 \, {y'} = -2 \, {y} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24%20%5C,%20%7Bx'%7D%20=%20-60%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D" alt="-24 \, {x'} = -60 \, {x} - 3 \, {x''}" title="-24 \, {x'} = -60 \, {x} - 3 \, {x''}" data-latex="-24 \, {x'} = -60 \, {x} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{t} + k_{2} e^{t}" alt="{y} = k_{1} t e^{t} + k_{2} e^{t}" title="{y} = k_{1} t e^{t} + k_{2} e^{t}" data-latex="{y} = k_{1} t e^{t} + k_{2} e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7Bt%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = k_{1} t e^{t} + k_{2} e^{t}" title="{y} = k_{1} t e^{t} + k_{2} e^{t}" data-latex="{y} = k_{1} t e^{t} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8271" title="C3 | Homogeneous second-order linear ODE | ver. 8271"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y'} - 10 \, {y} = 2 \, {y''}" alt="-8 \, {y'} - 10 \, {y} = 2 \, {y''}" title="-8 \, {y'} - 10 \, {y} = 2 \, {y''}" data-latex="-8 \, {y'} - 10 \, {y} = 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0" alt="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0" title="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0" data-latex="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By'%7D%20-%2010%20%5C,%20%7By%7D%20=%202%20%5C,%20%7By''%7D" alt="-8 \, {y'} - 10 \, {y} = 2 \, {y''}" title="-8 \, {y'} - 10 \, {y} = 2 \, {y''}" data-latex="-8 \, {y'} - 10 \, {y} = 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20-%2024%20%5C,%20%7Bx'%7D%20+%2048%20%5C,%20%7Bx%7D%20=%200" alt="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0" title="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0" data-latex="3 \, {x''} - 24 \, {x'} + 48 \, {x} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9224" title="C3 | Homogeneous second-order linear ODE | ver. 9224"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} + 2 \, {y''} = -26 \, {y}" alt="-12 \, {y'} + 2 \, {y''} = -26 \, {y}" title="-12 \, {y'} + 2 \, {y''} = -26 \, {y}" data-latex="-12 \, {y'} + 2 \, {y''} = -26 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0" alt="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0" title="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0" data-latex="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D%20=%20-26%20%5C,%20%7By%7D" alt="-12 \, {y'} + 2 \, {y''} = -26 \, {y}" title="-12 \, {y'} + 2 \, {y''} = -26 \, {y}" data-latex="-12 \, {y'} + 2 \, {y''} = -26 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20+%208%20%5C,%20%7Bx'%7D%20=%200" alt="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0" title="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0" data-latex="8 \, {x} + 2 \, {x''} + 8 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" alt="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1215" title="C3 | Homogeneous second-order linear ODE | ver. 1215"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-72 \, {y} = -24 \, {y'} + 2 \, {y''}" alt="-72 \, {y} = -24 \, {y'} + 2 \, {y''}" title="-72 \, {y} = -24 \, {y'} + 2 \, {y''}" data-latex="-72 \, {y} = -24 \, {y'} + 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} = -18 \, {x'} - 30 \, {x}" alt="3 \, {x''} = -18 \, {x'} - 30 \, {x}" title="3 \, {x''} = -18 \, {x'} - 30 \, {x}" data-latex="3 \, {x''} = -18 \, {x'} - 30 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-72%20%5C,%20%7By%7D%20=%20-24%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-72 \, {y} = -24 \, {y'} + 2 \, {y''}" title="-72 \, {y} = -24 \, {y'} + 2 \, {y''}" data-latex="-72 \, {y} = -24 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20=%20-18%20%5C,%20%7Bx'%7D%20-%2030%20%5C,%20%7Bx%7D" alt="3 \, {x''} = -18 \, {x'} - 30 \, {x}" title="3 \, {x''} = -18 \, {x'} - 30 \, {x}" data-latex="3 \, {x''} = -18 \, {x'} - 30 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" alt="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" title="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2062" title="C3 | Homogeneous second-order linear ODE | ver. 2062"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} + 300 \, {y} = -60 \, {y'}" alt="3 \, {y''} + 300 \, {y} = -60 \, {y'}" title="3 \, {y''} + 300 \, {y} = -60 \, {y'}" data-latex="3 \, {y''} + 300 \, {y} = -60 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}" alt="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}" title="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}" data-latex="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20+%20300%20%5C,%20%7By%7D%20=%20-60%20%5C,%20%7By'%7D" alt="3 \, {y''} + 300 \, {y} = -60 \, {y'}" title="3 \, {y''} + 300 \, {y} = -60 \, {y'}" data-latex="3 \, {y''} + 300 \, {y} = -60 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7Bx''%7D%20-%204%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7Bx'%7D" alt="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}" title="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}" data-latex="0 = -2 \, {x''} - 4 \, {x} + 4 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4987" title="C3 | Homogeneous second-order linear ODE | ver. 4987"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300 \, {y} = 60 \, {y'} - 3 \, {y''}" alt="300 \, {y} = 60 \, {y'} - 3 \, {y''}" title="300 \, {y} = 60 \, {y'} - 3 \, {y''}" data-latex="300 \, {y} = 60 \, {y'} - 3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0" alt="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0" title="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0" data-latex="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300%20%5C,%20%7By%7D%20=%2060%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By''%7D" alt="300 \, {y} = 60 \, {y'} - 3 \, {y''}" title="300 \, {y} = 60 \, {y'} - 3 \, {y''}" data-latex="300 \, {y} = 60 \, {y'} - 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?102%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D%20+%2018%20%5C,%20%7Bx'%7D%20=%200" alt="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0" title="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0" data-latex="102 \, {x} + 3 \, {x''} + 18 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8438" title="C3 | Homogeneous second-order linear ODE | ver. 8438"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?123 \, {y} - 30 \, {y'} = -3 \, {y''}" alt="123 \, {y} - 30 \, {y'} = -3 \, {y''}" title="123 \, {y} - 30 \, {y'} = -3 \, {y''}" data-latex="123 \, {y} - 30 \, {y'} = -3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 40 \, {x'} = 200 \, {x}" alt="-2 \, {x''} - 40 \, {x'} = 200 \, {x}" title="-2 \, {x''} - 40 \, {x'} = 200 \, {x}" data-latex="-2 \, {x''} - 40 \, {x'} = 200 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?123%20%5C,%20%7By%7D%20-%2030%20%5C,%20%7By'%7D%20=%20-3%20%5C,%20%7By''%7D" alt="123 \, {y} - 30 \, {y'} = -3 \, {y''}" title="123 \, {y} - 30 \, {y'} = -3 \, {y''}" data-latex="123 \, {y} - 30 \, {y'} = -3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%2040%20%5C,%20%7Bx'%7D%20=%20200%20%5C,%20%7Bx%7D" alt="-2 \, {x''} - 40 \, {x'} = 200 \, {x}" title="-2 \, {x''} - 40 \, {x'} = 200 \, {x}" data-latex="-2 \, {x''} - 40 \, {x'} = 200 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1842" title="C3 | Homogeneous second-order linear ODE | ver. 1842"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30 \, {y'} - 3 \, {y''} = 78 \, {y}" alt="30 \, {y'} - 3 \, {y''} = 78 \, {y}" title="30 \, {y'} - 3 \, {y''} = 78 \, {y}" data-latex="30 \, {y'} - 3 \, {y''} = 78 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} - 243 \, {x} = 54 \, {x'}" alt="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" title="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" data-latex="-3 \, {x''} - 243 \, {x} = 54 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By''%7D%20=%2078%20%5C,%20%7By%7D" alt="30 \, {y'} - 3 \, {y''} = 78 \, {y}" title="30 \, {y'} - 3 \, {y''} = 78 \, {y}" data-latex="30 \, {y'} - 3 \, {y''} = 78 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20-%20243%20%5C,%20%7Bx%7D%20=%2054%20%5C,%20%7Bx'%7D" alt="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" title="-3 \, {x''} - 243 \, {x} = 54 \, {x'}" data-latex="-3 \, {x''} - 243 \, {x} = 54 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7040" title="C3 | Homogeneous second-order linear ODE | ver. 7040"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?200 \, {x} = 40 \, {x'} - 2 \, {x''}" alt="200 \, {x} = 40 \, {x'} - 2 \, {x''}" title="200 \, {x} = 40 \, {x'} - 2 \, {x''}" data-latex="200 \, {x} = 40 \, {x'} - 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?40 \, {y} = -2 \, {y''} - 8 \, {y'}" alt="40 \, {y} = -2 \, {y''} - 8 \, {y'}" title="40 \, {y} = -2 \, {y''} - 8 \, {y'}" data-latex="40 \, {y} = -2 \, {y''} - 8 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?200%20%5C,%20%7Bx%7D%20=%2040%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D" alt="200 \, {x} = 40 \, {x'} - 2 \, {x''}" title="200 \, {x} = 40 \, {x'} - 2 \, {x''}" data-latex="200 \, {x} = 40 \, {x'} - 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?40%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By''%7D%20-%208%20%5C,%20%7By'%7D" alt="40 \, {y} = -2 \, {y''} - 8 \, {y'}" title="40 \, {y} = -2 \, {y''} - 8 \, {y'}" data-latex="40 \, {y} = -2 \, {y''} - 8 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1740" title="C3 | Homogeneous second-order linear ODE | ver. 1740"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0" alt="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0" title="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0" data-latex="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0" alt="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0" title="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0" data-latex="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20=%200" alt="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0" title="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0" data-latex="-6 \, {x'} - 3 \, {x} - 3 \, {x''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D%20+%204%20%5C,%20%7By'%7D%20=%200" alt="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0" title="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0" data-latex="4 \, {y} + 2 \, {y''} + 4 \, {y'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0766" title="C3 | Homogeneous second-order linear ODE | ver. 0766"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300 \, {x} + 3 \, {x''} = 60 \, {x'}" alt="300 \, {x} + 3 \, {x''} = 60 \, {x'}" title="300 \, {x} + 3 \, {x''} = 60 \, {x'}" data-latex="300 \, {x} + 3 \, {x''} = 60 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, {y'} + 3 \, {y''} = -75 \, {y}" alt="18 \, {y'} + 3 \, {y''} = -75 \, {y}" title="18 \, {y'} + 3 \, {y''} = -75 \, {y}" data-latex="18 \, {y'} + 3 \, {y''} = -75 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?300%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7Bx''%7D%20=%2060%20%5C,%20%7Bx'%7D" alt="300 \, {x} + 3 \, {x''} = 60 \, {x'}" title="300 \, {x} + 3 \, {x''} = 60 \, {x'}" data-latex="300 \, {x} + 3 \, {x''} = 60 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D%20=%20-75%20%5C,%20%7By%7D" alt="18 \, {y'} + 3 \, {y''} = -75 \, {y}" title="18 \, {y'} + 3 \, {y''} = -75 \, {y}" data-latex="18 \, {y'} + 3 \, {y''} = -75 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1875" title="C3 | Homogeneous second-order linear ODE | ver. 1875"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0" alt="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0" title="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0" data-latex="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = 30 \, {y'} - 78 \, {y}" alt="3 \, {y''} = 30 \, {y'} - 78 \, {y}" title="3 \, {y''} = 30 \, {y'} - 78 \, {y}" data-latex="3 \, {y''} = 30 \, {y'} - 78 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20%7Bx'%7D%20+%2050%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20=%200" alt="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0" title="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0" data-latex="-20 \, {x'} + 50 \, {x} + 2 \, {x''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%2030%20%5C,%20%7By'%7D%20-%2078%20%5C,%20%7By%7D" alt="3 \, {y''} = 30 \, {y'} - 78 \, {y}" title="3 \, {y''} = 30 \, {y'} - 78 \, {y}" data-latex="3 \, {y''} = 30 \, {y'} - 78 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8596" title="C3 | Homogeneous second-order linear ODE | ver. 8596"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?75 \, {y} = -3 \, {y''} - 24 \, {y'}" alt="75 \, {y} = -3 \, {y''} - 24 \, {y'}" title="75 \, {y} = -3 \, {y''} - 24 \, {y'}" data-latex="75 \, {y} = -3 \, {y''} - 24 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} - 50 \, {x} = -20 \, {x'}" alt="-2 \, {x''} - 50 \, {x} = -20 \, {x'}" title="-2 \, {x''} - 50 \, {x} = -20 \, {x'}" data-latex="-2 \, {x''} - 50 \, {x} = -20 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?75%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By''%7D%20-%2024%20%5C,%20%7By'%7D" alt="75 \, {y} = -3 \, {y''} - 24 \, {y'}" title="75 \, {y} = -3 \, {y''} - 24 \, {y'}" data-latex="75 \, {y} = -3 \, {y''} - 24 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20-%2050%20%5C,%20%7Bx%7D%20=%20-20%20%5C,%20%7Bx'%7D" alt="-2 \, {x''} - 50 \, {x} = -20 \, {x'}" title="-2 \, {x''} - 50 \, {x} = -20 \, {x'}" data-latex="-2 \, {x''} - 50 \, {x} = -20 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1901" title="C3 | Homogeneous second-order linear ODE | ver. 1901"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {x'} - 2 \, {x''} = 16 \, {x}" alt="-8 \, {x'} - 2 \, {x''} = 16 \, {x}" title="-8 \, {x'} - 2 \, {x''} = 16 \, {x}" data-latex="-8 \, {x'} - 2 \, {x''} = 16 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32 \, {y} - 2 \, {y''} = -16 \, {y'}" alt="-32 \, {y} - 2 \, {y''} = -16 \, {y'}" title="-32 \, {y} - 2 \, {y''} = -16 \, {y'}" data-latex="-32 \, {y} - 2 \, {y''} = -16 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D%20=%2016%20%5C,%20%7Bx%7D" alt="-8 \, {x'} - 2 \, {x''} = 16 \, {x}" title="-8 \, {x'} - 2 \, {x''} = 16 \, {x}" data-latex="-8 \, {x'} - 2 \, {x''} = 16 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D%20=%20-16%20%5C,%20%7By'%7D" alt="-32 \, {y} - 2 \, {y''} = -16 \, {y'}" title="-32 \, {y} - 2 \, {y''} = -16 \, {y'}" data-latex="-32 \, {y} - 2 \, {y''} = -16 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4049" title="C3 | Homogeneous second-order linear ODE | ver. 4049"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36 \, {y} + 12 \, {y'} = 2 \, {y''}" alt="-36 \, {y} + 12 \, {y'} = 2 \, {y''}" title="-36 \, {y} + 12 \, {y'} = 2 \, {y''}" data-latex="-36 \, {y} + 12 \, {y'} = 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} = 3 \, {x} + 6 \, {x'}" alt="-3 \, {x''} = 3 \, {x} + 6 \, {x'}" title="-3 \, {x''} = 3 \, {x} + 6 \, {x'}" data-latex="-3 \, {x''} = 3 \, {x} + 6 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36%20%5C,%20%7By%7D%20+%2012%20%5C,%20%7By'%7D%20=%202%20%5C,%20%7By''%7D" alt="-36 \, {y} + 12 \, {y'} = 2 \, {y''}" title="-36 \, {y} + 12 \, {y'} = 2 \, {y''}" data-latex="-36 \, {y} + 12 \, {y'} = 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20=%203%20%5C,%20%7Bx%7D%20+%206%20%5C,%20%7Bx'%7D" alt="-3 \, {x''} = 3 \, {x} + 6 \, {x'}" title="-3 \, {x''} = 3 \, {x} + 6 \, {x'}" data-latex="-3 \, {x''} = 3 \, {x} + 6 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1735" title="C3 | Homogeneous second-order linear ODE | ver. 1735"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} + 6 \, {x} = -6 \, {x'}" alt="3 \, {x''} + 6 \, {x} = -6 \, {x'}" title="3 \, {x''} + 6 \, {x} = -6 \, {x'}" data-latex="3 \, {x''} + 6 \, {x} = -6 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y''} + 36 \, {y'} = 162 \, {y}" alt="-2 \, {y''} + 36 \, {y'} = 162 \, {y}" title="-2 \, {y''} + 36 \, {y'} = 162 \, {y}" data-latex="-2 \, {y''} + 36 \, {y'} = 162 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20+%206%20%5C,%20%7Bx%7D%20=%20-6%20%5C,%20%7Bx'%7D" alt="3 \, {x''} + 6 \, {x} = -6 \, {x'}" title="3 \, {x''} + 6 \, {x} = -6 \, {x'}" data-latex="3 \, {x''} + 6 \, {x} = -6 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By''%7D%20+%2036%20%5C,%20%7By'%7D%20=%20162%20%5C,%20%7By%7D" alt="-2 \, {y''} + 36 \, {y'} = 162 \, {y}" title="-2 \, {y''} + 36 \, {y'} = 162 \, {y}" data-latex="-2 \, {y''} + 36 \, {y'} = 162 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}" alt="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}" title="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}" title="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2609" title="C3 | Homogeneous second-order linear ODE | ver. 2609"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {x'} = 50 \, {x} + 2 \, {x''}" alt="16 \, {x'} = 50 \, {x} + 2 \, {x''}" title="16 \, {x'} = 50 \, {x} + 2 \, {x''}" data-latex="16 \, {x'} = 50 \, {x} + 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y} + 3 \, {y''} = -6 \, {y'}" alt="3 \, {y} + 3 \, {y''} = -6 \, {y'}" title="3 \, {y} + 3 \, {y''} = -6 \, {y'}" data-latex="3 \, {y} + 3 \, {y''} = -6 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7Bx'%7D%20=%2050%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D" alt="16 \, {x'} = 50 \, {x} + 2 \, {x''}" title="16 \, {x'} = 50 \, {x} + 2 \, {x''}" data-latex="16 \, {x'} = 50 \, {x} + 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By''%7D%20=%20-6%20%5C,%20%7By'%7D" alt="3 \, {y} + 3 \, {y''} = -6 \, {y'}" title="3 \, {y} + 3 \, {y''} = -6 \, {y'}" data-latex="3 \, {y} + 3 \, {y''} = -6 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7318" title="C3 | Homogeneous second-order linear ODE | ver. 7318"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?123 \, {y} + 3 \, {y''} = 30 \, {y'}" alt="123 \, {y} + 3 \, {y''} = 30 \, {y'}" title="123 \, {y} + 3 \, {y''} = 30 \, {y'}" data-latex="123 \, {y} + 3 \, {y''} = 30 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?27 \, {x} + 18 \, {x'} = -3 \, {x''}" alt="27 \, {x} + 18 \, {x'} = -3 \, {x''}" title="27 \, {x} + 18 \, {x'} = -3 \, {x''}" data-latex="27 \, {x} + 18 \, {x'} = -3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?123%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By''%7D%20=%2030%20%5C,%20%7By'%7D" alt="123 \, {y} + 3 \, {y''} = 30 \, {y'}" title="123 \, {y} + 3 \, {y''} = 30 \, {y'}" data-latex="123 \, {y} + 3 \, {y''} = 30 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?27%20%5C,%20%7Bx%7D%20+%2018%20%5C,%20%7Bx'%7D%20=%20-3%20%5C,%20%7Bx''%7D" alt="27 \, {x} + 18 \, {x'} = -3 \, {x''}" title="27 \, {x} + 18 \, {x'} = -3 \, {x''}" data-latex="27 \, {x} + 18 \, {x'} = -3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9704" title="C3 | Homogeneous second-order linear ODE | ver. 9704"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} = -12 \, {y} - 3 \, {y''}" alt="-12 \, {y'} = -12 \, {y} - 3 \, {y''}" title="-12 \, {y'} = -12 \, {y} - 3 \, {y''}" data-latex="-12 \, {y'} = -12 \, {y} - 3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}" alt="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}" title="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}" data-latex="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20=%20-12%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By''%7D" alt="-12 \, {y'} = -12 \, {y} - 3 \, {y''}" title="-12 \, {y'} = -12 \, {y} - 3 \, {y''}" data-latex="-12 \, {y'} = -12 \, {y} - 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-34%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7Bx''%7D%20+%2016%20%5C,%20%7Bx'%7D" alt="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}" title="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}" data-latex="0 = -34 \, {x} - 2 \, {x''} + 16 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6991" title="C3 | Homogeneous second-order linear ODE | ver. 6991"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} + 82 \, {x} = -16 \, {x'}" alt="2 \, {x''} + 82 \, {x} = -16 \, {x'}" title="2 \, {x''} + 82 \, {x} = -16 \, {x'}" data-latex="2 \, {x''} + 82 \, {x} = -16 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32 \, {y} = -16 \, {y'} + 2 \, {y''}" alt="-32 \, {y} = -16 \, {y'} + 2 \, {y''}" title="-32 \, {y} = -16 \, {y'} + 2 \, {y''}" data-latex="-32 \, {y} = -16 \, {y'} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20+%2082%20%5C,%20%7Bx%7D%20=%20-16%20%5C,%20%7Bx'%7D" alt="2 \, {x''} + 82 \, {x} = -16 \, {x'}" title="2 \, {x''} + 82 \, {x} = -16 \, {x'}" data-latex="2 \, {x''} + 82 \, {x} = -16 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32%20%5C,%20%7By%7D%20=%20-16%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-32 \, {y} = -16 \, {y'} + 2 \, {y''}" title="-32 \, {y} = -16 \, {y'} + 2 \, {y''}" data-latex="-32 \, {y} = -16 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 4\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5211" title="C3 | Homogeneous second-order linear ODE | ver. 5211"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-128 \, {y} - 2 \, {y''} = -32 \, {y'}" alt="-128 \, {y} - 2 \, {y''} = -32 \, {y'}" title="-128 \, {y} - 2 \, {y''} = -32 \, {y'}" data-latex="-128 \, {y} - 2 \, {y''} = -32 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0" alt="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0" title="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0" data-latex="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-128%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D%20=%20-32%20%5C,%20%7By'%7D" alt="-128 \, {y} - 2 \, {y''} = -32 \, {y'}" title="-128 \, {y} - 2 \, {y''} = -32 \, {y'}" data-latex="-128 \, {y} - 2 \, {y''} = -32 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx'%7D%20+%2020%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20=%200" alt="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0" title="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0" data-latex="-4 \, {x'} + 20 \, {x} + 2 \, {x''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9559" title="C3 | Homogeneous second-order linear ODE | ver. 9559"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} + 24 \, {x'} = -48 \, {x}" alt="3 \, {x''} + 24 \, {x'} = -48 \, {x}" title="3 \, {x''} + 24 \, {x'} = -48 \, {x}" data-latex="3 \, {x''} + 24 \, {x'} = -48 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}" alt="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}" title="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}" data-latex="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20+%2024%20%5C,%20%7Bx'%7D%20=%20-48%20%5C,%20%7Bx%7D" alt="3 \, {x''} + 24 \, {x'} = -48 \, {x}" title="3 \, {x''} + 24 \, {x'} = -48 \, {x}" data-latex="3 \, {x''} + 24 \, {x'} = -48 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-40%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D%20+%208%20%5C,%20%7By'%7D" alt="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}" title="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}" data-latex="0 = -40 \, {y} - 2 \, {y''} + 8 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5867" title="C3 | Homogeneous second-order linear ODE | ver. 5867"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24 \, {x'} = -3 \, {x''} - 48 \, {x}" alt="24 \, {x'} = -3 \, {x''} - 48 \, {x}" title="24 \, {x'} = -3 \, {x''} - 48 \, {x}" data-latex="24 \, {x'} = -3 \, {x''} - 48 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64 \, {y} = -16 \, {y'} + 2 \, {y''}" alt="-64 \, {y} = -16 \, {y'} + 2 \, {y''}" title="-64 \, {y} = -16 \, {y'} + 2 \, {y''}" data-latex="-64 \, {y} = -16 \, {y'} + 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24%20%5C,%20%7Bx'%7D%20=%20-3%20%5C,%20%7Bx''%7D%20-%2048%20%5C,%20%7Bx%7D" alt="24 \, {x'} = -3 \, {x''} - 48 \, {x}" title="24 \, {x'} = -3 \, {x''} - 48 \, {x}" data-latex="24 \, {x'} = -3 \, {x''} - 48 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64%20%5C,%20%7By%7D%20=%20-16%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-64 \, {y} = -16 \, {y'} + 2 \, {y''}" title="-64 \, {y} = -16 \, {y'} + 2 \, {y''}" data-latex="-64 \, {y} = -16 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4941" title="C3 | Homogeneous second-order linear ODE | ver. 4941"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x} = 12 \, {x'} + 3 \, {x''}" alt="-12 \, {x} = 12 \, {x'} + 3 \, {x''}" title="-12 \, {x} = 12 \, {x'} + 3 \, {x''}" data-latex="-12 \, {x} = 12 \, {x'} + 3 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}" alt="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}" title="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}" data-latex="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx%7D%20=%2012%20%5C,%20%7Bx'%7D%20+%203%20%5C,%20%7Bx''%7D" alt="-12 \, {x} = 12 \, {x'} + 3 \, {x''}" title="-12 \, {x} = 12 \, {x'} + 3 \, {x''}" data-latex="-12 \, {x} = 12 \, {x'} + 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2016%20%5C,%20%7By'%7D%20-%2050%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D" alt="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}" title="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}" data-latex="0 = 16 \, {y'} - 50 \, {y} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" alt="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2948" title="C3 | Homogeneous second-order linear ODE | ver. 2948"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} = 12 \, {x'} + 15 \, {x}" alt="-3 \, {x''} = 12 \, {x'} + 15 \, {x}" title="-3 \, {x''} = 12 \, {x'} + 15 \, {x}" data-latex="-3 \, {x''} = 12 \, {x'} + 15 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24 \, {y'} + 3 \, {y''} = -48 \, {y}" alt="-24 \, {y'} + 3 \, {y''} = -48 \, {y}" title="-24 \, {y'} + 3 \, {y''} = -48 \, {y}" data-latex="-24 \, {y'} + 3 \, {y''} = -48 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20=%2012%20%5C,%20%7Bx'%7D%20+%2015%20%5C,%20%7Bx%7D" alt="-3 \, {x''} = 12 \, {x'} + 15 \, {x}" title="-3 \, {x''} = 12 \, {x'} + 15 \, {x}" data-latex="-3 \, {x''} = 12 \, {x'} + 15 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D%20=%20-48%20%5C,%20%7By%7D" alt="-24 \, {y'} + 3 \, {y''} = -48 \, {y}" title="-24 \, {y'} + 3 \, {y''} = -48 \, {y}" data-latex="-24 \, {y'} + 3 \, {y''} = -48 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1286" title="C3 | Homogeneous second-order linear ODE | ver. 1286"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0" alt="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0" title="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0" data-latex="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32 \, {y'} = -2 \, {y''} - 128 \, {y}" alt="-32 \, {y'} = -2 \, {y''} - 128 \, {y}" title="-32 \, {y'} = -2 \, {y''} - 128 \, {y}" data-latex="-32 \, {y'} = -2 \, {y''} - 128 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-51%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20-%206%20%5C,%20%7Bx'%7D%20=%200" alt="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0" title="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0" data-latex="-51 \, {x} - 3 \, {x''} - 6 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-32%20%5C,%20%7By'%7D%20=%20-2%20%5C,%20%7By''%7D%20-%20128%20%5C,%20%7By%7D" alt="-32 \, {y'} = -2 \, {y''} - 128 \, {y}" title="-32 \, {y'} = -2 \, {y''} - 128 \, {y}" data-latex="-32 \, {y'} = -2 \, {y''} - 128 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" title="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8473" title="C3 | Homogeneous second-order linear ODE | ver. 8473"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y'} = -3 \, {y''} - 3 \, {y}" alt="6 \, {y'} = -3 \, {y''} - 3 \, {y}" title="6 \, {y'} = -3 \, {y''} - 3 \, {y}" data-latex="6 \, {y'} = -3 \, {y''} - 3 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?58 \, {x} + 2 \, {x''} = 8 \, {x'}" alt="58 \, {x} + 2 \, {x''} = 8 \, {x'}" title="58 \, {x} + 2 \, {x''} = 8 \, {x'}" data-latex="58 \, {x} + 2 \, {x''} = 8 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By'%7D%20=%20-3%20%5C,%20%7By''%7D%20-%203%20%5C,%20%7By%7D" alt="6 \, {y'} = -3 \, {y''} - 3 \, {y}" title="6 \, {y'} = -3 \, {y''} - 3 \, {y}" data-latex="6 \, {y'} = -3 \, {y''} - 3 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?58%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20=%208%20%5C,%20%7Bx'%7D" alt="58 \, {x} + 2 \, {x''} = 8 \, {x'}" title="58 \, {x} + 2 \, {x''} = 8 \, {x'}" data-latex="58 \, {x} + 2 \, {x''} = 8 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4868" title="C3 | Homogeneous second-order linear ODE | ver. 4868"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = 18 \, {y'} - 27 \, {y}" alt="3 \, {y''} = 18 \, {y'} - 27 \, {y}" title="3 \, {y''} = 18 \, {y'} - 27 \, {y}" data-latex="3 \, {y''} = 18 \, {y'} - 27 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} = 8 \, {x'} + 26 \, {x}" alt="-2 \, {x''} = 8 \, {x'} + 26 \, {x}" title="-2 \, {x''} = 8 \, {x'} + 26 \, {x}" data-latex="-2 \, {x''} = 8 \, {x'} + 26 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%2018%20%5C,%20%7By'%7D%20-%2027%20%5C,%20%7By%7D" alt="3 \, {y''} = 18 \, {y'} - 27 \, {y}" title="3 \, {y''} = 18 \, {y'} - 27 \, {y}" data-latex="3 \, {y''} = 18 \, {y'} - 27 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20=%208%20%5C,%20%7Bx'%7D%20+%2026%20%5C,%20%7Bx%7D" alt="-2 \, {x''} = 8 \, {x'} + 26 \, {x}" title="-2 \, {x''} = 8 \, {x'} + 26 \, {x}" data-latex="-2 \, {x''} = 8 \, {x'} + 26 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(3 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3114" title="C3 | Homogeneous second-order linear ODE | ver. 3114"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?28 \, {x'} = 98 \, {x} + 2 \, {x''}" alt="28 \, {x'} = 98 \, {x} + 2 \, {x''}" title="28 \, {x'} = 98 \, {x} + 2 \, {x''}" data-latex="28 \, {x'} = 98 \, {x} + 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0" alt="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0" title="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0" data-latex="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?28%20%5C,%20%7Bx'%7D%20=%2098%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D" alt="28 \, {x'} = 98 \, {x} + 2 \, {x''}" title="28 \, {x'} = 98 \, {x} + 2 \, {x''}" data-latex="28 \, {x'} = 98 \, {x} + 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20+%2060%20%5C,%20%7By%7D%20-%2024%20%5C,%20%7By'%7D%20=%200" alt="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0" title="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0" data-latex="3 \, {y''} + 60 \, {y} - 24 \, {y'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%204%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%204%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i + 4\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 4\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" title="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2653" title="C3 | Homogeneous second-order linear ODE | ver. 2653"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, {y} = 12 \, {y'} + 2 \, {y''}" alt="-20 \, {y} = 12 \, {y'} + 2 \, {y''}" title="-20 \, {y} = 12 \, {y'} + 2 \, {y''}" data-latex="-20 \, {y} = 12 \, {y'} + 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, {x'} - 50 \, {x} = 2 \, {x''}" alt="-20 \, {x'} - 50 \, {x} = 2 \, {x''}" title="-20 \, {x'} - 50 \, {x} = 2 \, {x''}" data-latex="-20 \, {x'} - 50 \, {x} = 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20%7By%7D%20=%2012%20%5C,%20%7By'%7D%20+%202%20%5C,%20%7By''%7D" alt="-20 \, {y} = 12 \, {y'} + 2 \, {y''}" title="-20 \, {y} = 12 \, {y'} + 2 \, {y''}" data-latex="-20 \, {y} = 12 \, {y'} + 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20%7Bx'%7D%20-%2050%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D" alt="-20 \, {x'} - 50 \, {x} = 2 \, {x''}" title="-20 \, {x'} - 50 \, {x} = 2 \, {x''}" data-latex="-20 \, {x'} - 50 \, {x} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" alt="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" title="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-5 \, t\right)} + k_{2} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6895" title="C3 | Homogeneous second-order linear ODE | ver. 6895"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x''} = 30 \, {x'} - 75 \, {x}" alt="3 \, {x''} = 30 \, {x'} - 75 \, {x}" title="3 \, {x''} = 30 \, {x'} - 75 \, {x}" data-latex="3 \, {x''} = 30 \, {x'} - 75 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30 \, {y'} - 3 \, {y''} = 78 \, {y}" alt="30 \, {y'} - 3 \, {y''} = 78 \, {y}" title="30 \, {y'} - 3 \, {y''} = 78 \, {y}" data-latex="30 \, {y'} - 3 \, {y''} = 78 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx''%7D%20=%2030%20%5C,%20%7Bx'%7D%20-%2075%20%5C,%20%7Bx%7D" alt="3 \, {x''} = 30 \, {x'} - 75 \, {x}" title="3 \, {x''} = 30 \, {x'} - 75 \, {x}" data-latex="3 \, {x''} = 30 \, {x'} - 75 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By''%7D%20=%2078%20%5C,%20%7By%7D" alt="30 \, {y'} - 3 \, {y''} = 78 \, {y}" title="30 \, {y'} - 3 \, {y''} = 78 \, {y}" data-latex="30 \, {y'} - 3 \, {y''} = 78 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9861" title="C3 | Homogeneous second-order linear ODE | ver. 9861"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}" alt="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}" title="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}" data-latex="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}" alt="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}" title="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}" data-latex="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-12%20%5C,%20%7Bx'%7D%20+%202%20%5C,%20%7Bx''%7D%20+%2020%20%5C,%20%7Bx%7D" alt="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}" title="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}" data-latex="0 = -12 \, {x'} + 2 \, {x''} + 20 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2072%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D%20+%2024%20%5C,%20%7By'%7D" alt="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}" title="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}" data-latex="0 = 72 \, {y} + 2 \, {y''} + 24 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2916" title="C3 | Homogeneous second-order linear ODE | ver. 2916"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30 \, {y'} = -87 \, {y} - 3 \, {y''}" alt="30 \, {y'} = -87 \, {y} - 3 \, {y''}" title="30 \, {y'} = -87 \, {y} - 3 \, {y''}" data-latex="30 \, {y'} = -87 \, {y} - 3 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x'} - 2 \, {x} = 2 \, {x''}" alt="-4 \, {x'} - 2 \, {x} = 2 \, {x''}" title="-4 \, {x'} - 2 \, {x} = 2 \, {x''}" data-latex="-4 \, {x'} - 2 \, {x} = 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30%20%5C,%20%7By'%7D%20=%20-87%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By''%7D" alt="30 \, {y'} = -87 \, {y} - 3 \, {y''}" title="30 \, {y'} = -87 \, {y} - 3 \, {y''}" data-latex="30 \, {y'} = -87 \, {y} - 3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D" alt="-4 \, {x'} - 2 \, {x} = 2 \, {x''}" title="-4 \, {x'} - 2 \, {x} = 2 \, {x''}" data-latex="-4 \, {x'} - 2 \, {x} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 5\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-7408" title="C3 | Homogeneous second-order linear ODE | ver. 7408"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}" alt="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}" title="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}" data-latex="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0" alt="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0" title="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0" data-latex="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20200%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20-%2040%20%5C,%20%7Bx'%7D" alt="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}" title="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}" data-latex="0 = 200 \, {x} + 2 \, {x''} - 40 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By'%7D%20+%2051%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By''%7D%20=%200" alt="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0" title="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0" data-latex="6 \, {y'} + 51 \, {y} + 3 \, {y''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" title="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8718" title="C3 | Homogeneous second-order linear ODE | ver. 8718"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} - 2 \, {y''} = 50 \, {y}" alt="-12 \, {y'} - 2 \, {y''} = 50 \, {y}" title="-12 \, {y'} - 2 \, {y''} = 50 \, {y}" data-latex="-12 \, {y'} - 2 \, {y''} = 50 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} + 72 \, {x} = -24 \, {x'}" alt="2 \, {x''} + 72 \, {x} = -24 \, {x'}" title="2 \, {x''} + 72 \, {x} = -24 \, {x'}" data-latex="2 \, {x''} + 72 \, {x} = -24 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D%20=%2050%20%5C,%20%7By%7D" alt="-12 \, {y'} - 2 \, {y''} = 50 \, {y}" title="-12 \, {y'} - 2 \, {y''} = 50 \, {y}" data-latex="-12 \, {y'} - 2 \, {y''} = 50 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20+%2072%20%5C,%20%7Bx%7D%20=%20-24%20%5C,%20%7Bx'%7D" alt="2 \, {x''} + 72 \, {x} = -24 \, {x'}" title="2 \, {x''} + 72 \, {x} = -24 \, {x'}" data-latex="2 \, {x''} + 72 \, {x} = -24 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" alt="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(4 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9503" title="C3 | Homogeneous second-order linear ODE | ver. 9503"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = -243 \, {y} - 54 \, {y'}" alt="3 \, {y''} = -243 \, {y} - 54 \, {y'}" title="3 \, {y''} = -243 \, {y} - 54 \, {y'}" data-latex="3 \, {y''} = -243 \, {y} - 54 \, {y'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-52 \, {x} = 2 \, {x''} + 4 \, {x'}" alt="-52 \, {x} = 2 \, {x''} + 4 \, {x'}" title="-52 \, {x} = 2 \, {x''} + 4 \, {x'}" data-latex="-52 \, {x} = 2 \, {x''} + 4 \, {x'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%20-243%20%5C,%20%7By%7D%20-%2054%20%5C,%20%7By'%7D" alt="3 \, {y''} = -243 \, {y} - 54 \, {y'}" title="3 \, {y''} = -243 \, {y} - 54 \, {y'}" data-latex="3 \, {y''} = -243 \, {y} - 54 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-52%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D%20+%204%20%5C,%20%7Bx'%7D" alt="-52 \, {x} = 2 \, {x''} + 4 \, {x'}" title="-52 \, {x} = 2 \, {x''} + 4 \, {x'}" data-latex="-52 \, {x} = 2 \, {x''} + 4 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-9484" title="C3 | Homogeneous second-order linear ODE | ver. 9484"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0" alt="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0" title="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0" data-latex="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}" alt="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}" title="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}" data-latex="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx''%7D%20+%2012%20%5C,%20%7Bx'%7D%20=%200" alt="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0" title="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0" data-latex="-12 \, {x} - 3 \, {x''} + 12 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%203%20%5C,%20%7By''%7D%20-%2030%20%5C,%20%7By'%7D%20+%20123%20%5C,%20%7By%7D" alt="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}" title="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}" data-latex="0 = 3 \, {y''} - 30 \, {y'} + 123 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" alt="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(4%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(4%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(4 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(4 i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(4 \, t\right) + d_{2} \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4488" title="C3 | Homogeneous second-order linear ODE | ver. 4488"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0" alt="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0" title="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0" data-latex="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y} = -8 \, {y'} - 2 \, {y''}" alt="8 \, {y} = -8 \, {y'} - 2 \, {y''}" title="8 \, {y} = -8 \, {y'} - 2 \, {y''}" data-latex="8 \, {y} = -8 \, {y'} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx''%7D%20-%2024%20%5C,%20%7Bx%7D%20-%2012%20%5C,%20%7Bx'%7D%20=%200" alt="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0" title="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0" data-latex="-3 \, {x''} - 24 \, {x} - 12 \, {x'} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By%7D%20=%20-8%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D" alt="8 \, {y} = -8 \, {y'} - 2 \, {y''}" title="8 \, {y} = -8 \, {y'} - 2 \, {y''}" data-latex="8 \, {y} = -8 \, {y'} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8985" title="C3 | Homogeneous second-order linear ODE | ver. 8985"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} + 8 \, {y'} = 2 \, {y''}" alt="-8 \, {y} + 8 \, {y'} = 2 \, {y''}" title="-8 \, {y} + 8 \, {y'} = 2 \, {y''}" data-latex="-8 \, {y} + 8 \, {y'} = 2 \, {y''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}" alt="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}" title="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}" data-latex="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20+%208%20%5C,%20%7By'%7D%20=%202%20%5C,%20%7By''%7D" alt="-8 \, {y} + 8 \, {y'} = 2 \, {y''}" title="-8 \, {y} + 8 \, {y'} = 2 \, {y''}" data-latex="-8 \, {y} + 8 \, {y'} = 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-78%20%5C,%20%7Bx%7D%20-%206%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx''%7D" alt="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}" title="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}" data-latex="0 = -78 \, {x} - 6 \, {x'} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3136" title="C3 | Homogeneous second-order linear ODE | ver. 3136"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0" alt="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0" title="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0" data-latex="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}" alt="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}" title="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}" data-latex="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20%7Bx'%7D%20+%2032%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx''%7D%20=%200" alt="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0" title="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0" data-latex="-16 \, {x'} + 32 \, {x} + 2 \, {x''} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By''%7D%20-%2054%20%5C,%20%7By%7D%20+%2018%20%5C,%20%7By'%7D" alt="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}" title="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}" data-latex="0 = -3 \, {y''} - 54 \, {y} + 18 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(3 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-8344" title="C3 | Homogeneous second-order linear ODE | ver. 8344"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} + 20 \, {x} = 12 \, {x'}" alt="2 \, {x''} + 20 \, {x} = 12 \, {x'}" title="2 \, {x''} + 20 \, {x} = 12 \, {x'}" data-latex="2 \, {x''} + 20 \, {x} = 12 \, {x'}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} = -18 \, {y} - 2 \, {y''}" alt="-12 \, {y'} = -18 \, {y} - 2 \, {y''}" title="-12 \, {y'} = -18 \, {y} - 2 \, {y''}" data-latex="-12 \, {y'} = -18 \, {y} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20+%2020%20%5C,%20%7Bx%7D%20=%2012%20%5C,%20%7Bx'%7D" alt="2 \, {x''} + 20 \, {x} = 12 \, {x'}" title="2 \, {x''} + 20 \, {x} = 12 \, {x'}" data-latex="2 \, {x''} + 20 \, {x} = 12 \, {x'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20=%20-18%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By''%7D" alt="-12 \, {y'} = -18 \, {y} - 2 \, {y''}" title="-12 \, {y'} = -18 \, {y} - 2 \, {y''}" data-latex="-12 \, {y'} = -18 \, {y} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" title="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(3 \, t\right)} + k_{2} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4126" title="C3 | Homogeneous second-order linear ODE | ver. 4126"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0" alt="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0" title="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0" data-latex="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {x'} - 58 \, {x} = 2 \, {x''}" alt="-8 \, {x'} - 58 \, {x} = 2 \, {x''}" title="-8 \, {x'} - 58 \, {x} = 2 \, {x''}" data-latex="-8 \, {x'} - 58 \, {x} = 2 \, {x''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20+%2036%20%5C,%20%7By'%7D%20+%20108%20%5C,%20%7By%7D%20=%200" alt="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0" title="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0" data-latex="3 \, {y''} + 36 \, {y'} + 108 \, {y} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7Bx'%7D%20-%2058%20%5C,%20%7Bx%7D%20=%202%20%5C,%20%7Bx''%7D" alt="-8 \, {x'} - 58 \, {x} = 2 \, {x''}" title="-8 \, {x'} - 58 \, {x} = 2 \, {x''}" data-latex="-8 \, {x'} - 58 \, {x} = 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" title="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-3307" title="C3 | Homogeneous second-order linear ODE | ver. 3307"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} = 8 \, {x'} - 8 \, {x}" alt="2 \, {x''} = 8 \, {x'} - 8 \, {x}" title="2 \, {x''} = 8 \, {x'} - 8 \, {x}" data-latex="2 \, {x''} = 8 \, {x'} - 8 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}" alt="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}" title="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}" data-latex="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20=%208%20%5C,%20%7Bx'%7D%20-%208%20%5C,%20%7Bx%7D" alt="2 \, {x''} = 8 \, {x'} - 8 \, {x}" title="2 \, {x''} = 8 \, {x'} - 8 \, {x}" data-latex="2 \, {x''} = 8 \, {x'} - 8 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%203%20%5C,%20%7By''%7D%20+%2015%20%5C,%20%7By%7D%20+%206%20%5C,%20%7By'%7D" alt="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}" title="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}" data-latex="0 = 3 \, {y''} + 15 \, {y} + 6 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" alt="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" title="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(2 \, t\right)} + k_{2} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-5407" title="C3 | Homogeneous second-order linear ODE | ver. 5407"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {y'} + 3 \, {y''} = -15 \, {y}" alt="-12 \, {y'} + 3 \, {y''} = -15 \, {y}" title="-12 \, {y'} + 3 \, {y''} = -15 \, {y}" data-latex="-12 \, {y'} + 3 \, {y''} = -15 \, {y}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} = 16 \, {x'} + 32 \, {x}" alt="-2 \, {x''} = 16 \, {x'} + 32 \, {x}" title="-2 \, {x''} = 16 \, {x'} + 32 \, {x}" data-latex="-2 \, {x''} = 16 \, {x'} + 32 \, {x}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D%20=%20-15%20%5C,%20%7By%7D" alt="-12 \, {y'} + 3 \, {y''} = -15 \, {y}" title="-12 \, {y'} + 3 \, {y''} = -15 \, {y}" data-latex="-12 \, {y'} + 3 \, {y''} = -15 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20=%2016%20%5C,%20%7Bx'%7D%20+%2032%20%5C,%20%7Bx%7D" alt="-2 \, {x''} = 16 \, {x'} + 32 \, {x}" title="-2 \, {x''} = 16 \, {x'} + 32 \, {x}" data-latex="-2 \, {x''} = 16 \, {x'} + 32 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(i + 2\right) \, t\right)} + c_{2} e^{\left(-\left(i - 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-4 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2672" title="C3 | Homogeneous second-order linear ODE | ver. 2672"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-147 \, {x} - 42 \, {x'} = 3 \, {x''}" alt="-147 \, {x} - 42 \, {x'} = 3 \, {x''}" title="-147 \, {x} - 42 \, {x'} = 3 \, {x''}" data-latex="-147 \, {x} - 42 \, {x'} = 3 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30 \, {y'} + 150 \, {y} = -3 \, {y''}" alt="-30 \, {y'} + 150 \, {y} = -3 \, {y''}" title="-30 \, {y'} + 150 \, {y} = -3 \, {y''}" data-latex="-30 \, {y'} + 150 \, {y} = -3 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-147%20%5C,%20%7Bx%7D%20-%2042%20%5C,%20%7Bx'%7D%20=%203%20%5C,%20%7Bx''%7D" alt="-147 \, {x} - 42 \, {x'} = 3 \, {x''}" title="-147 \, {x} - 42 \, {x'} = 3 \, {x''}" data-latex="-147 \, {x} - 42 \, {x'} = 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30%20%5C,%20%7By'%7D%20+%20150%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By''%7D" alt="-30 \, {y'} + 150 \, {y} = -3 \, {y''}" title="-30 \, {y'} + 150 \, {y} = -3 \, {y''}" data-latex="-30 \, {y'} + 150 \, {y} = -3 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" alt="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" title="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20+%205%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20-%205%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(5 i + 5\right) \, t\right)} + c_{2} e^{\left(-\left(5 i - 5\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" title="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-7 \, t\right)} + k_{2} e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-2329" title="C3 | Homogeneous second-order linear ODE | ver. 2329"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + 4 \, {x'} = -2 \, {x''}" alt="2 \, {x} + 4 \, {x'} = -2 \, {x''}" title="2 \, {x} + 4 \, {x'} = -2 \, {x''}" data-latex="2 \, {x} + 4 \, {x'} = -2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30 \, {y} = -3 \, {y''} - 6 \, {y'}" alt="30 \, {y} = -3 \, {y''} - 6 \, {y'}" title="30 \, {y} = -3 \, {y''} - 6 \, {y'}" data-latex="30 \, {y} = -3 \, {y''} - 6 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7Bx'%7D%20=%20-2%20%5C,%20%7Bx''%7D" alt="2 \, {x} + 4 \, {x'} = -2 \, {x''}" title="2 \, {x} + 4 \, {x'} = -2 \, {x''}" data-latex="2 \, {x} + 4 \, {x'} = -2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?30%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By''%7D%20-%206%20%5C,%20%7By'%7D" alt="30 \, {y} = -3 \, {y''} - 6 \, {y'}" title="30 \, {y} = -3 \, {y''} - 6 \, {y'}" data-latex="30 \, {y} = -3 \, {y''} - 6 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(3%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(3%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(3 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(3 i + 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}" title="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(3 \, t\right) + d_{2} \sin\left(3 \, t\right)\right)} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{x} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-0623" title="C3 | Homogeneous second-order linear ODE | ver. 0623"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?200 \, {x} = -40 \, {x'} - 2 \, {x''}" alt="200 \, {x} = -40 \, {x'} - 2 \, {x''}" title="200 \, {x} = -40 \, {x'} - 2 \, {x''}" data-latex="200 \, {x} = -40 \, {x'} - 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?26 \, {y} = -12 \, {y'} - 2 \, {y''}" alt="26 \, {y} = -12 \, {y'} - 2 \, {y''}" title="26 \, {y} = -12 \, {y'} - 2 \, {y''}" data-latex="26 \, {y} = -12 \, {y'} - 2 \, {y''}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?200%20%5C,%20%7Bx%7D%20=%20-40%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx''%7D" alt="200 \, {x} = -40 \, {x'} - 2 \, {x''}" title="200 \, {x} = -40 \, {x'} - 2 \, {x''}" data-latex="200 \, {x} = -40 \, {x'} - 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?26%20%5C,%20%7By%7D%20=%20-12%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By''%7D" alt="26 \, {y} = -12 \, {y'} - 2 \, {y''}" title="26 \, {y} = -12 \, {y'} - 2 \, {y''}" data-latex="26 \, {y} = -12 \, {y'} - 2 \, {y''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}" alt="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}" title="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}" data-latex="{y} = c_{1} e^{\left(\left(2 i - 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}" title="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}" data-latex="{y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" title="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}" data-latex="{x} = k_{1} t e^{\left(-10 \, t\right)} + k_{2} e^{\left(-10 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-4712" title="C3 | Homogeneous second-order linear ODE | ver. 4712"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x''} = 4 \, {x'} - 4 \, {x}" alt="2 \, {x''} = 4 \, {x'} - 4 \, {x}" title="2 \, {x''} = 4 \, {x'} - 4 \, {x}" data-latex="2 \, {x''} = 4 \, {x'} - 4 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y''} + 16 \, {y'} = 32 \, {y}" alt="-2 \, {y''} + 16 \, {y'} = 32 \, {y}" title="-2 \, {y''} + 16 \, {y'} = 32 \, {y}" data-latex="-2 \, {y''} + 16 \, {y'} = 32 \, {y}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx''%7D%20=%204%20%5C,%20%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D" alt="2 \, {x''} = 4 \, {x'} - 4 \, {x}" title="2 \, {x''} = 4 \, {x'} - 4 \, {x}" data-latex="2 \, {x''} = 4 \, {x'} - 4 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By''%7D%20+%2016%20%5C,%20%7By'%7D%20=%2032%20%5C,%20%7By%7D" alt="-2 \, {y''} + 16 \, {y'} = 32 \, {y}" title="-2 \, {y''} + 16 \, {y'} = 32 \, {y}" data-latex="-2 \, {y''} + 16 \, {y'} = 32 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(i - 1\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(t\right) + d_{2} \sin\left(t\right)\right)} e^{t}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" title="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(4 \, t\right)} + k_{2} e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-1121" title="C3 | Homogeneous second-order linear ODE | ver. 1121"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x''} = 8 \, {x'} + 58 \, {x}" alt="-2 \, {x''} = 8 \, {x'} + 58 \, {x}" title="-2 \, {x''} = 8 \, {x'} + 58 \, {x}" data-latex="-2 \, {x''} = 8 \, {x'} + 58 \, {x}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y''} = -8 \, {y} - 8 \, {y'}" alt="2 \, {y''} = -8 \, {y} - 8 \, {y'}" title="2 \, {y''} = -8 \, {y} - 8 \, {y'}" data-latex="2 \, {y''} = -8 \, {y} - 8 \, {y'}"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx''%7D%20=%208%20%5C,%20%7Bx'%7D%20+%2058%20%5C,%20%7Bx%7D" alt="-2 \, {x''} = 8 \, {x'} + 58 \, {x}" title="-2 \, {x''} = 8 \, {x'} + 58 \, {x}" data-latex="-2 \, {x''} = 8 \, {x'} + 58 \, {x}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By''%7D%20=%20-8%20%5C,%20%7By%7D%20-%208%20%5C,%20%7By'%7D" alt="2 \, {y''} = -8 \, {y} - 8 \, {y'}" title="2 \, {y''} = -8 \, {y} - 8 \, {y'}" data-latex="2 \, {y''} = -8 \, {y} - 8 \, {y'}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(5%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(5%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(5 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 2\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" title="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C3-6228" title="C3 | Homogeneous second-order linear ODE | ver. 6228"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C3.</strong></p><p>Explain how to find the general solution to each given ODE using exponential functions.</p><p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x'} = -26 \, {x} - 2 \, {x''}" alt="-12 \, {x'} = -26 \, {x} - 2 \, {x''}" title="-12 \, {x'} = -26 \, {x} - 2 \, {x''}" data-latex="-12 \, {x'} = -26 \, {x} - 2 \, {x''}"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0" alt="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0" title="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0" data-latex="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx'%7D%20=%20-26%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7Bx''%7D" alt="-12 \, {x'} = -26 \, {x} - 2 \, {x''}" title="-12 \, {x'} = -26 \, {x} - 2 \, {x''}" data-latex="-12 \, {x'} = -26 \, {x} - 2 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;p style="text-align:center;"&gt;
        &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-30%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D%20+%2075%20%5C,%20%7By%7D%20=%200" alt="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0" title="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0" data-latex="-30 \, {y'} + 3 \, {y''} + 75 \, {y} = 0"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" alt="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%203%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%203%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 3\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 3\right) \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" title="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(5 \, t\right)} + k_{2} e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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</questestinterop>