C5.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="C5">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- C5</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="C5-9755" title="C5 | Non-homogeneous second-order linear ODE | ver. 9755"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" alt="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" title="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" data-latex="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%202%20%5C,%20%7By'%7D%20-%207%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%203%20%5C,%20%7By%7D" alt="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" title="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" data-latex="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2016" title="C5 | Non-homogeneous second-order linear ODE | ver. 2016"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}" alt="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}" title="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}" data-latex="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%2036%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20-%20%7By''%7D%20-%202%20%5C,%20%7By'%7D%20+%2024%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}" title="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}" data-latex="0 = 36 \, t e^{\left(3 \, t\right)} + 3 \, {y} - {y''} - 2 \, {y'} + 24 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" alt="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1377" title="C5 | Non-homogeneous second-order linear ODE | ver. 1377"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" alt="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" title="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" data-latex="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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=%20-16%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20%7By%7D%20-%2012%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" title="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" data-latex="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&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} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} 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?%7By%7D%20=%20-2%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5185" title="C5 | Non-homogeneous second-order linear ODE | ver. 5185"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}" alt="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}" title="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}" data-latex="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20-%2054%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2010%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%20-2%20%5C,%20%7By%7D" alt="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}" title="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}" data-latex="-{y''} - {y'} - 54 \, \cos\left(5 \, t\right) - 10 \, \sin\left(5 \, t\right) = -2 \, {y}"&gt;
  &lt;/p&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} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} + 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0396" title="C5 | Non-homogeneous second-order linear ODE | ver. 0396"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}" alt="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}" title="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}" data-latex="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20+%204%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%2032%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%20-12%20%5C,%20%7By%7D" alt="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}" title="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}" data-latex="-{y''} - {y'} + 4 \, \cos\left(2 \, t\right) - 32 \, \sin\left(2 \, t\right) = -12 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0095" title="C5 | Non-homogeneous second-order linear ODE | ver. 0095"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)" alt="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)" title="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20%7By''%7D%20+%206%20%5C,%20%7By'%7D%20+%208%20%5C,%20%7By%7D%20-%2017%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2030%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)" title="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} + 6 \, {y'} + 8 \, {y} - 17 \, \cos\left(5 \, t\right) - 30 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1702" title="C5 | Non-homogeneous second-order linear ODE | ver. 1702"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)" alt="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)" title="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)" data-latex="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?2%20%5C,%20%7By'%7D%20+%20%7By''%7D%20=%208%20%5C,%20%7By%7D%20+%2018%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%2051%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)" title="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)" data-latex="2 \, {y'} + {y''} = 8 \, {y} + 18 \, \cos\left(3 \, t\right) - 51 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5831" title="C5 | Non-homogeneous second-order linear ODE | ver. 5831"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" alt="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?18%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20+%20%7By'%7D%20-%2015%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20-12%20%5C,%20%7By%7D" alt="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" alt="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} 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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6305" title="C5 | Non-homogeneous second-order linear ODE | ver. 6305"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)" alt="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)" title="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)" data-latex="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-5%20%5C,%20%7By'%7D%20+%20%7By''%7D%20=%20-6%20%5C,%20%7By%7D%20-%2057%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2075%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)" title="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)" data-latex="-5 \, {y'} + {y''} = -6 \, {y} - 57 \, \cos\left(5 \, t\right) + 75 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5246" title="C5 | Non-homogeneous second-order linear ODE | ver. 5246"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)" alt="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)" title="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)" data-latex="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20%7By'%7D%20-%206%20%5C,%20%7By%7D%20+%20%7By''%7D%20+%2062%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2010%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)" title="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)" data-latex="0 = {y'} - 6 \, {y} + {y''} + 62 \, \cos\left(5 \, t\right) + 10 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8710" title="C5 | Non-homogeneous second-order linear ODE | ver. 8710"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)" alt="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)" title="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)" data-latex="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%208%20%5C,%20%7By%7D%20-%206%20%5C,%20%7By'%7D%20+%20%7By''%7D%20-%2060%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2034%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)" title="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)" data-latex="0 = 8 \, {y} - 6 \, {y'} + {y''} - 60 \, \cos\left(5 \, t\right) - 34 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3235" title="C5 | Non-homogeneous second-order linear ODE | ver. 3235"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0" alt="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0" title="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0" data-latex="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%204%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By'%7D%20+%2030%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2058%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%200" alt="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0" title="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0" data-latex="-{y''} + 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) - 58 \, \sin\left(5 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 2 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9855" title="C5 | Non-homogeneous second-order linear ODE | ver. 9855"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)" alt="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)" title="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)" data-latex="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%206%20%5C,%20%7By'%7D%20=%208%20%5C,%20%7By%7D%20-%2051%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2090%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)" title="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)" data-latex="-{y''} + 6 \, {y'} = 8 \, {y} - 51 \, \cos\left(5 \, t\right) + 90 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0208" title="C5 | Non-homogeneous second-order linear ODE | ver. 0208"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}" alt="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}" title="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}" data-latex="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?4%20%5C,%20%7By'%7D%20+%20%7By''%7D%20=%20-72%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%7By%7D%20-%2030%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}" title="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}" data-latex="4 \, {y'} + {y''} = -72 \, t e^{\left(3 \, t\right)} - 3 \, {y} - 30 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&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} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2858" title="C5 | Non-homogeneous second-order linear ODE | ver. 2858"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0" alt="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0" title="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0" data-latex="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?27%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20+%20%7By''%7D%20+%2012%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%200" alt="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0" title="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0" data-latex="27 \, t e^{\left(5 \, t\right)} + 2 \, {y'} - 8 \, {y} + {y''} + 12 \, e^{\left(5 \, t\right)} = 0"&gt;
  &lt;/p&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} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, 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=%20-t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0284" title="C5 | Non-homogeneous second-order linear ODE | ver. 0284"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)" alt="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)" title="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)" data-latex="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-2%20%5C,%20%7By%7D%20=%20-%7By'%7D%20-%20%7By''%7D%20-%206%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%2022%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)" title="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)" data-latex="-2 \, {y} = -{y'} - {y''} - 6 \, \cos\left(3 \, t\right) + 22 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3032" title="C5 | Non-homogeneous second-order linear ODE | ver. 3032"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}" alt="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}" title="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}" data-latex="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?20%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20=%204%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%205%20%5C,%20%7By'%7D" alt="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}" title="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}" data-latex="20 \, \cos\left(2 \, t\right) = 4 \, {y} + {y''} - 5 \, {y'}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} - 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2448" title="C5 | Non-homogeneous second-order linear ODE | ver. 2448"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" alt="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" title="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" data-latex="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20-%2012%20%5C,%20%7By%7D%20=%20-32%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" title="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" data-latex="{y''} + {y'} - 12 \, {y} = -32 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6880" title="C5 | Non-homogeneous second-order linear ODE | ver. 6880"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}" alt="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}" title="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}" data-latex="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-6%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20+%207%20%5C,%20%7By'%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%2012%20%5C,%20%7By%7D" alt="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}" title="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}" data-latex="-6 \, t e^{\left(5 \, t\right)} - {y''} + 7 \, {y'} - 9 \, e^{\left(5 \, t\right)} = 12 \, {y}"&gt;
  &lt;/p&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} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} 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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8956" title="C5 | Non-homogeneous second-order linear ODE | ver. 8956"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}" alt="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}" title="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}" data-latex="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%2045%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20-%204%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By%7D%20+%2024%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}" title="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}" data-latex="0 = 45 \, t e^{\left(2 \, t\right)} - {y''} - 4 \, {y'} - 3 \, {y} + 24 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} 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=%203%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7704" title="C5 | Non-homogeneous second-order linear ODE | ver. 7704"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)" alt="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)" title="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)" data-latex="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20%7By%7D%20=%20-%7By''%7D%20-%20%7By'%7D%20-%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%2016%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)" title="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)" data-latex="-12 \, {y} = -{y''} - {y'} - 2 \, \cos\left(2 \, t\right) + 16 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2222" title="C5 | Non-homogeneous second-order linear ODE | ver. 2222"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)" alt="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)" title="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)" data-latex="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?2%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By%7D%20-%20%7By''%7D%20=%20-8%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%2014%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)" title="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)" data-latex="2 \, {y'} + 3 \, {y} - {y''} = -8 \, \cos\left(2 \, t\right) - 14 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9189" title="C5 | Non-homogeneous second-order linear ODE | ver. 9189"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}" alt="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}" title="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}" data-latex="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-8%20%5C,%20%7By%7D%20=%2054%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20-%202%20%5C,%20%7By'%7D%20+%2024%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}" title="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}" data-latex="-8 \, {y} = 54 \, t e^{\left(5 \, t\right)} - {y''} - 2 \, {y'} + 24 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, 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=%202%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = 2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3756" title="C5 | Non-homogeneous second-order linear ODE | ver. 3756"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}" alt="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}" title="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}" data-latex="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?39%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%2027%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%204%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By'%7D%20-%20%7By''%7D" alt="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}" title="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}" data-latex="39 \, \cos\left(3 \, t\right) + 27 \, \sin\left(3 \, t\right) = 4 \, {y} - 3 \, {y'} - {y''}"&gt;
  &lt;/p&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} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0495" title="C5 | Non-homogeneous second-order linear ODE | ver. 0495"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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} = {y''} - 26 \, \sin\left(5 \, t\right)" alt="{y} = {y''} - 26 \, \sin\left(5 \, t\right)" title="{y} = {y''} - 26 \, \sin\left(5 \, t\right)" data-latex="{y} = {y''} - 26 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By''%7D%20-%2026%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = {y''} - 26 \, \sin\left(5 \, t\right)" title="{y} = {y''} - 26 \, \sin\left(5 \, t\right)" data-latex="{y} = {y''} - 26 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D%20-%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} - \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2210" title="C5 | Non-homogeneous second-order linear ODE | ver. 2210"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}" alt="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}" title="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}" data-latex="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-5%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%20-168%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%206%20%5C,%20%7By%7D%20-%2045%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}" title="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}" data-latex="-5 \, {y'} - {y''} = -168 \, t e^{\left(5 \, t\right)} + 6 \, {y} - 45 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, 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=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1611" title="C5 | Non-homogeneous second-order linear ODE | ver. 1611"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)" alt="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)" title="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)" data-latex="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-2%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By'%7D%20+%20%7By''%7D%20+%2069%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2045%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)" title="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)" data-latex="-2 \, {y} = -3 \, {y'} + {y''} + 69 \, \cos\left(5 \, t\right) - 45 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7215" title="C5 | Non-homogeneous second-order linear ODE | ver. 7215"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)" alt="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)" title="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)" data-latex="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20-8%20%5C,%20%7By%7D%20+%206%20%5C,%20%7By'%7D%20-%20%7By''%7D%20+%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%2018%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)" title="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)" data-latex="0 = -8 \, {y} + 6 \, {y'} - {y''} + \cos\left(3 \, t\right) - 18 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6136" title="C5 | Non-homogeneous second-order linear ODE | ver. 6136"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)" alt="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)" title="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)" data-latex="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?9%20%5C,%20%7By%7D%20-%20%7By''%7D%20=%20102%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)" title="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)" data-latex="9 \, {y} - {y''} = 102 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0901" title="C5 | Non-homogeneous second-order linear ODE | ver. 0901"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}" alt="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}" title="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}" data-latex="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-24%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20-%2018%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%204%20%5C,%20%7By'%7D%20-%20%7By''%7D" alt="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}" title="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}" data-latex="-24 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y'} - {y''}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-4741" title="C5 | Non-homogeneous second-order linear ODE | ver. 4741"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0" alt="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0" title="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0" data-latex="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%208%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20-%2017%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%206%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%200" alt="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0" title="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0" data-latex="{y''} - 8 \, {y} + 2 \, {y'} - 17 \, \cos\left(3 \, t\right) - 6 \, \sin\left(3 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} - \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2635" title="C5 | Non-homogeneous second-order linear ODE | ver. 2635"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0" alt="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0" title="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0" data-latex="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%209%20%5C,%20%7By%7D%20-%2039%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20=%200" alt="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0" title="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0" data-latex="-{y''} + 9 \, {y} - 39 \, \cos\left(2 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + 3 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8686" title="C5 | Non-homogeneous second-order linear ODE | ver. 8686"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}" alt="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}" title="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}" data-latex="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%203%20%5C,%20%7By'%7D%20=%2028%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20%7By%7D%20+%2018%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}" title="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}" data-latex="-{y''} - 3 \, {y'} = 28 \, t e^{\left(3 \, t\right)} - 4 \, {y} + 18 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&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} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} 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?%7By%7D%20=%20-2%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5938" title="C5 | Non-homogeneous second-order linear ODE | ver. 5938"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}" alt="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20%7By'%7D%20-%20%7By''%7D%20+%2010%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20-12%20%5C,%20%7By%7D" alt="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="-12 \, t e^{\left(3 \, t\right)} + {y'} - {y''} + 10 \, e^{\left(3 \, t\right)} = -12 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + 2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = k_{1} e^{\left(4 \, t\right)} + 2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = k_{1} e^{\left(4 \, t\right)} + 2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + 2 \, 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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%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} e^{\left(4 \, t\right)} + 2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = k_{1} e^{\left(4 \, t\right)} + 2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + 2 \, 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="C5-4471" title="C5 | Non-homogeneous second-order linear ODE | ver. 4471"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}" alt="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}" title="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}" data-latex="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-8%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%2014%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%20-3%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20+%20%7By''%7D" alt="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}" title="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}" data-latex="-8 \, \cos\left(2 \, t\right) + 14 \, \sin\left(2 \, t\right) = -3 \, {y} + 2 \, {y'} + {y''}"&gt;
  &lt;/p&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} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{t} - 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9958" title="C5 | Non-homogeneous second-order linear ODE | ver. 9958"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0" alt="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0" title="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0" data-latex="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?6%20%5C,%20%7By%7D%20-%205%20%5C,%20%7By'%7D%20+%20%7By''%7D%20+%2050%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2038%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%200" alt="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0" title="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0" data-latex="6 \, {y} - 5 \, {y'} + {y''} + 50 \, \cos\left(5 \, t\right) + 38 \, \sin\left(5 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + 2 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7365" title="C5 | Non-homogeneous second-order linear ODE | ver. 7365"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}" alt="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}" title="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}" data-latex="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?6%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%2042%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%20%7By''%7D%20-%20%7By'%7D%20-%2012%20%5C,%20%7By%7D" alt="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}" title="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}" data-latex="6 \, \cos\left(3 \, t\right) + 42 \, \sin\left(3 \, t\right) = {y''} - {y'} - 12 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 2 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8606" title="C5 | Non-homogeneous second-order linear ODE | ver. 8606"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)" alt="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)" title="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20%7By''%7D%20+%202%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20-%2020%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2066%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)" title="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} + 2 \, {y'} - 8 \, {y} - 20 \, \cos\left(5 \, t\right) + 66 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 2 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3006" title="C5 | Non-homogeneous second-order linear ODE | ver. 3006"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}" alt="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}" title="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}" data-latex="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%206%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%202%20%5C,%20%7By'%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}" title="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}" data-latex="0 = 6 \, t e^{\left(2 \, t\right)} - 3 \, {y} + {y''} - 2 \, {y'} - 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}" alt="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} 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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2274" title="C5 | Non-homogeneous second-order linear ODE | ver. 2274"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}" alt="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}" title="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}" data-latex="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?5%20%5C,%20%7By'%7D%20+%20%7By''%7D%20-%2075%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2063%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%20-4%20%5C,%20%7By%7D" alt="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}" title="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}" data-latex="5 \, {y'} + {y''} - 75 \, \cos\left(5 \, t\right) + 63 \, \sin\left(5 \, t\right) = -4 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6546" title="C5 | Non-homogeneous second-order linear ODE | ver. 6546"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}" alt="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}" title="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}" data-latex="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20%7By%7D%20-%2039%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20105%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%20%7By''%7D%20-%207%20%5C,%20%7By'%7D" alt="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}" title="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}" data-latex="-12 \, {y} - 39 \, \cos\left(5 \, t\right) + 105 \, \sin\left(5 \, t\right) = {y''} - 7 \, {y'}"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0874" title="C5 | Non-homogeneous second-order linear ODE | ver. 0874"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}" alt="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}" title="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}" data-latex="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?8%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%206%20%5C,%20%7By'%7D%20=%209%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%2012%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}" title="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}" data-latex="8 \, {y} + {y''} - 6 \, {y'} = 9 \, t e^{\left(5 \, t\right)} + 12 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} 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=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5980" title="C5 | Non-homogeneous second-order linear ODE | ver. 5980"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}" alt="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}" title="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}" data-latex="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-3%20%5C,%20%7By%7D%20=%20-2%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20%7By''%7D%20-%204%20%5C,%20%7By'%7D" alt="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}" title="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}" data-latex="-3 \, {y} = -2 \, t e^{\left(2 \, t\right)} + {y''} - 4 \, {y'}"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}" alt="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}" title="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} 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?%7By%7D%20=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}" title="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8021" title="C5 | Non-homogeneous second-order linear ODE | ver. 8021"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)" alt="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)" title="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)" data-latex="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%205%20%5C,%20%7By'%7D%20=%206%20%5C,%20%7By%7D%20-%2075%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2057%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)" title="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)" data-latex="-{y''} + 5 \, {y'} = 6 \, {y} - 75 \, \cos\left(5 \, t\right) - 57 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7685" title="C5 | Non-homogeneous second-order linear ODE | ver. 7685"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}" alt="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}" title="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}" data-latex="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By''%7D%20=%20-8%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%206%20%5C,%20%7By%7D%20+%206%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}" title="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}" data-latex="{y'} - {y''} = -8 \, t e^{\left(2 \, t\right)} - 6 \, {y} + 6 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" alt="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} 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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} - 2 \, t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9726" title="C5 | Non-homogeneous second-order linear ODE | ver. 9726"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)" alt="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)" title="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)" data-latex="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%206%20%5C,%20%7By%7D%20-%205%20%5C,%20%7By'%7D%20=%20-38%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2050%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)" title="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)" data-latex="-{y''} - 6 \, {y} - 5 \, {y'} = -38 \, \cos\left(5 \, t\right) - 50 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3002" title="C5 | Non-homogeneous second-order linear ODE | ver. 3002"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}" alt="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}" title="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}" data-latex="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?12%20%5C,%20%7By%7D%20+%209%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%2063%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%20-7%20%5C,%20%7By'%7D%20-%20%7By''%7D" alt="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}" title="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}" data-latex="12 \, {y} + 9 \, \cos\left(3 \, t\right) - 63 \, \sin\left(3 \, t\right) = -7 \, {y'} - {y''}"&gt;
  &lt;/p&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} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} - 3 \, \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0066" title="C5 | Non-homogeneous second-order linear ODE | ver. 0066"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)" alt="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)" title="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)" data-latex="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?3%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%20-4%20%5C,%20%7By%7D%20-%2045%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2087%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)" title="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)" data-latex="3 \, {y'} - {y''} = -4 \, {y} - 45 \, \cos\left(5 \, t\right) - 87 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0807" title="C5 | Non-homogeneous second-order linear ODE | ver. 0807"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)" alt="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)" title="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)" data-latex="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20-5%20%5C,%20%7By'%7D%20-%20%7By''%7D%20-%204%20%5C,%20%7By%7D%20-%2030%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)" title="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)" data-latex="0 = -5 \, {y'} - {y''} - 4 \, {y} - 30 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-4 \, t\right)} + 3 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9874" title="C5 | Non-homogeneous second-order linear ODE | ver. 9874"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}" alt="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}" title="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}" data-latex="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-3%20%5C,%20%7By%7D%20=%20-24%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20%7By''%7D%20-%204%20%5C,%20%7By'%7D%20-%2018%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}" title="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}" data-latex="-3 \, {y} = -24 \, t e^{\left(5 \, t\right)} + {y''} - 4 \, {y'} - 18 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2156" title="C5 | Non-homogeneous second-order linear ODE | ver. 2156"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" alt="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" title="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" data-latex="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-6%20%5C,%20%7By%7D%20=%20-%7By''%7D%20-%20%7By'%7D%20-%2015%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" title="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" data-latex="-6 \, {y} = -{y''} - {y'} - 15 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-4576" title="C5 | Non-homogeneous second-order linear ODE | ver. 4576"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" alt="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" title="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" data-latex="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?2%20%5C,%20%7By%7D%20=%20%7By''%7D%20+%20%7By'%7D%20-%2011%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" title="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)" data-latex="2 \, {y} = {y''} + {y'} - 11 \, \cos\left(3 \, t\right) - 3 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0622" title="C5 | Non-homogeneous second-order linear ODE | ver. 0622"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0" alt="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0" title="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0" data-latex="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?96%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By'%7D%20-%20%7By''%7D%20+%2036%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%200" alt="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0" title="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0" data-latex="96 \, t e^{\left(5 \, t\right)} + 3 \, {y} - 2 \, {y'} - {y''} + 36 \, e^{\left(5 \, t\right)} = 0"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1088" title="C5 | Non-homogeneous second-order linear ODE | ver. 1088"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}" alt="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}" title="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}" data-latex="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-4%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%2048%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20+%2014%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}" title="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}" data-latex="-4 \, {y'} - {y''} = 48 \, t e^{\left(5 \, t\right)} + 3 \, {y} + 14 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-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=%20-t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9671" title="C5 | Non-homogeneous second-order linear ODE | ver. 9671"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" alt="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20-6%20%5C,%20%7By%7D%20-%20%7By'%7D%20+%20%7By''%7D%20-%2020%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="0 = -6 \, {y} - {y'} + {y''} - 20 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - 2 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1735" title="C5 | Non-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>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0" alt="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0" title="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0" data-latex="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?36%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20%7By''%7D%20+%202%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D%20+%2021%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%200" alt="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0" title="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0" data-latex="36 \, t e^{\left(2 \, t\right)} + {y''} + 2 \, {y} + 3 \, {y'} + 21 \, e^{\left(2 \, t\right)} = 0"&gt;
  &lt;/p&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} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" alt="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} 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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5479" title="C5 | Non-homogeneous second-order linear ODE | ver. 5479"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0" alt="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0" title="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0" data-latex="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-4%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%206%20%5C,%20%7By%7D%20+%20%7By'%7D%20-%20%7By''%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%200" alt="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0" title="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0" data-latex="-4 \, t e^{\left(2 \, t\right)} + 6 \, {y} + {y'} - {y''} + 3 \, e^{\left(2 \, t\right)} = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" alt="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} 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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + t e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5624" title="C5 | Non-homogeneous second-order linear ODE | ver. 5624"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}" alt="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}" title="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}" data-latex="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20%7By'%7D%20+%20%7By''%7D%20-%2015%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%202%20%5C,%20%7By%7D" alt="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}" title="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}" data-latex="-12 \, t e^{\left(2 \, t\right)} + {y'} + {y''} - 15 \, e^{\left(2 \, t\right)} = 2 \, {y}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} 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?%7By%7D%20=%203%20%5C,%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%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0795" title="C5 | Non-homogeneous second-order linear ODE | ver. 0795"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}" alt="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}" title="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}" data-latex="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By''%7D%20+%2032%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%2012%20%5C,%20%7By%7D" alt="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}" title="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}" data-latex="{y'} + {y''} + 32 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 12 \, {y}"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(-4 \, t\right)} + 2 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8092" title="C5 | Non-homogeneous second-order linear ODE | ver. 8092"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}" alt="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}" title="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}" data-latex="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%202%20%5C,%20%7By'%7D%20+%2012%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%2036%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%208%20%5C,%20%7By%7D" alt="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}" title="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}" data-latex="{y''} - 2 \, {y'} + 12 \, \cos\left(2 \, t\right) + 36 \, \sin\left(2 \, t\right) = 8 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3196" title="C5 | Non-homogeneous second-order linear ODE | ver. 3196"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}" alt="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}" title="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}" data-latex="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-60%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20-%203%20%5C,%20%7By'%7D%20-%2027%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%202%20%5C,%20%7By%7D" alt="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}" title="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}" data-latex="-60 \, t e^{\left(3 \, t\right)} - {y''} - 3 \, {y'} - 27 \, e^{\left(3 \, t\right)} = 2 \, {y}"&gt;
  &lt;/p&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} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" alt="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} 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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" title="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6899" title="C5 | Non-homogeneous second-order linear ODE | ver. 6899"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)" alt="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)" title="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)" data-latex="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20-%7By'%7D%20-%206%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%205%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2031%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)" title="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)" data-latex="0 = -{y'} - 6 \, {y} + {y''} - 5 \, \cos\left(5 \, t\right) - 31 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} - \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3978" title="C5 | Non-homogeneous second-order linear ODE | ver. 3978"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}" alt="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}" title="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}" data-latex="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?24%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20+%20%7By'%7D%20+%2027%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-12%20%5C,%20%7By%7D" alt="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}" title="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}" data-latex="24 \, t e^{\left(5 \, t\right)} - {y''} + {y'} + 27 \, e^{\left(5 \, t\right)} = -12 \, {y}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} 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=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8601" title="C5 | Non-homogeneous second-order linear ODE | ver. 8601"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)" alt="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)" title="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20%7By''%7D%20-%204%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By'%7D%20+%2030%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2058%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)" title="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)" data-latex="0 = {y''} - 4 \, {y} - 3 \, {y'} + 30 \, \cos\left(5 \, t\right) + 58 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-t\right)} + 2 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7318" title="C5 | Non-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>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}" alt="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}" title="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}" data-latex="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-16%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%2028%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%2012%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%207%20%5C,%20%7By'%7D" alt="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}" title="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}" data-latex="-16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right) = 12 \, {y} + {y''} - 7 \, {y'}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} - 2 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8330" title="C5 | Non-homogeneous second-order linear ODE | ver. 8330"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}" alt="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}" title="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}" data-latex="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%20102%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20=%209%20%5C,%20%7By%7D" alt="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}" title="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}" data-latex="{y''} - 102 \, \cos\left(5 \, t\right) = 9 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - 3 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1225" title="C5 | Non-homogeneous second-order linear ODE | ver. 1225"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0" alt="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0" title="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0" data-latex="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%2012%20%5C,%20%7By%7D%20+%207%20%5C,%20%7By'%7D%20+%2013%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2035%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%200" alt="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0" title="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0" data-latex="-{y''} - 12 \, {y} + 7 \, {y'} + 13 \, \cos\left(5 \, t\right) - 35 \, \sin\left(5 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7481" title="C5 | Non-homogeneous second-order linear ODE | ver. 7481"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)" alt="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)" title="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)" data-latex="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-8%20%5C,%20%7By%7D%20=%20-%7By''%7D%20-%202%20%5C,%20%7By'%7D%20+%2010%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2033%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)" title="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)" data-latex="-8 \, {y} = -{y''} - 2 \, {y'} + 10 \, \cos\left(5 \, t\right) - 33 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3823" title="C5 | Non-homogeneous second-order linear ODE | ver. 3823"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}" alt="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}" title="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}" data-latex="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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-%202%20%5C,%20%7By'%7D%20=%20-6%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}" title="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}" data-latex="{y''} - 2 \, {y'} = -6 \, t e^{\left(2 \, t\right)} + 3 \, {y} + 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, 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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + 2 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5563" title="C5 | Non-homogeneous second-order linear ODE | ver. 5563"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)" alt="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)" title="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)" data-latex="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?3%20%5C,%20%7By%7D%20=%20-%7By''%7D%20+%204%20%5C,%20%7By'%7D%20-%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%208%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)" title="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)" data-latex="3 \, {y} = -{y''} + 4 \, {y'} - \cos\left(2 \, t\right) + 8 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)" title="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{t} + \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-1529" title="C5 | Non-homogeneous second-order linear ODE | ver. 1529"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0" alt="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0" title="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0" data-latex="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%2012%20%5C,%20%7By%7D%20+%20%7By'%7D%20-%2063%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%209%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%200" alt="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0" title="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0" data-latex="-{y''} + 12 \, {y} + {y'} - 63 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 3 \, \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8786" title="C5 | Non-homogeneous second-order linear ODE | ver. 8786"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}" alt="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}" title="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}" data-latex="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20+%2033%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%209%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%202%20%5C,%20%7By%7D" alt="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}" title="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}" data-latex="{y''} - {y'} + 33 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right) = 2 \, {y}"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)} + 3 \, \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8694" title="C5 | Non-homogeneous second-order linear ODE | ver. 8694"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}" alt="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}" title="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}" data-latex="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?27%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%208%20%5C,%20%7By%7D%20+%2012%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-%7By''%7D%20-%202%20%5C,%20%7By'%7D" alt="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}" title="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}" data-latex="27 \, t e^{\left(5 \, t\right)} - 8 \, {y} + 12 \, e^{\left(5 \, t\right)} = -{y''} - 2 \, {y'}"&gt;
  &lt;/p&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} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}" alt="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} 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=%20-t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-4121" title="C5 | Non-homogeneous second-order linear ODE | ver. 4121"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}" alt="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}" title="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}" data-latex="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?54%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%2024%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-2%20%5C,%20%7By'%7D%20+%208%20%5C,%20%7By%7D%20-%20%7By''%7D" alt="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}" title="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}" data-latex="54 \, t e^{\left(5 \, t\right)} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'} + 8 \, {y} - {y''}"&gt;
  &lt;/p&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} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, 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=%20-2%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7339" title="C5 | Non-homogeneous second-order linear ODE | ver. 7339"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}" alt="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}" title="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}" data-latex="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?12%20%5C,%20%7By%7D%20=%20-8%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20%7By''%7D%20-%20%7By'%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}" title="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}" data-latex="12 \, {y} = -8 \, t e^{\left(5 \, t\right)} + {y''} - {y'} - 9 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, 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=%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = t e^{\left(5 \, t\right)} + k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3107" title="C5 | Non-homogeneous second-order linear ODE | ver. 3107"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)" alt="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)" title="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)" data-latex="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?12%20%5C,%20%7By%7D%20=%20-%7By'%7D%20+%20%7By''%7D%20+%204%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%2032%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)" title="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)" data-latex="12 \, {y} = -{y'} + {y''} + 4 \, \cos\left(2 \, t\right) + 32 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} + 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3334" title="C5 | Non-homogeneous second-order linear ODE | ver. 3334"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)" alt="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)" title="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)" data-latex="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?4%20%5C,%20%7By%7D%20-%205%20%5C,%20%7By'%7D%20+%20%7By''%7D%20=%20-21%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2025%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)" title="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)" data-latex="4 \, {y} - 5 \, {y'} + {y''} = -21 \, \cos\left(5 \, t\right) + 25 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D%20+%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{t} + \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9030" title="C5 | Non-homogeneous second-order linear ODE | ver. 9030"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}" alt="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}" title="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}" data-latex="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-5%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%20-60%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%206%20%5C,%20%7By%7D%20-%2022%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}" title="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}" data-latex="-5 \, {y'} - {y''} = -60 \, t e^{\left(3 \, t\right)} + 6 \, {y} - 22 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&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} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" alt="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, 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=%202%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" title="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-5585" title="C5 | Non-homogeneous second-order linear ODE | ver. 5585"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}" alt="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}" title="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}" data-latex="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-84%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%7By%7D%20-%2033%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-%7By'%7D%20-%20%7By''%7D" alt="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}" title="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}" data-latex="-84 \, t e^{\left(5 \, t\right)} - 2 \, {y} - 33 \, e^{\left(5 \, t\right)} = -{y'} - {y''}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}" title="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}" data-latex="{y} = 3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} + k_{2} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6142" title="C5 | Non-homogeneous second-order linear ODE | ver. 6142"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0" alt="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0" title="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0" data-latex="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-8%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20+%20%7By''%7D%20-%2010%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2033%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%200" alt="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0" title="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0" data-latex="-8 \, {y} + 2 \, {y'} + {y''} - 10 \, \cos\left(5 \, t\right) + 33 \, \sin\left(5 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7375" title="C5 | Non-homogeneous second-order linear ODE | ver. 7375"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}" alt="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}" title="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}" data-latex="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?15%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2081%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%202%20%5C,%20%7By%7D%20-%20%7By''%7D%20-%20%7By'%7D" alt="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}" title="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}" data-latex="15 \, \cos\left(5 \, t\right) - 81 \, \sin\left(5 \, t\right) = 2 \, {y} - {y''} - {y'}"&gt;
  &lt;/p&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} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(-2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9091" title="C5 | Non-homogeneous second-order linear ODE | ver. 9091"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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} + 52 \, \cos\left(5 \, t\right) = -{y''}" alt="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}" title="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}" data-latex="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%2052%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20=%20-%7By''%7D" alt="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}" title="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}" data-latex="-{y} + 52 \, \cos\left(5 \, t\right) = -{y''}"&gt;
  &lt;/p&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} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7Bt%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{t} + 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3453" title="C5 | Non-homogeneous second-order linear ODE | ver. 3453"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}" alt="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}" title="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}" data-latex="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?54%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%7By%7D%20+%2027%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-%7By''%7D%20+%20%7By'%7D" alt="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}" title="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}" data-latex="54 \, t e^{\left(5 \, t\right)} - 2 \, {y} + 27 \, e^{\left(5 \, t\right)} = -{y''} + {y'}"&gt;
  &lt;/p&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} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, 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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2296" title="C5 | Non-homogeneous second-order linear ODE | ver. 2296"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)" alt="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)" title="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)" data-latex="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-2%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7By'%7D%20+%20%7By''%7D%20+%207%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%209%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)" title="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)" data-latex="-2 \, {y} = -3 \, {y'} + {y''} + 7 \, \cos\left(3 \, t\right) - 9 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20+%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} + \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2599" title="C5 | Non-homogeneous second-order linear ODE | ver. 2599"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)" alt="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)" title="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)" data-latex="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%207%20%5C,%20%7By'%7D%20-%2012%20%5C,%20%7By%7D%20=%20-16%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%2028%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)" title="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)" data-latex="-{y''} + 7 \, {y'} - 12 \, {y} = -16 \, \cos\left(2 \, t\right) - 28 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6948" title="C5 | Non-homogeneous second-order linear ODE | ver. 6948"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)" alt="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)" title="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)" data-latex="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%202%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%203%20%5C,%20%7By'%7D%20-%2045%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2069%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)" title="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)" data-latex="0 = 2 \, {y} + {y''} - 3 \, {y'} - 45 \, \cos\left(5 \, t\right) - 69 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-8704" title="C5 | Non-homogeneous second-order linear ODE | ver. 8704"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)" alt="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)" title="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)" data-latex="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20=%206%20%5C,%20%7By%7D%20+%2030%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%206%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)" title="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)" data-latex="{y''} + {y'} = 6 \, {y} + 30 \, \cos\left(3 \, t\right) + 6 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 2 \, \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9057" title="C5 | Non-homogeneous second-order linear ODE | ver. 9057"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}" alt="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}" title="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}" data-latex="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-4%20%5C,%20%7By%7D%20-%2030%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%2010%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%20%7By''%7D%20+%205%20%5C,%20%7By'%7D" alt="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}" title="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}" data-latex="-4 \, {y} - 30 \, \cos\left(3 \, t\right) + 10 \, \sin\left(3 \, t\right) = {y''} + 5 \, {y'}"&gt;
  &lt;/p&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} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-4 \, t\right)} - 2 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-4533" title="C5 | Non-homogeneous second-order linear ODE | ver. 4533"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)" alt="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)" title="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)" data-latex="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?4%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%20-13%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%209%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)" title="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)" data-latex="4 \, {y} + 3 \, {y'} - {y''} = -13 \, \cos\left(3 \, t\right) + 9 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} - \cos\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2512" title="C5 | Non-homogeneous second-order linear ODE | ver. 2512"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</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''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}" alt="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}" title="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}" data-latex="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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+%203%20%5C,%20%7By'%7D%20+%2023%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%2015%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%202%20%5C,%20%7By%7D" alt="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}" title="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}" data-latex="-{y''} + 3 \, {y'} + 23 \, \cos\left(5 \, t\right) - 15 \, \sin\left(5 \, t\right) = 2 \, {y}"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-3058" title="C5 | Non-homogeneous second-order linear ODE | ver. 3058"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)" alt="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)" title="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)" data-latex="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?7%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%2012%20%5C,%20%7By%7D%20+%2028%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%2016%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)" title="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)" data-latex="7 \, {y'} - {y''} = 12 \, {y} + 28 \, \cos\left(2 \, t\right) - 16 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)} + 2 \, \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9113" title="C5 | Non-homogeneous second-order linear ODE | ver. 9113"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)" alt="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)" title="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)" data-latex="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?0%20=%20-%7By''%7D%20+%205%20%5C,%20%7By'%7D%20-%206%20%5C,%20%7By%7D%20-%2038%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2050%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)" title="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)" data-latex="0 = -{y''} + 5 \, {y'} - 6 \, {y} - 38 \, \cos\left(5 \, t\right) + 50 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-7948" title="C5 | Non-homogeneous second-order linear ODE | ver. 7948"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0" alt="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0" title="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0" data-latex="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?4%20%5C,%20%7By%7D%20-%20%7By''%7D%20+%203%20%5C,%20%7By'%7D%20-%2018%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%2026%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20=%200" alt="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0" title="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0" data-latex="4 \, {y} - {y''} + 3 \, {y'} - 18 \, \cos\left(3 \, t\right) - 26 \, \sin\left(3 \, t\right) = 0"&gt;
  &lt;/p&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} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)" title="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)} + 2 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9283" title="C5 | Non-homogeneous second-order linear ODE | ver. 9283"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" alt="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20%7By%7D%20-%20%7By''%7D%20+%207%20%5C,%20%7By'%7D%20=%202%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="-12 \, {y} - {y''} + 7 \, {y'} = 2 \, t e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" alt="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} 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=%20-t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" title="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}" data-latex="{y} = -t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9532" title="C5 | Non-homogeneous second-order linear ODE | ver. 9532"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}" alt="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}" title="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}" data-latex="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?21%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%2016%20%5C,%20%7By%7D%20-%2018%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20%7By''%7D" alt="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}" title="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}" data-latex="21 \, t e^{\left(3 \, t\right)} + 16 \, {y} - 18 \, e^{\left(3 \, t\right)} = {y''}"&gt;
  &lt;/p&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} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" alt="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, 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=%20k_%7B1%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" title="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}" data-latex="{y} = k_{1} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-0533" title="C5 | Non-homogeneous second-order linear ODE | ver. 0533"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)" alt="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)" title="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)" data-latex="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-12%20%5C,%20%7By%7D%20=%20-%7By''%7D%20-%20%7By'%7D%20+%2015%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%20111%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)" title="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)" data-latex="-12 \, {y} = -{y''} - {y'} + 15 \, \cos\left(5 \, t\right) - 111 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&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} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + 3 \, \sin\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2112" title="C5 | Non-homogeneous second-order linear ODE | ver. 2112"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}" alt="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}" title="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}" data-latex="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?18%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%7By'%7D%20-%20%7By''%7D%20+%2021%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%20-4%20%5C,%20%7By%7D" alt="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}" title="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}" data-latex="18 \, t e^{\left(2 \, t\right)} - 3 \, {y'} - {y''} + 21 \, e^{\left(2 \, t\right)} = -4 \, {y}"&gt;
  &lt;/p&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} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} 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?%7By%7D%20=%203%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" title="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}" data-latex="{y} = 3 \, t e^{\left(2 \, t\right)} + k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-9175" title="C5 | Non-homogeneous second-order linear ODE | ver. 9175"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}" alt="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}" title="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}" data-latex="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?2%20%5C,%20%7By%7D%20+%2046%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%2030%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20=%20-%7By''%7D%20-%203%20%5C,%20%7By'%7D" alt="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}" title="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}" data-latex="2 \, {y} + 46 \, \cos\left(5 \, t\right) + 30 \, \sin\left(5 \, t\right) = -{y''} - 3 \, {y'}"&gt;
  &lt;/p&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} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\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=%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" title="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)" data-latex="{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, \cos\left(5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-6681" title="C5 | Non-homogeneous second-order linear ODE | ver. 6681"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)" alt="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)" title="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)" data-latex="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-3%20%5C,%20%7By'%7D%20+%20%7By''%7D%20+%202%20%5C,%20%7By%7D%20=%209%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%207%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)" title="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)" data-latex="-3 \, {y'} + {y''} + 2 \, {y} = 9 \, \cos\left(3 \, t\right) + 7 \, \sin\left(3 \, t\right)"&gt;
  &lt;/p&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} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\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=%20k_%7B2%7D%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D%20-%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)" title="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)" data-latex="{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{t} - \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C5-2197" title="C5 | Non-homogeneous second-order linear ODE | ver. 2197"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C5.</strong></p><p>Explain how to find the general solution to the given ODE.</p><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} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}" alt="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}" title="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}" data-latex="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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?-4%20%5C,%20%7By%7D%20=%20-18%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By'%7D%20-%20%7By''%7D%20-%2021%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}" title="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}" data-latex="-4 \, {y} = -18 \, t e^{\left(5 \, t\right)} + 3 \, {y'} - {y''} - 21 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&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} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} 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=%20-3%20%5C,%20t%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}" title="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}" data-latex="{y} = -3 \, t e^{\left(5 \, t\right)} + k_{2} e^{\left(4 \, t\right)} + k_{1} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>