CoCalc -- C2.qti

masterit / banks / clontz-diff-eq / build / qti-bank / C2.qti
²⁰⁸⁶⁵ views
<?xml version='1.0' encoding='UTF-8'?>
<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="C2">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- C2</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="C2-4619" title="C2 | Non-homogeneous first-order linear ODE | ver. 4619"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}" alt="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}" title="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}" data-latex="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-4%20%5C,%20%7By'%7D%20+%2020%20%5C,%20%7By%7D%20+%2048%20%5C,%20e%5E%7Bt%7D" alt="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}" title="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}" data-latex="0 = -4 \, {y'} + 20 \, {y} + 48 \, e^{t}"&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 e^{\left(5 \, t\right)} - 3 \, e^{t}" alt="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{t}" title="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{t}" data-latex="{y} = k e^{\left(5 \, t\right)} - 3 \, 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%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{t}" title="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{t}" data-latex="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9329" title="C2 | Non-homogeneous first-order linear ODE | ver. 9329"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 3 \, e^{\left(-t\right)}" alt="-{y} = {y'} + 3 \, e^{\left(-t\right)}" title="-{y} = {y'} + 3 \, e^{\left(-t\right)}" data-latex="-{y} = {y'} + 3 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="-{y} = {y'} + 3 \, e^{\left(-t\right)}" title="-{y} = {y'} + 3 \, e^{\left(-t\right)}" data-latex="-{y} = {y'} + 3 \, e^{\left(-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 e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}" alt="{y} = k e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}" title="{y} = k e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 3 \, t 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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}" title="{y} = k e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 3 \, t e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2716" title="C2 | Non-homogeneous first-order linear ODE | ver. 2716"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}" alt="12 \, {y} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}" title="12 \, {y} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}" data-latex="12 \, {y} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-3%20%5C,%20%7By'%7D%20-%206%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="12 \, {y} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}" title="12 \, {y} = -3 \, {y'} - 6 \, e^{\left(-5 \, t\right)}" data-latex="12 \, {y} = -3 \, {y'} - 6 \, 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} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-1562" title="C2 | Non-homogeneous first-order linear ODE | ver. 1562"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'}" alt="2 \, {y} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'}" title="2 \, {y} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'}" data-latex="2 \, {y} + 24 \, e^{\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;C2.&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+%2024%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20-2%20%5C,%20%7By'%7D" alt="2 \, {y} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'}" title="2 \, {y} + 24 \, e^{\left(5 \, t\right)} = -2 \, {y'}" data-latex="2 \, {y} + 24 \, e^{\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 e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5395" title="C2 | Non-homogeneous first-order linear ODE | ver. 5395"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}" alt="48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}" title="48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}" data-latex="48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?48%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20=%20-12%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D" alt="48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}" title="48 \, e^{\left(-4 \, t\right)} = -12 \, {y} + 3 \, {y'}" data-latex="48 \, e^{\left(-4 \, t\right)} = -12 \, {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 e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-1061" title="C2 | Non-homogeneous first-order linear ODE | ver. 1061"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}" alt="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}" title="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}" data-latex="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?48%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)%20-%2016%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7By'%7D" alt="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}" title="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {y} = -4 \, {y'}" data-latex="48 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right) - 16 \, {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 e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 3 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6322" title="C2 | Non-homogeneous first-order linear ODE | ver. 6322"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}" alt="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}" title="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}" data-latex="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?32%20%5C,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D%20=%204%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D" alt="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}" title="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {y} - 4 \, {y'}" data-latex="32 \, \cos\left(-4 \, t\right) e^{t} = 4 \, {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 e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)" alt="{y} = k e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} + 2 \, e^{t} \sin\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%20e%5E%7Bt%7D%20+%202%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} + 2 \, e^{t} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6873" title="C2 | Non-homogeneous first-order linear ODE | ver. 6873"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} + 14 \, e^{\left(-4 \, t\right)}" alt="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, t\right)}" title="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, t\right)}" data-latex="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%2014%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, t\right)}" title="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, t\right)}" data-latex="{y'} = 3 \, {y} + 14 \, e^{\left(-4 \, 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 e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)} - 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%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4881" title="C2 | Non-homogeneous first-order linear ODE | ver. 4881"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}" alt="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}" title="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}" data-latex="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%2018%20%5C,%20%5Ccos%5Cleft(-2%20%5C,%20t%5Cright)%20e%5E%7Bt%7D%20+%203%20%5C,%20%7By'%7D" alt="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}" title="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 3 \, {y'}" data-latex="3 \, {y} = 18 \, \cos\left(-2 \, t\right) e^{t} + 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 e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \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%20e%5E%7Bt%7D%20+%203%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(-2%20%5C,%20t%5Cright)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \sin\left(-2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8534" title="C2 | Non-homogeneous first-order linear ODE | ver. 8534"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y}" alt="2 \, {y'} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y}" title="2 \, {y'} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y}" data-latex="2 \, {y'} - 18 \, e^{\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;C2.&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-%2018%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%204%20%5C,%20%7By%7D" alt="2 \, {y'} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y}" title="2 \, {y'} - 18 \, e^{\left(5 \, t\right)} = 4 \, {y}" data-latex="2 \, {y'} - 18 \, e^{\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 e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6859" title="C2 | Non-homogeneous first-order linear ODE | ver. 6859"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}" alt="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}" title="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}" data-latex="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20=%202%20%5C,%20%7By'%7D%20+%2010%20%5C,%20%7By%7D" alt="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}" title="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {y}" data-latex="-16 \, e^{\left(-t\right)} = 2 \, {y'} + 10 \, {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 e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 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%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7895" title="C2 | Non-homogeneous first-order linear ODE | ver. 7895"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -2 \, {y'} + 4 \, e^{\left(-2 \, t\right)}" alt="2 \, {y} = -2 \, {y'} + 4 \, e^{\left(-2 \, t\right)}" title="2 \, {y} = -2 \, {y'} + 4 \, e^{\left(-2 \, t\right)}" data-latex="2 \, {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;C2.&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=%20-2%20%5C,%20%7By'%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="2 \, {y} = -2 \, {y'} + 4 \, e^{\left(-2 \, t\right)}" title="2 \, {y} = -2 \, {y'} + 4 \, e^{\left(-2 \, t\right)}" data-latex="2 \, {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 e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0061" title="C2 | Non-homogeneous first-order linear ODE | ver. 0061"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}" alt="-6 \, {y} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}" title="-6 \, {y} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}" data-latex="-6 \, {y} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-3%20%5C,%20%7By'%7D%20+%2027%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="-6 \, {y} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}" title="-6 \, {y} = -3 \, {y'} + 27 \, e^{\left(5 \, t\right)}" data-latex="-6 \, {y} = -3 \, {y'} + 27 \, 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} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8546" title="C2 | Non-homogeneous first-order linear ODE | ver. 8546"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}" alt="-12 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}" title="-12 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}" data-latex="-12 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D%20-%20%7By%7D%20=%20-%7By'%7D" alt="-12 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}" title="-12 \, \cos\left(-4 \, t\right) e^{t} - {y} = -{y'}" data-latex="-12 \, \cos\left(-4 \, t\right) e^{t} - {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 e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)" alt="{y} = k e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} - 3 \, e^{t} \sin\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%20e%5E%7Bt%7D%20-%203%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} - 3 \, e^{t} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4173" title="C2 | Non-homogeneous first-order linear ODE | ver. 4173"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" alt="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" title="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" data-latex="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%203%20%5C,%20%7By'%7D%20=%20-36%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" title="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" data-latex="-6 \, {y} - 3 \, {y'} = -36 \, \cos\left(4 \, t\right) 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 e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6804" title="C2 | Non-homogeneous first-order linear ODE | ver. 6804"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}" alt="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}" title="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}" data-latex="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-16%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20+%2012%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}" title="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, t\right)}" data-latex="0 = -16 \, {y} - 4 \, {y'} + 12 \, e^{\left(-4 \, 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 e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, t 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%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, t e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0737" title="C2 | Non-homogeneous first-order linear ODE | ver. 0737"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}" alt="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}" title="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}" data-latex="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%2015%20%5C,%20%7By%7D%20=%2036%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}" title="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}" data-latex="3 \, {y'} - 15 \, {y} = 36 \, \cos\left(4 \, t\right) 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} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)" alt="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\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%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3364" title="C2 | Non-homogeneous first-order linear ODE | ver. 3364"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}" alt="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}" title="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}" data-latex="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-4%20%5C,%20%7By'%7D%20+%2016%20%5C,%20%7By%7D%20+%2064%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}" title="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, t\right)}" data-latex="0 = -4 \, {y'} + 16 \, {y} + 64 \, e^{\left(-4 \, 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 e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} - 2 \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3183" title="C2 | Non-homogeneous first-order linear ODE | ver. 3183"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)" alt="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)" title="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)" data-latex="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%204%20%5C,%20%7By%7D%20=%20-60%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)" title="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \sin\left(5 \, t\right)" data-latex="-4 \, {y'} + 4 \, {y} = -60 \, e^{t} \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 e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}" alt="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}" title="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) 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%20e%5E%7Bt%7D%20-%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20e%5E%7Bt%7D" alt="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}" title="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8706" title="C2 | Non-homogeneous first-order linear ODE | ver. 8706"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{t} = {y'} - {y}" alt="-2 \, e^{t} = {y'} - {y}" title="-2 \, e^{t} = {y'} - {y}" data-latex="-2 \, e^{t} = {y'} - {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7Bt%7D%20=%20%7By'%7D%20-%20%7By%7D" alt="-2 \, e^{t} = {y'} - {y}" title="-2 \, e^{t} = {y'} - {y}" data-latex="-2 \, e^{t} = {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 e^{t} - 2 \, t e^{t}" alt="{y} = k e^{t} - 2 \, t e^{t}" title="{y} = k e^{t} - 2 \, t e^{t}" data-latex="{y} = k e^{t} - 2 \, t 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%20e%5E%7Bt%7D%20-%202%20%5C,%20t%20e%5E%7Bt%7D" alt="{y} = k e^{t} - 2 \, t e^{t}" title="{y} = k e^{t} - 2 \, t e^{t}" data-latex="{y} = k e^{t} - 2 \, t e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8539" title="C2 | Non-homogeneous first-order linear ODE | ver. 8539"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0" alt="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0" title="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0" data-latex="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%7By'%7D%20+%204%20%5C,%20%7By%7D%20=%200" alt="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0" title="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 0" data-latex="18 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)} + 2 \, {y'} + 4 \, {y} = 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 e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5293" title="C2 | Non-homogeneous first-order linear ODE | ver. 5293"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 2 \, {y} + 18 \, e^{\left(-2 \, t\right)} = 0" alt="-2 \, {y'} + 2 \, {y} + 18 \, e^{\left(-2 \, t\right)} = 0" title="-2 \, {y'} + 2 \, {y} + 18 \, e^{\left(-2 \, t\right)} = 0" data-latex="-2 \, {y'} + 2 \, {y} + 18 \, 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;C2.&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+%202%20%5C,%20%7By%7D%20+%2018%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20=%200" alt="-2 \, {y'} + 2 \, {y} + 18 \, e^{\left(-2 \, t\right)} = 0" title="-2 \, {y'} + 2 \, {y} + 18 \, e^{\left(-2 \, t\right)} = 0" data-latex="-2 \, {y'} + 2 \, {y} + 18 \, 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 e^{t} - 3 \, e^{\left(-2 \, t\right)}" alt="{y} = k e^{t} - 3 \, e^{\left(-2 \, t\right)}" title="{y} = k e^{t} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{t} - 3 \, 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%20e%5E%7Bt%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{t} - 3 \, e^{\left(-2 \, t\right)}" title="{y} = k e^{t} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{t} - 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4469" title="C2 | Non-homogeneous first-order linear ODE | ver. 4469"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}" alt="2 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}" title="2 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}" data-latex="2 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%205%20%5C,%20%7By%7D%20+%20%7By'%7D" alt="2 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}" title="2 \, e^{\left(-5 \, t\right)} = 5 \, {y} + {y'}" data-latex="2 \, e^{\left(-5 \, t\right)} = 5 \, {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 e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 2 \, t e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9751" title="C2 | Non-homogeneous first-order linear ODE | ver. 9751"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}" alt="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}" title="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}" data-latex="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-18%20%5C,%20%5Ccos%5Cleft(-3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%208%20%5C,%20%7By%7D" alt="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}" title="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, t\right)} + 8 \, {y}" data-latex="2 \, {y'} = -18 \, \cos\left(-3 \, t\right) e^{\left(4 \, 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 e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-3%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, e^{\left(4 \, t\right)} \sin\left(-3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8796" title="C2 | Non-homogeneous first-order linear ODE | ver. 8796"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)" alt="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)" title="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)" data-latex="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%2020%20%5C,%20%7By%7D%20=%20-48%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)" title="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)" data-latex="4 \, {y'} + 20 \, {y} = -48 \, e^{\left(-5 \, t\right)} \sin\left(4 \, 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 e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2342" title="C2 | Non-homogeneous first-order linear ODE | ver. 2342"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-3 \, t\right)}" alt="-3 \, {y} = {y'} + 2 \, e^{\left(-3 \, t\right)}" title="-3 \, {y} = {y'} + 2 \, e^{\left(-3 \, t\right)}" data-latex="-3 \, {y} = {y'} + 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="-3 \, {y} = {y'} + 2 \, e^{\left(-3 \, t\right)}" title="-3 \, {y} = {y'} + 2 \, e^{\left(-3 \, t\right)}" data-latex="-3 \, {y} = {y'} + 2 \, 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} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, t 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, t e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6668" title="C2 | Non-homogeneous first-order linear ODE | ver. 6668"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}" alt="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}" title="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}" data-latex="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-3%20%5C,%20t%5Cright)%20+%205%20%5C,%20%7By%7D%20=%20-%7By'%7D" alt="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}" title="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {y} = -{y'}" data-latex="-6 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right) + 5 \, {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 e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(-3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 2 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5471" title="C2 | Non-homogeneous first-order linear ODE | ver. 5471"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}" alt="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}" title="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}" data-latex="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%7By'%7D%20+%208%20%5C,%20%7By%7D" alt="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}" title="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 8 \, {y}" data-latex="0 = 8 \, \cos\left(2 \, t\right) e^{\left(4 \, t\right)} - 2 \, {y'} + 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 e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right)" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 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="C2-0227" title="C2 | Non-homogeneous first-order linear ODE | ver. 0227"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}" alt="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}" title="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}" data-latex="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%2015%20%5C,%20%7By%7D%20=%2045%20%5C,%20%5Ccos%5Cleft(-5%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}" title="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) e^{\left(5 \, t\right)}" data-latex="3 \, {y'} - 15 \, {y} = 45 \, \cos\left(-5 \, t\right) 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} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)" alt="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, 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%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-5%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} - 3 \, e^{\left(5 \, t\right)} \sin\left(-5 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5789" title="C2 | Non-homogeneous first-order linear ODE | ver. 5789"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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) e^{\left(-t\right)} - {y} = {y'}" alt="-6 \, \cos\left(3 \, t\right) e^{\left(-t\right)} - {y} = {y'}" title="-6 \, \cos\left(3 \, t\right) e^{\left(-t\right)} - {y} = {y'}" data-latex="-6 \, \cos\left(3 \, t\right) e^{\left(-t\right)} - {y} = {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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)%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%7By%7D%20=%20%7By'%7D" alt="-6 \, \cos\left(3 \, t\right) e^{\left(-t\right)} - {y} = {y'}" title="-6 \, \cos\left(3 \, t\right) e^{\left(-t\right)} - {y} = {y'}" data-latex="-6 \, \cos\left(3 \, t\right) e^{\left(-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} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8763" title="C2 | Non-homogeneous first-order linear ODE | ver. 8763"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}" alt="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}" title="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}" data-latex="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)%20+%20%7By'%7D%20-%205%20%5C,%20%7By%7D" alt="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}" title="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right) + {y'} - 5 \, {y}" data-latex="0 = -8 \, e^{\left(5 \, t\right)} \sin\left(-4 \, 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 e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0833" title="C2 | Non-homogeneous first-order linear ODE | ver. 0833"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}" alt="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}" title="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}" data-latex="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-6%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(t%5Cright)%20+%209%20%5C,%20%7By%7D" alt="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}" title="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(t\right) + 9 \, {y}" data-latex="3 \, {y'} = -6 \, e^{\left(3 \, t\right)} \sin\left(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 e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}" alt="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) 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%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(t%5Cright)%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0355" title="C2 | Non-homogeneous first-order linear ODE | ver. 0355"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}" alt="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}" title="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}" data-latex="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-4%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-2%20%5C,%20t%5Cright)%20-%20%7By'%7D" alt="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}" title="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right) - {y'}" data-latex="5 \, {y} = -4 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, 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 e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(-2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8496" title="C2 | Non-homogeneous first-order linear ODE | ver. 8496"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)" alt="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)" title="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)" data-latex="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%204%20%5C,%20%7By'%7D%20=%2032%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)" title="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, t\right)" data-latex="12 \, {y} + 4 \, {y'} = 32 \, e^{\left(-3 \, t\right)} \sin\left(4 \, 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 e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4620" title="C2 | Non-homogeneous first-order linear ODE | ver. 4620"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 2 \, e^{\left(5 \, t\right)} = 0" alt="4 \, {y} - {y'} + 2 \, e^{\left(5 \, t\right)} = 0" title="4 \, {y} - {y'} + 2 \, e^{\left(5 \, t\right)} = 0" data-latex="4 \, {y} - {y'} + 2 \, 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;C2.&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+%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%200" alt="4 \, {y} - {y'} + 2 \, e^{\left(5 \, t\right)} = 0" title="4 \, {y} - {y'} + 2 \, e^{\left(5 \, t\right)} = 0" data-latex="4 \, {y} - {y'} + 2 \, 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} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2624" title="C2 | Non-homogeneous first-order linear ODE | ver. 2624"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}" alt="9 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}" title="9 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}" data-latex="9 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%2015%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D" alt="9 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}" title="9 \, e^{\left(-5 \, t\right)} = 15 \, {y} + 3 \, {y'}" data-latex="9 \, e^{\left(-5 \, t\right)} = 15 \, {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 e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0682" title="C2 | Non-homogeneous first-order linear ODE | ver. 0682"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}" alt="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}" title="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}" data-latex="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20=%203%20%5C,%20%7By'%7D%20+%206%20%5C,%20%7By%7D" alt="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}" title="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {y}" data-latex="-9 \, e^{\left(-2 \, t\right)} = 3 \, {y'} + 6 \, {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 e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0078" title="C2 | Non-homogeneous first-order linear ODE | ver. 0078"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}" alt="10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}" title="10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}" data-latex="10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?10%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20=%206%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}" title="10 \, {y} + 2 \, {y'} = 6 \, e^{\left(-5 \, t\right)}" data-latex="10 \, {y} + 2 \, {y'} = 6 \, 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} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3940" title="C2 | Non-homogeneous first-order linear ODE | ver. 3940"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} + 12 \, e^{\left(2 \, t\right)} = -4 \, {y'}" alt="-8 \, {y} + 12 \, e^{\left(2 \, t\right)} = -4 \, {y'}" title="-8 \, {y} + 12 \, e^{\left(2 \, t\right)} = -4 \, {y'}" data-latex="-8 \, {y} + 12 \, 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;C2.&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+%2012%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%20-4%20%5C,%20%7By'%7D" alt="-8 \, {y} + 12 \, e^{\left(2 \, t\right)} = -4 \, {y'}" title="-8 \, {y} + 12 \, e^{\left(2 \, t\right)} = -4 \, {y'}" data-latex="-8 \, {y} + 12 \, 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} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} - 3 \, t 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} - 3 \, t e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9457" title="C2 | Non-homogeneous first-order linear ODE | ver. 9457"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}" alt="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}" title="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}" data-latex="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-9%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-t%5Cright)%20-%203%20%5C,%20%7By'%7D" alt="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}" title="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-t\right) - 3 \, {y'}" data-latex="-6 \, {y} = -9 \, e^{\left(2 \, t\right)} \sin\left(-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 e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(-t%5Cright)%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} - 3 \, \cos\left(-t\right) e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3838" title="C2 | Non-homogeneous first-order linear ODE | ver. 3838"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}" alt="9 \, {y} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}" title="9 \, {y} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}" data-latex="9 \, {y} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-3%20%5C,%20%7By'%7D%20+%2030%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="9 \, {y} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}" title="9 \, {y} = -3 \, {y'} + 30 \, e^{\left(2 \, t\right)}" data-latex="9 \, {y} = -3 \, {y'} + 30 \, 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 e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5183" title="C2 | Non-homogeneous first-order linear ODE | ver. 5183"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0" alt="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0" title="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0" data-latex="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20-%202%20%5C,%20%7By'%7D%20-%204%20%5C,%20%7By%7D%20=%200" alt="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0" title="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 0" data-latex="-18 \, e^{\left(-2 \, t\right)} \sin\left(3 \, t\right) - 2 \, {y'} - 4 \, {y} = 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 e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, \cos\left(3 \, t\right) e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-1674" title="C2 | Non-homogeneous first-order linear ODE | ver. 1674"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}" alt="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}" title="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}" data-latex="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-24%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20+%2015%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D" alt="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}" title="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {y} + 3 \, {y'}" data-latex="0 = -24 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right) + 15 \, {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 e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2513" title="C2 | Non-homogeneous first-order linear ODE | ver. 2513"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right) + {y'} = 5 \, {y}" alt="6 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right) + {y'} = 5 \, {y}" title="6 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right) + {y'} = 5 \, {y}" data-latex="6 \, e^{\left(5 \, t\right)} \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;C2.&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,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-3%20%5C,%20t%5Cright)%20+%20%7By'%7D%20=%205%20%5C,%20%7By%7D" alt="6 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right) + {y'} = 5 \, {y}" title="6 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right) + {y'} = 5 \, {y}" data-latex="6 \, e^{\left(5 \, t\right)} \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 e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20%5Ccos%5Cleft(-3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} - 2 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3889" title="C2 | Non-homogeneous first-order linear ODE | ver. 3889"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}" alt="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}" title="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}" data-latex="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%7By'%7D%20=%20-12%20%5C,%20%7By%7D" alt="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}" title="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {y'} = -12 \, {y}" data-latex="-24 \, \cos\left(-4 \, t\right) e^{\left(4 \, t\right)} - 3 \, {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 e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, e^{\left(4 \, t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9444" title="C2 | Non-homogeneous first-order linear ODE | ver. 9444"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" alt="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" title="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" data-latex="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%7By%7D%20+%203%20%5C,%20%7By'%7D%20=%20-12%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" title="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" data-latex="-15 \, {y} + 3 \, {y'} = -12 \, e^{\left(5 \, t\right)} \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 e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}" title="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2085" title="C2 | Non-homogeneous first-order linear ODE | ver. 2085"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}" alt="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}" title="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}" data-latex="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%206%20%5C,%20%7By%7D%20=%202%20%5C,%20%7By'%7D" alt="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}" title="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {y} = 2 \, {y'}" data-latex="16 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 6 \, {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} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\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%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} - 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0064" title="C2 | Non-homogeneous first-order linear ODE | ver. 0064"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}" alt="-12 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}" title="-12 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}" data-latex="-12 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D%20+%20%7By%7D%20=%20%7By'%7D" alt="-12 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}" title="-12 \, \cos\left(-4 \, t\right) e^{t} + {y} = {y'}" data-latex="-12 \, \cos\left(-4 \, t\right) e^{t} + {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 e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \sin\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%20e%5E%7Bt%7D%20+%203%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6057" title="C2 | Non-homogeneous first-order linear ODE | ver. 6057"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}" alt="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}" title="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}" data-latex="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-30%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-5%20%5C,%20t%5Cright)%20-%208%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D" alt="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}" title="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {y} + 2 \, {y'}" data-latex="0 = -30 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right) - 8 \, {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} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(-5%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4056" title="C2 | Non-homogeneous first-order linear ODE | ver. 4056"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0" alt="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0" title="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0" data-latex="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7Bt%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20+%203%20%5C,%20%7By'%7D%20-%203%20%5C,%20%7By%7D%20=%200" alt="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0" title="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 0" data-latex="-24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'} - 3 \, {y} = 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 e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" alt="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" title="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) 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%20e%5E%7Bt%7D%20-%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D" alt="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" title="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0693" title="C2 | Non-homogeneous first-order linear ODE | ver. 0693"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}" alt="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}" title="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}" data-latex="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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(-t%5Cright)%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20-6%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D" alt="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}" title="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {y} + 2 \, {y'}" data-latex="6 \, \cos\left(-t\right) e^{\left(3 \, t\right)} = -6 \, {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} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)" alt="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)" title="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\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%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-t%5Cright)" alt="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)" title="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} - 3 \, e^{\left(3 \, t\right)} \sin\left(-t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3150" title="C2 | Non-homogeneous first-order linear ODE | ver. 3150"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = -8 \, {y} + 36 \, e^{t}" alt="4 \, {y'} = -8 \, {y} + 36 \, e^{t}" title="4 \, {y'} = -8 \, {y} + 36 \, e^{t}" data-latex="4 \, {y'} = -8 \, {y} + 36 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-8%20%5C,%20%7By%7D%20+%2036%20%5C,%20e%5E%7Bt%7D" alt="4 \, {y'} = -8 \, {y} + 36 \, e^{t}" title="4 \, {y'} = -8 \, {y} + 36 \, e^{t}" data-latex="4 \, {y'} = -8 \, {y} + 36 \, e^{t}"&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 e^{\left(-2 \, t\right)} + 3 \, e^{t}" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{t}" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{t}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{t}" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{t}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-5009" title="C2 | Non-homogeneous first-order linear ODE | ver. 5009"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0" alt="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0" title="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0" data-latex="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%7By%7D%20+%204%20%5C,%20%7By'%7D%20-%20108%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20=%200" alt="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0" title="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, t\right)} = 0" data-latex="20 \, {y} + 4 \, {y'} - 108 \, e^{\left(4 \, 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 e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, 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%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2691" title="C2 | Non-homogeneous first-order linear ODE | ver. 2691"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}" alt="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}" title="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}" data-latex="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%5Ccos%5Cleft(t%5Cright)%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%7By%7D%20=%20-%7By'%7D" alt="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}" title="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 3 \, {y} = -{y'}" data-latex="2 \, \cos\left(t\right) e^{\left(-3 \, t\right)} + 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 e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(t%5Cright)" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3443" title="C2 | Non-homogeneous first-order linear ODE | ver. 3443"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}" alt="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}" title="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}" data-latex="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-2%20%5C,%20t%5Cright)%20-%20%7By'%7D%20=%203%20%5C,%20%7By%7D" alt="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}" title="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, t\right) - {y'} = 3 \, {y}" data-latex="4 \, e^{\left(-3 \, t\right)} \sin\left(-2 \, 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 e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(-2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 2 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3784" title="C2 | Non-homogeneous first-order linear ODE | ver. 3784"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-2 \, t\right)}" alt="0 = -2 \, {y} - {y'} - 3 \, e^{\left(-2 \, t\right)}" title="0 = -2 \, {y} - {y'} - 3 \, e^{\left(-2 \, t\right)}" data-latex="0 = -2 \, {y} - {y'} - 3 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-2%20%5C,%20%7By%7D%20-%20%7By'%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="0 = -2 \, {y} - {y'} - 3 \, e^{\left(-2 \, t\right)}" title="0 = -2 \, {y} - {y'} - 3 \, e^{\left(-2 \, t\right)}" data-latex="0 = -2 \, {y} - {y'} - 3 \, 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 e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6279" title="C2 | Non-homogeneous first-order linear ODE | ver. 6279"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-5 \, t\right)}" alt="0 = {y'} + 4 \, {y} + 3 \, e^{\left(-5 \, t\right)}" title="0 = {y'} + 4 \, {y} + 3 \, e^{\left(-5 \, t\right)}" data-latex="0 = {y'} + 4 \, {y} + 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;C2.&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,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="0 = {y'} + 4 \, {y} + 3 \, e^{\left(-5 \, t\right)}" title="0 = {y'} + 4 \, {y} + 3 \, e^{\left(-5 \, t\right)}" data-latex="0 = {y'} + 4 \, {y} + 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} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2485" title="C2 | Non-homogeneous first-order linear ODE | ver. 2485"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}" alt="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}" title="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}" data-latex="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%2012%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7By'%7D" alt="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}" title="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {y} = -4 \, {y'}" data-latex="24 \, \cos\left(2 \, t\right) e^{\left(-3 \, t\right)} + 12 \, {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 e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)" alt="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-3 \, t\right)} \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6545" title="C2 | Non-homogeneous first-order linear ODE | ver. 6545"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 6 \, {y} + 8 \, e^{\left(-t\right)}" alt="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-t\right)}" title="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-t\right)}" data-latex="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%206%20%5C,%20%7By%7D%20+%208%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-t\right)}" title="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-t\right)}" data-latex="0 = 2 \, {y'} + 6 \, {y} + 8 \, e^{\left(-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 e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6138" title="C2 | Non-homogeneous first-order linear ODE | ver. 6138"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}" alt="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}" title="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}" data-latex="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%2016%20%5C,%20%5Ccos%5Cleft(-2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%204%20%5C,%20%7By%7D" alt="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}" title="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-t\right)} - 4 \, {y}" data-latex="4 \, {y'} = 16 \, \cos\left(-2 \, t\right) e^{\left(-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 e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Csin%5Cleft(-2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7114" title="C2 | Non-homogeneous first-order linear ODE | ver. 7114"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}" alt="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}" title="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}" data-latex="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%204%20%5C,%20%7By%7D%20=%20-12%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}" title="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}" data-latex="2 \, {y'} - 4 \, {y} = -12 \, \cos\left(3 \, t\right) 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 e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)" alt="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)" title="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)" data-latex="{y} = k e^{\left(2 \, t\right)} - 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9346" title="C2 | Non-homogeneous first-order linear ODE | ver. 9346"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}" alt="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}" title="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}" data-latex="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-36%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%206%20%5C,%20%7By%7D" alt="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}" title="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {y}" data-latex="-3 \, {y'} = -36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} + 6 \, {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 e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6108" title="C2 | Non-homogeneous first-order linear ODE | ver. 6108"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0" alt="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0" title="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0" data-latex="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%5Ccos%5Cleft(-2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%7By'%7D%20-%205%20%5C,%20%7By%7D%20=%200" alt="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0" title="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 0" data-latex="4 \, \cos\left(-2 \, t\right) e^{\left(-5 \, t\right)} - {y'} - 5 \, {y} = 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 e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, 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%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9765" title="C2 | Non-homogeneous first-order linear ODE | ver. 9765"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} + 2 \, {y'} + 28 \, e^{\left(5 \, t\right)} = 0" alt="4 \, {y} + 2 \, {y'} + 28 \, e^{\left(5 \, t\right)} = 0" title="4 \, {y} + 2 \, {y'} + 28 \, e^{\left(5 \, t\right)} = 0" data-latex="4 \, {y} + 2 \, {y'} + 28 \, 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;C2.&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+%202%20%5C,%20%7By'%7D%20+%2028%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%200" alt="4 \, {y} + 2 \, {y'} + 28 \, e^{\left(5 \, t\right)} = 0" title="4 \, {y} + 2 \, {y'} + 28 \, e^{\left(5 \, t\right)} = 0" data-latex="4 \, {y} + 2 \, {y'} + 28 \, 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} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8877" title="C2 | Non-homogeneous first-order linear ODE | ver. 8877"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}" alt="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}" title="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}" data-latex="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20=%204%20%5C,%20%7By%7D%20+%20%7By'%7D" alt="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}" title="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, t\right) = 4 \, {y} + {y'}" data-latex="-12 \, e^{\left(-4 \, t\right)} \sin\left(4 \, 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} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) 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%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} + 3 \, \cos\left(4 \, t\right) e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6485" title="C2 | Non-homogeneous first-order linear ODE | ver. 6485"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}" alt="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}" title="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}" data-latex="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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(-t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%208%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7By'%7D" alt="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}" title="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {y} = -4 \, {y'}" data-latex="-8 \, \cos\left(-t\right) e^{\left(-2 \, t\right)} + 8 \, {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 e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, e^{\left(-2 \, t\right)} \sin\left(-t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3818" title="C2 | Non-homogeneous first-order linear ODE | ver. 3818"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = 2 \, {y} - 6 \, e^{\left(-t\right)}" alt="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-t\right)}" title="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-t\right)}" data-latex="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%202%20%5C,%20%7By%7D%20-%206%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-t\right)}" title="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-t\right)}" data-latex="-2 \, {y'} = 2 \, {y} - 6 \, e^{\left(-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 e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}" alt="{y} = k e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}" title="{y} = k e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-t\right)} + 3 \, t 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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}" title="{y} = k e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}" data-latex="{y} = k e^{\left(-t\right)} + 3 \, t e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2855" title="C2 | Non-homogeneous first-order linear ODE | ver. 2855"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}" alt="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}" title="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}" data-latex="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-12%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%2010%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D" alt="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}" title="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {y} + 2 \, {y'}" data-latex="0 = -12 \, \cos\left(2 \, t\right) e^{\left(5 \, t\right)} - 10 \, {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} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" alt="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, 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%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2878" title="C2 | Non-homogeneous first-order linear ODE | ver. 2878"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}" alt="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}" title="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}" data-latex="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%2024%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20+%206%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D" alt="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}" title="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {y} + 3 \, {y'}" data-latex="0 = 24 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right) + 6 \, {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 e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2203" title="C2 | Non-homogeneous first-order linear ODE | ver. 2203"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}" alt="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}" title="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}" data-latex="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20=%202%20%5C,%20%7By%7D%20-%20%7By'%7D" alt="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}" title="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 2 \, {y} - {y'}" data-latex="8 \, e^{\left(2 \, t\right)} \sin\left(4 \, t\right) = 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 e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(4 \, t\right) e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-4594" title="C2 | Non-homogeneous first-order linear ODE | ver. 4594"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0" alt="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0" title="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0" data-latex="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%203%20%5C,%20%7By'%7D%20-%2036%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20=%200" alt="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0" title="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, t\right)} = 0" data-latex="6 \, {y} - 3 \, {y'} - 36 \, e^{\left(-4 \, 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 e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8021" title="C2 | Non-homogeneous first-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>C2.</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 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}" alt="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}" title="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}" data-latex="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20%7By%7D" alt="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}" title="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, t\right)} + 9 \, {y}" data-latex="3 \, {y'} = -24 \, \cos\left(-4 \, t\right) e^{\left(3 \, 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 e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\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%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(3 \, t\right)} + 2 \, e^{\left(3 \, t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6542" title="C2 | Non-homogeneous first-order linear ODE | ver. 6542"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" alt="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" title="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" data-latex="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?10%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20=%2012%20%5C,%20%5Ccos%5Cleft(-3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" title="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) e^{\left(-5 \, t\right)}" data-latex="10 \, {y} + 2 \, {y'} = 12 \, \cos\left(-3 \, t\right) 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} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \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%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-3%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)" title="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)" data-latex="{y} = k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-3 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7160" title="C2 | Non-homogeneous first-order linear ODE | ver. 7160"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}" alt="-3 \, {y} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}" title="-3 \, {y} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}" data-latex="-3 \, {y} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-3%20%5C,%20%7By'%7D%20-%2036%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="-3 \, {y} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}" title="-3 \, {y} = -3 \, {y'} - 36 \, e^{\left(-5 \, t\right)}" data-latex="-3 \, {y} = -3 \, {y'} - 36 \, 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} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}" alt="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7Bt%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{t} + 2 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3229" title="C2 | Non-homogeneous first-order linear ODE | ver. 3229"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}" alt="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}" title="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}" data-latex="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-t%5Cright)%7D%20=%203%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By%7D" alt="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}" title="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {y'} + 3 \, {y}" data-latex="36 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} = 3 \, {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 e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-2299" title="C2 | Non-homogeneous first-order linear ODE | ver. 2299"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}" alt="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}" title="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}" data-latex="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%202%20%5C,%20%7By'%7D%20-%206%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}" title="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, t\right)}" data-latex="0 = -6 \, {y} - 2 \, {y'} - 6 \, e^{\left(-4 \, 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 e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 3 \, 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%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7202" title="C2 | Non-homogeneous first-order linear ODE | ver. 7202"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}" alt="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}" title="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}" data-latex="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%204%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D" alt="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}" title="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 8 \, {y}" data-latex="8 \, e^{\left(2 \, t\right)} = 4 \, {y'} - 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 e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, t 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, t e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6285" title="C2 | Non-homogeneous first-order linear ODE | ver. 6285"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -3 \, {y'} + 6 \, e^{\left(-2 \, t\right)}" alt="6 \, {y} = -3 \, {y'} + 6 \, e^{\left(-2 \, t\right)}" title="6 \, {y} = -3 \, {y'} + 6 \, e^{\left(-2 \, t\right)}" data-latex="6 \, {y} = -3 \, {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;C2.&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=%20-3%20%5C,%20%7By'%7D%20+%206%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="6 \, {y} = -3 \, {y'} + 6 \, e^{\left(-2 \, t\right)}" title="6 \, {y} = -3 \, {y'} + 6 \, e^{\left(-2 \, t\right)}" data-latex="6 \, {y} = -3 \, {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 e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, t 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, t e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-1789" title="C2 | Non-homogeneous first-order linear ODE | ver. 1789"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 6 \, {y} = -12 \, e^{\left(-5 \, t\right)}" alt="-2 \, {y'} - 6 \, {y} = -12 \, e^{\left(-5 \, t\right)}" title="-2 \, {y'} - 6 \, {y} = -12 \, e^{\left(-5 \, t\right)}" data-latex="-2 \, {y'} - 6 \, {y} = -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;C2.&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-%206%20%5C,%20%7By%7D%20=%20-12%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="-2 \, {y'} - 6 \, {y} = -12 \, e^{\left(-5 \, t\right)}" title="-2 \, {y'} - 6 \, {y} = -12 \, e^{\left(-5 \, t\right)}" data-latex="-2 \, {y'} - 6 \, {y} = -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} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3578" title="C2 | Non-homogeneous first-order linear ODE | ver. 3578"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}" alt="3 \, {y} = -24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}" title="3 \, {y} = -24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}" data-latex="3 \, {y} = -24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7Bt%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)%20+%203%20%5C,%20%7By'%7D" alt="3 \, {y} = -24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}" title="3 \, {y} = -24 \, e^{t} \sin\left(4 \, t\right) + 3 \, {y'}" data-latex="3 \, {y} = -24 \, e^{t} \sin\left(4 \, 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 e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" alt="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" title="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) 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%20e%5E%7Bt%7D%20-%202%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D" alt="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" title="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}" data-latex="{y} = k e^{t} - 2 \, \cos\left(4 \, t\right) e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8045" title="C2 | Non-homogeneous first-order linear ODE | ver. 8045"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0" alt="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0" title="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0" data-latex="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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?32%20%5C,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%2020%20%5C,%20%7By%7D%20+%204%20%5C,%20%7By'%7D%20=%200" alt="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0" title="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 0" data-latex="32 \, \cos\left(-4 \, t\right) e^{\left(5 \, t\right)} - 20 \, {y} + 4 \, {y'} = 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 e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\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%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(5 \, t\right)} + 2 \, e^{\left(5 \, t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9898" title="C2 | Non-homogeneous first-order linear ODE | ver. 9898"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0" alt="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0" title="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0" data-latex="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%7By%7D%20-%204%20%5C,%20%7By'%7D%20-%208%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20=%200" alt="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0" title="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, t\right)} = 0" data-latex="-16 \, {y} - 4 \, {y'} - 8 \, e^{\left(-4 \, 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 e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} - 2 \, t 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%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7820" title="C2 | Non-homogeneous first-order linear ODE | ver. 7820"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 2 \, e^{\left(-2 \, t\right)} = 0" alt="-2 \, {y} - {y'} - 2 \, e^{\left(-2 \, t\right)} = 0" title="-2 \, {y} - {y'} - 2 \, e^{\left(-2 \, t\right)} = 0" data-latex="-2 \, {y} - {y'} - 2 \, 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;C2.&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-%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20=%200" alt="-2 \, {y} - {y'} - 2 \, e^{\left(-2 \, t\right)} = 0" title="-2 \, {y} - {y'} - 2 \, e^{\left(-2 \, t\right)} = 0" data-latex="-2 \, {y} - {y'} - 2 \, 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 e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, t 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 2 \, t e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7606" title="C2 | Non-homogeneous first-order linear ODE | ver. 7606"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}" alt="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}" title="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}" data-latex="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20=%20-6%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By'%7D" alt="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}" title="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {y} - 3 \, {y'}" data-latex="-36 \, \cos\left(4 \, t\right) e^{\left(-2 \, t\right)} = -6 \, {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 e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} + 3 \, e^{\left(-2 \, t\right)} \sin\left(4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6836" title="C2 | Non-homogeneous first-order linear ODE | ver. 6836"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 2 \, {y} = -4 \, e^{t}" alt="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}" title="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}" data-latex="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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+%202%20%5C,%20%7By%7D%20=%20-4%20%5C,%20e%5E%7Bt%7D" alt="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}" title="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}" data-latex="-2 \, {y'} + 2 \, {y} = -4 \, e^{t}"&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 e^{t} + 2 \, t e^{t}" alt="{y} = k e^{t} + 2 \, t e^{t}" title="{y} = k e^{t} + 2 \, t e^{t}" data-latex="{y} = k e^{t} + 2 \, t 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%20e%5E%7Bt%7D%20+%202%20%5C,%20t%20e%5E%7Bt%7D" alt="{y} = k e^{t} + 2 \, t e^{t}" title="{y} = k e^{t} + 2 \, t e^{t}" data-latex="{y} = k e^{t} + 2 \, t e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6283" title="C2 | Non-homogeneous first-order linear ODE | ver. 6283"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}" alt="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}" title="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}" data-latex="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-8%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%7By'%7D" alt="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}" title="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, t\right)} + 2 \, {y'}" data-latex="-8 \, {y} = -8 \, \cos\left(2 \, t\right) e^{\left(-4 \, 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 e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)" alt="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(-4 \, t\right)} + 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%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)" title="{y} = k e^{\left(-4 \, t\right)} + 2 \, e^{\left(-4 \, t\right)} \sin\left(2 \, t\right)" data-latex="{y} = k e^{\left(-4 \, t\right)} + 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="C2-3322" title="C2 | Non-homogeneous first-order linear ODE | ver. 3322"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}" alt="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}" title="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}" data-latex="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%7By%7D%20+%204%20%5C,%20%7By'%7D%20=%20-8%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}" title="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, t\right)}" data-latex="16 \, {y} + 4 \, {y'} = -8 \, e^{\left(-4 \, 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 e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" alt="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} - 2 \, t 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%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20t%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" title="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}" data-latex="{y} = k e^{\left(-4 \, t\right)} - 2 \, t e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6420" title="C2 | Non-homogeneous first-order linear ODE | ver. 6420"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}" alt="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}" title="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}" data-latex="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)%20=%202%20%5C,%20%7By%7D%20-%20%7By'%7D" alt="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}" title="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 2 \, {y} - {y'}" data-latex="-8 \, e^{\left(2 \, t\right)} \sin\left(-4 \, t\right) = 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 e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(-4 \, t\right) e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-1361" title="C2 | Non-homogeneous first-order linear ODE | ver. 1361"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}" alt="-8 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}" title="-8 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}" data-latex="-8 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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(-4%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20%7By'%7D%20=%20-%7By%7D" alt="-8 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}" title="-8 \, \cos\left(-4 \, t\right) e^{\left(-t\right)} + {y'} = -{y}" data-latex="-8 \, \cos\left(-4 \, t\right) e^{\left(-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} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\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%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Csin%5Cleft(-4%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" title="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)" data-latex="{y} = k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6872" title="C2 | Non-homogeneous first-order linear ODE | ver. 6872"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}" alt="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}" title="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}" data-latex="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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%7By%7D%20+%204%20%5C,%20%7By'%7D%20=%208%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}" title="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, t\right)}" data-latex="-16 \, {y} + 4 \, {y'} = 8 \, e^{\left(4 \, 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 e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, t 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 2 \, t e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-7000" title="C2 | Non-homogeneous first-order linear ODE | ver. 7000"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}" alt="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}" title="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}" data-latex="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-45%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20-%2015%20%5C,%20%7By%7D" alt="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}" title="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {y}" data-latex="3 \, {y'} = -45 \, e^{\left(-5 \, t\right)} \sin\left(5 \, t\right) - 15 \, {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 e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, \cos\left(5 \, t\right) e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3499" title="C2 | Non-homogeneous first-order linear ODE | ver. 3499"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} + 15 \, {y} = 9 \, e^{\left(-5 \, t\right)}" alt="3 \, {y'} + 15 \, {y} = 9 \, e^{\left(-5 \, t\right)}" title="3 \, {y'} + 15 \, {y} = 9 \, e^{\left(-5 \, t\right)}" data-latex="3 \, {y'} + 15 \, {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;C2.&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+%2015%20%5C,%20%7By%7D%20=%209%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="3 \, {y'} + 15 \, {y} = 9 \, e^{\left(-5 \, t\right)}" title="3 \, {y'} + 15 \, {y} = 9 \, e^{\left(-5 \, t\right)}" data-latex="3 \, {y'} + 15 \, {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} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(-5 \, t\right)} + 3 \, t e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3252" title="C2 | Non-homogeneous first-order linear ODE | ver. 3252"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0" alt="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0" title="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0" data-latex="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-3%20%5C,%20t%5Cright)%20-%208%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20=%200" alt="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0" title="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 0" data-latex="-36 \, e^{\left(-2 \, t\right)} \sin\left(-3 \, t\right) - 8 \, {y} - 4 \, {y'} = 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 e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%5Ccos%5Cleft(-3%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, \cos\left(-3 \, t\right) e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3760" title="C2 | Non-homogeneous first-order linear ODE | ver. 3760"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} - 3 \, {y'} - 63 \, e^{\left(-5 \, t\right)} = 0" alt="6 \, {y} - 3 \, {y'} - 63 \, e^{\left(-5 \, t\right)} = 0" title="6 \, {y} - 3 \, {y'} - 63 \, e^{\left(-5 \, t\right)} = 0" data-latex="6 \, {y} - 3 \, {y'} - 63 \, 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;C2.&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-%203%20%5C,%20%7By'%7D%20-%2063%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%200" alt="6 \, {y} - 3 \, {y'} - 63 \, e^{\left(-5 \, t\right)} = 0" title="6 \, {y} - 3 \, {y'} - 63 \, e^{\left(-5 \, t\right)} = 0" data-latex="6 \, {y} - 3 \, {y'} - 63 \, 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} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-8242" title="C2 | Non-homogeneous first-order linear ODE | ver. 8242"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}" alt="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}" title="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}" data-latex="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%2024%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20e%5E%7Bt%7D%20-%202%20%5C,%20%7By'%7D" alt="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}" title="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 2 \, {y'}" data-latex="-2 \, {y} = 24 \, \cos\left(4 \, t\right) e^{t} - 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 e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \sin\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%20e%5E%7Bt%7D%20+%203%20%5C,%20e%5E%7Bt%7D%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = k e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)" title="{y} = k e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)" data-latex="{y} = k e^{t} + 3 \, e^{t} \sin\left(4 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-6520" title="C2 | Non-homogeneous first-order linear ODE | ver. 6520"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)" alt="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)" title="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)" data-latex="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%203%20%5C,%20%7By'%7D%20=%2012%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)" title="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \sin\left(2 \, t\right)" data-latex="-6 \, {y} - 3 \, {y'} = 12 \, e^{\left(-2 \, t\right)} \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 e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-3074" title="C2 | Non-homogeneous first-order linear ODE | ver. 3074"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}" alt="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}" title="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}" data-latex="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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-%204%20%5C,%20%7By%7D%20=%20-30%20%5C,%20%5Ccos%5Cleft(-5%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}" title="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}" data-latex="-2 \, {y'} - 4 \, {y} = -30 \, \cos\left(-5 \, t\right) 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 e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(-5 \, t\right)" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(-5 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(-5 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(-5%20%5C,%20t%5Cright)" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(-5 \, t\right)" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)} \sin\left(-5 \, t\right)" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, 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="C2-7656" title="C2 | Non-homogeneous first-order linear ODE | ver. 7656"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}" alt="0 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}" title="0 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}" data-latex="0 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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=%20-4%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20-%2012%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="0 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}" title="0 = -4 \, {y'} - 8 \, {y} - 12 \, e^{\left(-2 \, t\right)}" data-latex="0 = -4 \, {y'} - 8 \, {y} - 12 \, 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 e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t 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%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)} - 3 \, t e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-9757" title="C2 | Non-homogeneous first-order linear ODE | ver. 9757"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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 \, e^{\left(4 \, t\right)} = 3 \, {y'}" alt="12 \, {y} + 9 \, e^{\left(4 \, t\right)} = 3 \, {y'}" title="12 \, {y} + 9 \, e^{\left(4 \, t\right)} = 3 \, {y'}" data-latex="12 \, {y} + 9 \, e^{\left(4 \, t\right)} = 3 \, {y'}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C2.&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,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20=%203%20%5C,%20%7By'%7D" alt="12 \, {y} + 9 \, e^{\left(4 \, t\right)} = 3 \, {y'}" title="12 \, {y} + 9 \, e^{\left(4 \, t\right)} = 3 \, {y'}" data-latex="12 \, {y} + 9 \, e^{\left(4 \, 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 e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, t 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%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20t%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}" title="{y} = k e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}" data-latex="{y} = k e^{\left(4 \, t\right)} + 3 \, t e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C2-0332" title="C2 | Non-homogeneous first-order linear ODE | ver. 0332"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C2.</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?10 \, e^{\left(2 \, t\right)} \sin\left(5 \, t\right) + {y'} = 2 \, {y}" alt="10 \, e^{\left(2 \, t\right)} \sin\left(5 \, t\right) + {y'} = 2 \, {y}" title="10 \, e^{\left(2 \, t\right)} \sin\left(5 \, t\right) + {y'} = 2 \, {y}" data-latex="10 \, e^{\left(2 \, t\right)} \sin\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;C2.&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?10%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)%20+%20%7By'%7D%20=%202%20%5C,%20%7By%7D" alt="10 \, e^{\left(2 \, t\right)} \sin\left(5 \, t\right) + {y'} = 2 \, {y}" title="10 \, e^{\left(2 \, t\right)} \sin\left(5 \, t\right) + {y'} = 2 \, {y}" data-latex="10 \, e^{\left(2 \, t\right)} \sin\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} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) 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%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)} + 2 \, \cos\left(5 \, t\right) e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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

Resources

Company
