CoCalc -- C1.qti

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<?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="C1">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- C1</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="C1-8782" title="C1 | Homogeneous first-order linear IVP | ver. 8782"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" alt="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%205%20%5C,%20%7By'%7D%20-%2010%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2016" alt="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="0 = 5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-0439" title="C1 | Homogeneous first-order linear IVP | ver. 0439"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2012%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="4 \, {y'} + 12 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9148" title="C1 | Homogeneous first-order linear IVP | ver. 9148"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" alt="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" title="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" data-latex="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-4%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2027" alt="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" title="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" data-latex="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(2 \, t\right)}" alt="{y} = 3 \, e^{\left(2 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(2 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4066" title="C1 | Homogeneous first-order linear IVP | ver. 4066"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" alt="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%205%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2016" alt="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5397" title="C1 | Homogeneous first-order linear IVP | ver. 5397"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" alt="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20-%2012%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-24" alt="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="0 = 4 \, {y'} - 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(3 \, t\right)}" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7062" title="C1 | Homogeneous first-order linear IVP | ver. 7062"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" alt="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" title="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" data-latex="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B3%7D%7B8%7D" alt="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" title="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" data-latex="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-3 \, t\right)}" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8522" title="C1 | Homogeneous first-order linear IVP | ver. 8522"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" alt="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%208%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B4%7D" alt="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-4 \, {y'} = 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-2 \, t\right)}" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8364" title="C1 | Homogeneous first-order linear IVP | ver. 8364"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" alt="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%203%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-16" alt="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(3 \, t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4366" title="C1 | Homogeneous first-order linear IVP | ver. 4366"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" alt="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%202%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-16" alt="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(3 \, t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-0953" title="C1 | Homogeneous first-order linear IVP | ver. 0953"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%209%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1437" title="C1 | Homogeneous first-order linear IVP | ver. 1437"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-18" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(2 \, t\right)}" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9120" title="C1 | Homogeneous first-order linear IVP | ver. 9120"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" alt="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B27%7D" alt="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-3 \, t\right)}" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7169" title="C1 | Homogeneous first-order linear IVP | ver. 7169"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" alt="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2010%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-12" alt="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8238" title="C1 | Homogeneous first-order linear IVP | ver. 8238"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" alt="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%209%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2032" alt="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="3 \, {y'} = 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(3 \, t\right)}" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9176" title="C1 | Homogeneous first-order linear IVP | ver. 9176"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" alt="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%208%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B9%7D" alt="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="-4 \, {y'} - 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-2 \, t\right)}" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4591" title="C1 | Homogeneous first-order linear IVP | ver. 4591"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" alt="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" title="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" data-latex="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-15%20%5C,%20%7By%7D%20-%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B4%7D%7B27%7D" alt="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" title="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" data-latex="0 = -15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-3 \, t\right)}" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-3175" title="C1 | Homogeneous first-order linear IVP | ver. 3175"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" alt="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B2%7D%7B9%7D" alt="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="-8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-2 \, t\right)}" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6696" title="C1 | Homogeneous first-order linear IVP | ver. 6696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}" alt="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}" title="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}" data-latex="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B4%7D%7B9%7D" alt="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}" title="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}" data-latex="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-2 \, t\right)}" alt="{y} = -4 \, e^{\left(-2 \, t\right)}" title="{y} = -4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-2 \, t\right)}" title="{y} = -4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5607" title="C1 | Homogeneous first-order linear IVP | ver. 5607"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-12" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-3198" title="C1 | Homogeneous first-order linear IVP | ver. 3198"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" alt="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" title="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" data-latex="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%205%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-81" alt="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" title="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" data-latex="15 \, {y} - 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(3 \, t\right)}" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6517" title="C1 | Homogeneous first-order linear IVP | ver. 6517"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" alt="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" title="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" data-latex="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%209%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-81" alt="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" title="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81" data-latex="-3 \, {y'} + 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -81"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(3 \, t\right)}" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4619" title="C1 | Homogeneous first-order linear IVP | 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>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" alt="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B9%7D" alt="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="2 \, {y'} + 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-3 \, t\right)}" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6886" title="C1 | Homogeneous first-order linear IVP | ver. 6886"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-12%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="0 = -12 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5547" title="C1 | Homogeneous first-order linear IVP | ver. 5547"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" alt="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By'%7D%20=%20-6%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-16" alt="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="-3 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(2 \, t\right)}" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2469" title="C1 | Homogeneous first-order linear IVP | ver. 2469"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" alt="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-5%20%5C,%20%7By'%7D%20-%2010%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B9%7D" alt="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="0 = -5 \, {y'} - 10 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-2 \, t\right)}" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1443" title="C1 | Homogeneous first-order linear IVP | ver. 1443"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" alt="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-9%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B27%7D" alt="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-3 \, t\right)}" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-3771" title="C1 | Homogeneous first-order linear IVP | ver. 3771"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" alt="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" title="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" data-latex="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%203%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-8" alt="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" title="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" data-latex="0 = -6 \, {y} + 3 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(2 \, t\right)}" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9775" title="C1 | Homogeneous first-order linear IVP | ver. 9775"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" alt="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" title="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" data-latex="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B2%7D%7B9%7D" alt="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" title="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" data-latex="4 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-2 \, t\right)}" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1660" title="C1 | Homogeneous first-order linear IVP | ver. 1660"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" alt="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-5%20%5C,%20%7By'%7D%20+%2015%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2032" alt="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="0 = -5 \, {y'} + 15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(3 \, t\right)}" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2525" title="C1 | Homogeneous first-order linear IVP | ver. 2525"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" alt="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" title="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" data-latex="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-4%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2081" alt="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" title="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" data-latex="-12 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(3 \, t\right)}" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9498" title="C1 | Homogeneous first-order linear IVP | ver. 9498"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2012%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B4%7D%7B27%7D" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{4}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-3 \, t\right)}" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2858" title="C1 | Homogeneous first-order linear IVP | ver. 2858"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-15%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-32" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8234" title="C1 | Homogeneous first-order linear IVP | ver. 8234"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" alt="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2015%20%5C,%20%7By%7D%20-%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2032" alt="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="0 = 15 \, {y} - 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(3 \, t\right)}" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4862" title="C1 | Homogeneous first-order linear IVP | ver. 4862"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54" alt="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54" title="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54" data-latex="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%203%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2054" alt="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54" title="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54" data-latex="9 \, {y} = 3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 54"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(3 \, t\right)}" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8997" title="C1 | Homogeneous first-order linear IVP | ver. 8997"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" alt="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%202%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2016" alt="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-6 \, {y} + 2 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(3 \, t\right)}" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2850" title="C1 | Homogeneous first-order linear IVP | ver. 2850"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" alt="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" title="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" data-latex="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By'%7D%20+%209%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2024" alt="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" title="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" data-latex="0 = -3 \, {y'} + 9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(3 \, t\right)}" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-0827" title="C1 | Homogeneous first-order linear IVP | ver. 0827"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" alt="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-10%20%5C,%20%7By%7D%20+%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-16" alt="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(2 \, t\right)}" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8736" title="C1 | Homogeneous first-order linear IVP | ver. 8736"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}" alt="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}" title="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}" data-latex="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B2%7D%7B27%7D" alt="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}" title="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}" data-latex="9 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-3 \, t\right)}" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9432" title="C1 | Homogeneous first-order linear IVP | ver. 9432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" alt="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2016" alt="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8471" title="C1 | Homogeneous first-order linear IVP | ver. 8471"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2010%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4663" title="C1 | Homogeneous first-order linear IVP | ver. 4663"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-9%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="3 \, {y'} = -9 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2894" title="C1 | Homogeneous first-order linear IVP | ver. 2894"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" alt="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-16" alt="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" title="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16" data-latex="-2 \, {y'} + 4 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(2 \, t\right)}" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1902" title="C1 | Homogeneous first-order linear IVP | ver. 1902"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54" alt="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54" title="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54" data-latex="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-54" alt="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54" title="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54" data-latex="2 \, {y'} - 6 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -54"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(3 \, t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1671" title="C1 | Homogeneous first-order linear IVP | ver. 1671"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" alt="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B4%7D" alt="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-2 \, t\right)}" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6380" title="C1 | Homogeneous first-order linear IVP | ver. 6380"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" alt="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B2%7D" alt="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="6 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-2 \, t\right)}" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6311" title="C1 | Homogeneous first-order linear IVP | ver. 6311"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}" alt="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}" title="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}" data-latex="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B3%7D" alt="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}" title="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}" data-latex="10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{1}{3}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-2 \, t\right)}" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6041" title="C1 | Homogeneous first-order linear IVP | ver. 6041"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" alt="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2032" alt="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(3 \, t\right)}" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7955" title="C1 | Homogeneous first-order linear IVP | ver. 7955"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4527" title="C1 | Homogeneous first-order linear IVP | ver. 4527"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" alt="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2015%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-24" alt="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="-5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(3 \, t\right)}" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5243" title="C1 | Homogeneous first-order linear IVP | ver. 5243"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" alt="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B4%7D" alt="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-10 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-2 \, t\right)}" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7948" title="C1 | Homogeneous first-order linear IVP | ver. 7948"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" alt="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B9%7D" alt="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="0 = 2 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-3 \, t\right)}" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7448" title="C1 | Homogeneous first-order linear IVP | ver. 7448"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-15%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-24" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(3 \, t\right)}" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(3 \, t\right)}" title="{y} = -3 \, e^{\left(3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4047" title="C1 | Homogeneous first-order linear IVP | ver. 4047"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-15%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2081" alt="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" title="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81" data-latex="-5 \, {y'} = -15 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 81"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(3 \, t\right)}" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7989" title="C1 | Homogeneous first-order linear IVP | ver. 7989"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" alt="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" title="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" data-latex="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By'%7D%20+%206%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2027" alt="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" title="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27" data-latex="0 = -3 \, {y'} + 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 27"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(2 \, t\right)}" alt="{y} = 3 \, e^{\left(2 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(2 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2785" title="C1 | Homogeneous first-order linear IVP | ver. 2785"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" alt="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%204%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B9%7D" alt="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" title="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}" data-latex="0 = -2 \, {y'} - 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-2 \, t\right)}" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-2 \, t\right)}" title="{y} = 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4464" title="C1 | Homogeneous first-order linear IVP | ver. 4464"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-8" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(2 \, t\right)}" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(2 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5260" title="C1 | Homogeneous first-order linear IVP | ver. 5260"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" alt="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-108" alt="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="-15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5141" title="C1 | Homogeneous first-order linear IVP | ver. 5141"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}" alt="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}" title="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}" data-latex="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B3%7D" alt="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}" title="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}" data-latex="6 \, {y} = -3 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{3}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-2 \, t\right)}" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9333" title="C1 | Homogeneous first-order linear IVP | ver. 9333"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" alt="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%208%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B2%7D%7B9%7D" alt="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="4 \, {y'} + 8 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-2 \, t\right)}" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4680" title="C1 | Homogeneous first-order linear IVP | ver. 4680"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%208" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(2 \, t\right)}" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5788" title="C1 | Homogeneous first-order linear IVP | ver. 5788"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" alt="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2010%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%208" alt="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(2 \, t\right)}" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5670" title="C1 | Homogeneous first-order linear IVP | ver. 5670"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%205%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="-10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4251" title="C1 | Homogeneous first-order linear IVP | ver. 4251"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" alt="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B2%7D%7B27%7D" alt="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2023" title="C1 | Homogeneous first-order linear IVP | ver. 2023"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" alt="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" title="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" data-latex="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-8%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B3%7D%7B4%7D" alt="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" title="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" data-latex="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-2 \, t\right)}" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2446" title="C1 | Homogeneous first-order linear IVP | ver. 2446"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" alt="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-6%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-108" alt="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="-2 \, {y'} = -6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1922" title="C1 | Homogeneous first-order linear IVP | ver. 1922"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" alt="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-4%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-12" alt="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2360" title="C1 | Homogeneous first-order linear IVP | ver. 2360"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" alt="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%208" alt="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" title="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8" data-latex="-10 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 8"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(2 \, t\right)}" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6555" title="C1 | Homogeneous first-order linear IVP | ver. 6555"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" alt="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%206%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B2%7D%7B27%7D" alt="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="0 = -2 \, {y'} - 6 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2248" title="C1 | Homogeneous first-order linear IVP | ver. 2248"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-8%20%5C,%20%7By%7D%20+%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="0 = -8 \, {y} + 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6825" title="C1 | Homogeneous first-order linear IVP | ver. 6825"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" alt="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2010%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B2%7D" alt="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="-5 \, {y'} = 10 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-2 \, t\right)}" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5759" title="C1 | Homogeneous first-order linear IVP | ver. 5759"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-36" alt="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" title="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" data-latex="0 = 4 \, {y'} - 8 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(2 \, t\right)}" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7793" title="C1 | Homogeneous first-order linear IVP | ver. 7793"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" alt="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" title="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" data-latex="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%206%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B3%7D%7B4%7D" alt="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" title="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}" data-latex="-3 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-2 \, t\right)}" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9461" title="C1 | Homogeneous first-order linear IVP | ver. 9461"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" alt="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-5%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B4%7D%7B27%7D" alt="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" title="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}" data-latex="15 \, {y} = -5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{4}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(-3 \, t\right)}" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(-3 \, t\right)}" title="{y} = 4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5557" title="C1 | Homogeneous first-order linear IVP | ver. 5557"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="6 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4613" title="C1 | Homogeneous first-order linear IVP | ver. 4613"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" alt="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" title="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" data-latex="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-36" alt="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" title="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36" data-latex="8 \, {y} = 4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(2 \, t\right)}" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(2 \, t\right)}" title="{y} = -4 \, e^{\left(2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2150" title="C1 | Homogeneous first-order linear IVP | ver. 2150"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" alt="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B4%7D" alt="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" title="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}" data-latex="-4 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-2 \, t\right)}" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-2 \, t\right)}" title="{y} = -3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9907" title="C1 | Homogeneous first-order linear IVP | ver. 9907"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2012%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B4%7D" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4251" title="C1 | Homogeneous first-order linear IVP | ver. 4251"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" alt="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B2%7D%7B27%7D" alt="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" title="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}" data-latex="-12 \, {y} - 4 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{27}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6094" title="C1 | Homogeneous first-order linear IVP | ver. 6094"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" alt="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B2%7D%7B9%7D" alt="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" title="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}" data-latex="4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{2}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-2 \, t\right)}" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-0496" title="C1 | Homogeneous first-order linear IVP | ver. 0496"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="2 \, {y'} = 4 \, {y} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8248" title="C1 | Homogeneous first-order linear IVP | ver. 8248"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" alt="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" title="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" data-latex="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2015%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B4%7D" alt="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" title="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}" data-latex="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8633" title="C1 | Homogeneous first-order linear IVP | ver. 8633"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" alt="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-10%20%5C,%20%7By%7D%20+%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2036" alt="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" title="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36" data-latex="0 = -10 \, {y} + 5 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= 36"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(2 \, t\right)}" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(2 \, t\right)}" title="{y} = 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-1654" title="C1 | Homogeneous first-order linear IVP | ver. 1654"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}" alt="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}" title="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}" data-latex="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B2%7D" alt="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}" title="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}" data-latex="4 \, {y'} = -8 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-2 \, t\right)}" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-2 \, t\right)}" title="{y} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9755" title="C1 | Homogeneous first-order linear IVP | ver. 9755"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" alt="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%205%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B2%7D" alt="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="10 \, {y} + 5 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-2 \, t\right)}" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5361" title="C1 | Homogeneous first-order linear IVP | ver. 5361"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1" alt="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1" title="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1" data-latex="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-8%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-1" alt="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1" title="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1" data-latex="0 = -8 \, {y} - 4 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -1"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-2 \, t\right)}" alt="{y} = -4 \, e^{\left(-2 \, t\right)}" title="{y} = -4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-2 \, t\right)}" title="{y} = -4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-7128" title="C1 | Homogeneous first-order linear IVP | ver. 7128"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" alt="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" title="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" data-latex="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%203%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2024" alt="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" title="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24" data-latex="-9 \, {y} + 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= 24"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(3 \, t\right)}" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4822" title="C1 | Homogeneous first-order linear IVP | ver. 4822"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" alt="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-12%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2016" alt="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" title="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16" data-latex="-4 \, {y'} = -12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= 16"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(3 \, t\right)}" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(3 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4271" title="C1 | Homogeneous first-order linear IVP | ver. 4271"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2010%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-12" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -12"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-6431" title="C1 | Homogeneous first-order linear IVP | ver. 6431"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%2012%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B4%7D" alt="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" title="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" data-latex="-4 \, {y'} = 12 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-3 \, t\right)}" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9424" title="C1 | Homogeneous first-order linear IVP | ver. 9424"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" alt="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%203%20%5C,%20%7By'%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B9%7D" alt="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" title="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}" data-latex="-9 \, {y} - 3 \, {y'} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= \frac{1}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-3 \, t\right)}" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2999" title="C1 | Homogeneous first-order linear IVP | ver. 2999"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" alt="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2015%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B2%7D" alt="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="5 \, {y'} + 15 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-3 \, t\right)}" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2658" title="C1 | Homogeneous first-order linear IVP | ver. 2658"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" alt="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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-%209%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B3%7D%7B8%7D" alt="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" title="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}" data-latex="-3 \, {y'} - 9 \, {y} = 0 ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(-3 \, t\right)}" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-3 \, t\right)}" title="{y} = -3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5259" title="C1 | Homogeneous first-order linear IVP | ver. 5259"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" alt="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" title="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" data-latex="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%206%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B1%7D%7B4%7D" alt="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" title="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}" data-latex="0 = 6 \, {y} + 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{1}{4}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(-3 \, t\right)}" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-3 \, t\right)}" title="{y} = 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-9868" title="C1 | Homogeneous first-order linear IVP | ver. 9868"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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+%2010%20%5C,%20%7By%7D%20=%200%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%2018" alt="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18" title="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18" data-latex="-5 \, {y'} + 10 \, {y} = 0 ,\hspace{1em} y\big( \log\left(3\right) \big)= 18"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(2 \, t\right)}" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 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(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-2554" title="C1 | Homogeneous first-order linear IVP | ver. 2554"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" alt="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" title="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" data-latex="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%205%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-32" alt="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" title="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32" data-latex="15 \, {y} = 5 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-0914" title="C1 | Homogeneous first-order linear IVP | ver. 0914"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" alt="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%202%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-108" alt="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" title="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108" data-latex="6 \, {y} = 2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -108"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-8980" title="C1 | Homogeneous first-order linear IVP | ver. 8980"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" alt="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" title="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" data-latex="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-4%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-%5Cfrac%7B2%7D%7B9%7D" alt="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" title="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}" data-latex="8 \, {y} = -4 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -\frac{2}{9}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(-2 \, t\right)}" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -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(-2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-2 \, t\right)}" title="{y} = k e^{\left(-2 \, t\right)}" data-latex="{y} = k e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-2 \, t\right)}" title="{y} = -2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-5619" title="C1 | Homogeneous first-order linear IVP | ver. 5619"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(3%5Cright)%20%5Cbig)=%20-27" alt="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27" title="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27" data-latex="-4 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(3\right) \big)= -27"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -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%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(2 \, t\right)}" title="{y} = k e^{\left(2 \, t\right)}" data-latex="{y} = k e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-4756" title="C1 | Homogeneous first-order linear IVP | ver. 4756"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" alt="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20-%5Cfrac%7B1%7D%7B2%7D" alt="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" title="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}" data-latex="0 = -6 \, {y} - 2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= -\frac{1}{2}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-3 \, t\right)}" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, 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" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="C1-3571" title="C1 | Homogeneous first-order linear IVP | ver. 3571"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C1.</strong></p><p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </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} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" alt="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"/></p><p>Then show how to verify that your particular solution is correct.</p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p 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=%20-2%20%5C,%20%7By'%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%2032" alt="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" title="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32" data-latex="-6 \, {y} = -2 \, {y'} ,\hspace{1em} y\big( \log\left(2\right) \big)= 32"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&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)}" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, e^{\left(3 \, t\right)}" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, 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" alt="{y} = k e^{\left(3 \, t\right)}" title="{y} = k e^{\left(3 \, t\right)}" data-latex="{y} = k e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(3 \, t\right)}" title="{y} = 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 4 \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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