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<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
<objectbank ident="C4">
<qtimetadata>
<qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- C4</fieldentry></qtimetadatafield>
</qtimetadata>
<item ident="C4-9124" title="C4 | Homogeneous second-order linear IVP | ver. 9124"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" alt="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" title="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" data-latex="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%208" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%2012%20%5C,%20%7By%7D%20=%207%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%202" alt="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" title="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" data-latex="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\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(-3 \, t\right)} + e^{\left(-4 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\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?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9455" title="C4 | Homogeneous second-order linear IVP | ver. 9455"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12" alt="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12" title="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12" data-latex="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" alt="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" title="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" data-latex="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?36%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%20-12" alt="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12" title="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12" data-latex="36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2010%20%5C,%20%7By%7D%20+%20%7By''%7D%20-%207%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%20-13" alt="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" title="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" data-latex="0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" alt="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)"/></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(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)">
</p>
<p style="text-align:center;">
<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(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5644" title="C4 | Homogeneous second-order linear IVP | ver. 5644"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" alt="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" title="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" data-latex="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" alt="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" title="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" data-latex="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%20-15" alt="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" title="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" data-latex="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By'%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%20-3" alt="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" title="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" data-latex="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" alt="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5611" title="C4 | Homogeneous second-order linear IVP | ver. 5611"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8" alt="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8" title="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8" data-latex="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3" alt="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3" title="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3" data-latex="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx''%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-8" alt="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8" title="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8" data-latex="0 = {x''} + 4 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By'%7D%20=%20%7By''%7D%20+%2010%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-3" alt="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3" title="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3" data-latex="7 \, {y'} = {y''} + 10 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -3">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" alt="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" title="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" data-latex="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\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} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" alt="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" title="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" title="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\left(2 \, t\right)" data-latex="{x} = 3 \, \cos\left(2 \, t\right) - 4 \, \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" title="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = e^{\left(5 \, t\right)} - 4 \, e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1999" title="C4 | Homogeneous second-order linear IVP | ver. 1999"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19" alt="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19" title="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19" data-latex="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5" alt="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5" title="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5" data-latex="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20-%202%20%5C,%20%7Bx'%7D%20=%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%20-19" alt="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19" title="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19" data-latex="{x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-25%20%5C,%20%7By%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%205" alt="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5" title="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5" data-latex="0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" alt="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)" alt="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)" title="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)" data-latex="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)" title="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)" data-latex="{y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2691" title="C4 | Homogeneous second-order linear IVP | ver. 2691"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10" alt="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10" title="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10" data-latex="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30" alt="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30" title="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30" data-latex="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx''%7D%20-%203%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%2010" alt="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10" title="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10" data-latex="0 = -{x''} - 3 \, {x} + 4 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%20-30" alt="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30" title="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30" data-latex="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -30">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}" alt="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}" title="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}" data-latex="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" alt="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}" title="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}" data-latex="{x} = 3 \, e^{\left(3 \, t\right)} + e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = 3 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4984" title="C4 | Homogeneous second-order linear IVP | ver. 4984"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10" alt="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10" title="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10" data-latex="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4" alt="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4" title="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4" data-latex="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%2012%20%5C,%20%7Bx%7D%20+%207%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%2010" alt="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10" title="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10" data-latex="-{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%2016%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-4" alt="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4" title="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4" data-latex="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}" alt="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}" title="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)" alt="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}" title="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6940" title="C4 | Homogeneous second-order linear IVP | ver. 6940"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2" alt="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2" title="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2" data-latex="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18" alt="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18" title="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18" data-latex="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%20%7Bx'%7D%20=%206%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%209%20,%20%7Bx%7D%20'(0)=%20-2" alt="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2" title="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2" data-latex="{x''} + {x'} = 6 \, {x} \hspace{2em} {x} (0)= 9 , {x} '(0)= -2">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?36%20%5C,%20%7By%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%2018" alt="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18" title="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18" data-latex="36 \, {y} = -{y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, 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 \, \sin\left(6 \, t\right)" alt="{y} = 3 \, \sin\left(6 \, t\right)" title="{y} = 3 \, \sin\left(6 \, t\right)" data-latex="{y} = 3 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="{x} = 5 \, e^{\left(2 \, t\right)} + 4 \, 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?%7By%7D%20=%203%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = 3 \, \sin\left(6 \, t\right)" title="{y} = 3 \, \sin\left(6 \, t\right)" data-latex="{y} = 3 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6494" title="C4 | Homogeneous second-order linear IVP | ver. 6494"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50" alt="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50" title="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50" data-latex="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14" alt="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14" title="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14" data-latex="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-50" alt="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50" title="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50" data-latex="100 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -50">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%203%20%5C,%20%7By'%7D%20=%2010%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%207%20,%20%7By%7D%20'(0)=%20-14" alt="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14" title="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14" data-latex="{y''} + 3 \, {y'} = 10 \, {y} \hspace{2em} {y} (0)= 7 , {y} '(0)= -14">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)" alt="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)" title="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)" data-latex="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)"/></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)} + 4 \, e^{\left(-5 \, t\right)}" alt="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)" title="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)" data-latex="{x} = -2 \, \cos\left(10 \, t\right) - 5 \, \sin\left(10 \, t\right)">
</p>
<p style="text-align:center;">
<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%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3832" title="C4 | Homogeneous second-order linear IVP | ver. 3832"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45" alt="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45" title="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45" data-latex="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12" alt="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12" title="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12" data-latex="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx''%7D%20-%2081%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%20-45" alt="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45" title="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45" data-latex="0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-12" alt="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12" title="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12" data-latex="-20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)" alt="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)" title="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)" data-latex="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)"/></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(4 \, t\right)}" alt="{y} = -3 \, e^{\left(4 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)" title="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)" data-latex="{x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<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(4%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(4 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7786" title="C4 | Homogeneous second-order linear IVP | ver. 7786"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" alt="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" title="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" data-latex="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" alt="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx'%7D%20-%208%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%2010" alt="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" title="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" data-latex="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%20-32" alt="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" alt="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" alt="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" title="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" data-latex="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" title="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" data-latex="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1809" title="C4 | Homogeneous second-order linear IVP | ver. 1809"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0" alt="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0" title="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0" data-latex="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13" alt="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13" title="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13" data-latex="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%200" alt="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0" title="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0" data-latex="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 0">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-%7By''%7D%20+%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-8%20,%20%7By%7D%20'(0)=%20-13" alt="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13" title="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13" data-latex="2 \, {y} = -{y''} + 3 \, {y'} \hspace{2em} {y} (0)= -8 , {y} '(0)= -13">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -4 \, \cos\left(2 \, t\right)" alt="{x} = -4 \, \cos\left(2 \, t\right)" title="{x} = -4 \, \cos\left(2 \, t\right)" data-latex="{x} = -4 \, \cos\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} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}" alt="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}" title="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-4%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = -4 \, \cos\left(2 \, t\right)" title="{x} = -4 \, \cos\left(2 \, t\right)" data-latex="{x} = -4 \, \cos\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?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}" title="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} - 3 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3277" title="C4 | Homogeneous second-order linear IVP | ver. 3277"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45" alt="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45" title="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45" data-latex="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22" alt="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22" title="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22" data-latex="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%2045" alt="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45" title="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45" data-latex="-81 \, {x} = {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 45">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20-%205%20%5C,%20%7By%7D%20-%206%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%20-22" alt="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22" title="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22" data-latex="0 = -{y''} - 5 \, {y} - 6 \, {y'} \hspace{2em} {y} (0)= 2 , {y} '(0)= -22">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)" alt="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)" title="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)" data-latex="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)"/></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(-t\right)} + 5 \, e^{\left(-5 \, t\right)}" alt="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}" title="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%205%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)" title="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)" data-latex="{x} = 4 \, \cos\left(9 \, t\right) + 5 \, \sin\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<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(-t%5Cright)%7D%20+%205%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}" title="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = -3 \, e^{\left(-t\right)} + 5 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5569" title="C4 | Homogeneous second-order linear IVP | ver. 5569"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4" alt="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4" title="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4" data-latex="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9" alt="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9" title="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9" data-latex="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%20-4" alt="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4" title="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4" data-latex="0 = -4 \, {x} - {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By'%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%20-9" alt="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9" title="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9" data-latex="-3 \, {y} + 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= -9">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" alt="{x} = \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) - 2 \, \sin\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(-3 \, t\right)} - 3 \, e^{t}" alt="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}" title="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) - 2 \, \sin\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?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}" title="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}" data-latex="{y} = 2 \, e^{\left(-3 \, t\right)} - 3 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3469" title="C4 | Homogeneous second-order linear IVP | ver. 3469"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12" alt="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12" title="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12" data-latex="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6" alt="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6" title="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6" data-latex="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%2016%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%2012" alt="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12" title="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12" data-latex="{x''} + 16 \, {x} = 0 \hspace{2em} {x} (0)= -1 , {x} '(0)= 12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%204%20%5C,%20%7By%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%20-6" alt="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6" title="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6" data-latex="0 = 4 \, {y} - {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -6">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" alt="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" title="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" data-latex="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 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"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" title="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" data-latex="{x} = -\cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%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)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4104" title="C4 | Homogeneous second-order linear IVP | ver. 4104"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4" alt="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4" title="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4" data-latex="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24" alt="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24" title="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24" data-latex="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20+%204%20%5C,%20%7Bx%7D%20=%20-3%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-6%20,%20%7Bx%7D%20'(0)=%20-4" alt="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4" title="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4" data-latex="-{x''} + 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= -6 , {x} '(0)= -4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-36%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%2024" alt="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24" title="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24" data-latex="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 24">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" alt="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)" alt="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)" title="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)" data-latex="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)" title="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)" data-latex="{y} = -\cos\left(6 \, t\right) + 4 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9890" title="C4 | Homogeneous second-order linear IVP | ver. 9890"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31" alt="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31" title="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31" data-latex="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16" alt="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16" title="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16" data-latex="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2010%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20-%207%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-8%20,%20%7Bx%7D%20'(0)=%20-31" alt="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31" title="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31" data-latex="0 = 10 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= -8 , {x} '(0)= -31">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-16%20%5C,%20%7By%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-16" alt="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16" title="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16" data-latex="0 = -16 \, {y} - {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(2 \, t\right)}" alt="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(2 \, t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(2 \, t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, 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} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" alt="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" title="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" data-latex="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(2 \, t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(2 \, t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} - 3 \, 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?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" title="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" data-latex="{y} = 5 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8385" title="C4 | Homogeneous second-order linear IVP | ver. 8385"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40" alt="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40" title="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40" data-latex="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5" alt="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5" title="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5" data-latex="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%2064%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-40" alt="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40" title="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40" data-latex="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -40">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20-6%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%205" alt="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5" title="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5" data-latex="-{y'} = -6 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 5">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)" alt="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)" title="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)" data-latex="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)"/></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)} - 3 \, e^{\left(-3 \, t\right)}" alt="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)" title="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)" data-latex="{x} = 2 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)">
</p>
<p style="text-align:center;">
<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%20-%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" title="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1561" title="C4 | Homogeneous second-order linear IVP | ver. 1561"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7" alt="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7" title="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7" data-latex="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14" alt="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14" title="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14" data-latex="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx'%7D%20-%202%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%20-7" alt="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7" title="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7" data-latex="3 \, {x'} - 2 \, {x} = {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -7">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-49%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%2014" alt="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14" title="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14" data-latex="-49 \, {y} = {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 14">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}" alt="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}" title="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}" data-latex="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)" alt="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)" title="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)" data-latex="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}" title="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}" data-latex="{x} = -3 \, e^{\left(2 \, t\right)} - e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)" title="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)" data-latex="{y} = -3 \, \cos\left(7 \, t\right) + 2 \, \sin\left(7 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5281" title="C4 | Homogeneous second-order linear IVP | ver. 5281"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18" alt="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18" title="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18" data-latex="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6" alt="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6" title="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6" data-latex="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-81%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%20-18" alt="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18" title="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18" data-latex="{x''} = -81 \, {x} \hspace{2em} {x} (0)= -5 , {x} '(0)= -18">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%202%20%5C,%20%7By'%7D%20=%20-8%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-9%20,%20%7By%7D%20'(0)=%206" alt="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6" title="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6" data-latex="-{y''} - 2 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -9 , {y} '(0)= 6">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)" alt="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)" title="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)" data-latex="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" alt="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" title="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)" title="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)" data-latex="{x} = -5 \, \cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" title="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8800" title="C4 | Homogeneous second-order linear IVP | ver. 8800"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0" alt="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0" title="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0" data-latex="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5" alt="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5" title="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5" data-latex="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%200" alt="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0" title="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0" data-latex="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 0">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%2020%20%5C,%20%7By%7D%20+%209%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%205" alt="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5" title="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5" data-latex="-{y''} = 20 \, {y} + 9 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= 5">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(4 \, t\right)" alt="{x} = -3 \, \cos\left(4 \, t\right)" title="{x} = -3 \, \cos\left(4 \, t\right)" data-latex="{x} = -3 \, \cos\left(4 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(4 \, t\right)" title="{x} = -3 \, \cos\left(4 \, t\right)" data-latex="{x} = -3 \, \cos\left(4 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="{y} = -5 \, e^{\left(-4 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4120" title="C4 | Homogeneous second-order linear IVP | ver. 4120"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11" alt="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11" title="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11" data-latex="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12" alt="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12" title="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12" data-latex="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%20-3%20%5C,%20%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-6%20,%20%7Bx%7D%20'(0)=%20-11" alt="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11" title="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11" data-latex="-{x''} = -3 \, {x'} + 2 \, {x} \hspace{2em} {x} (0)= -6 , {x} '(0)= -11">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%2036%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%2012" alt="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12" title="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12" data-latex="-{y''} = 36 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}" alt="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}" title="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}" data-latex="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" alt="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}" title="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}" data-latex="{x} = -5 \, e^{\left(2 \, t\right)} - e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1574" title="C4 | Homogeneous second-order linear IVP | ver. 1574"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1" alt="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1" title="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1" data-latex="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" alt="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" title="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" data-latex="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7Bx%7D%20=%20-4%20%5C,%20%7Bx'%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%201" alt="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1" title="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1" data-latex="5 \, {x} = -4 \, {x'} + {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= 1">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-36%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%2012" alt="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" title="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" data-latex="{y''} = -36 \, {y} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -e^{\left(-t\right)}" alt="{x} = -e^{\left(-t\right)}" title="{x} = -e^{\left(-t\right)}" data-latex="{x} = -e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" alt="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -e^{\left(-t\right)}" title="{x} = -e^{\left(-t\right)}" data-latex="{x} = -e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = -2 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7704" title="C4 | Homogeneous second-order linear IVP | ver. 7704"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32" alt="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32" title="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32" data-latex="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16" alt="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16" title="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16" data-latex="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%2064%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%2032" alt="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32" title="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32" data-latex="-{x''} = 64 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 32">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%204%20%5C,%20%7By%7D%20+%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%209%20,%20%7By%7D%20'(0)=%2016" alt="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16" title="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16" data-latex="{y''} = 4 \, {y} + 3 \, {y'} \hspace{2em} {y} (0)= 9 , {y} '(0)= 16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)" alt="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)" title="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)" data-latex="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}" alt="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)" title="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)" data-latex="{x} = \cos\left(8 \, t\right) + 4 \, \sin\left(8 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = 5 \, e^{\left(4 \, t\right)} + 4 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7138" title="C4 | Homogeneous second-order linear IVP | ver. 7138"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4" alt="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4" title="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4" data-latex="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18" alt="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18" title="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18" data-latex="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20+%209%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%20-4" alt="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4" title="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4" data-latex="20 \, {x} = -{x''} + 9 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= -4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20-%2081%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%2018" alt="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18" title="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18" data-latex="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}" alt="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}" title="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}" data-latex="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" alt="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" title="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" data-latex="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}" title="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}" data-latex="{x} = -4 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" title="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" data-latex="{y} = -5 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9961" title="C4 | Homogeneous second-order linear IVP | ver. 9961"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13" alt="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13" title="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13" data-latex="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25" alt="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25" title="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25" data-latex="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%205%20%5C,%20%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%207%20,%20%7Bx%7D%20'(0)=%2013" alt="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13" title="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13" data-latex="{x''} = 5 \, {x'} - 4 \, {x} \hspace{2em} {x} (0)= 7 , {x} '(0)= 13">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2025%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%20-25" alt="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25" title="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25" data-latex="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 2 , {y} '(0)= -25">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}" alt="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}" title="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" alt="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7Bt%7D" alt="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}" title="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + 5 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{y} = 2 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8233" title="C4 | Homogeneous second-order linear IVP | ver. 8233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24" alt="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24" title="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24" data-latex="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12" alt="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12" title="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12" data-latex="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%2024" alt="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24" title="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24" data-latex="16 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -2 , {x} '(0)= 24">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2036%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%20-12" alt="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12" title="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12" data-latex="0 = 36 \, {y} + {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= -12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" alt="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" alt="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{y} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8891" title="C4 | Homogeneous second-order linear IVP | ver. 8891"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" alt="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" title="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" data-latex="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9" alt="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9" title="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9" data-latex="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-2" alt="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" title="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" data-latex="-3 \, {x'} = -2 \, {x} - {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2081%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-9" alt="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9" title="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9" data-latex="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= -9">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{t}" alt="{x} = -2 \, e^{t}" title="{x} = -2 \, e^{t}" data-latex="{x} = -2 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" alt="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7Bt%7D" alt="{x} = -2 \, e^{t}" title="{x} = -2 \, e^{t}" data-latex="{x} = -2 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7567" title="C4 | Homogeneous second-order linear IVP | ver. 7567"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10" alt="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10" title="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10" data-latex="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11" alt="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11" title="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11" data-latex="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%20-10" alt="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10" title="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10" data-latex="4 \, {x} = -{x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By''%7D%20-%2012%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-6%20,%20%7By%7D%20'(0)=%20-11" alt="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11" title="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11" data-latex="0 = {y''} - 12 \, {y} + {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= -11">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -\cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" alt="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" title="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" data-latex="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\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} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}" alt="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}" title="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" title="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" data-latex="{x} = -\cos\left(2 \, t\right) - 5 \, \sin\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?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}" title="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{y} = -5 \, e^{\left(3 \, t\right)} - e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0999" title="C4 | Homogeneous second-order linear IVP | ver. 0999"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20" alt="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20" title="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20" data-latex="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" alt="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" title="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" data-latex="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx''%7D%20-%20100%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%20-20" alt="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20" title="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20" data-latex="0 = -{x''} - 100 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= -20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20+%2012%20%5C,%20%7By%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%2012" alt="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" title="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" data-latex="-{y''} + 12 \, {y} = -{y'} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" alt="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" title="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" data-latex="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)"/></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(4 \, t\right)}" alt="{y} = 3 \, e^{\left(4 \, t\right)}" title="{y} = 3 \, e^{\left(4 \, t\right)}" data-latex="{y} = 3 \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" title="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" data-latex="{x} = \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)">
</p>
<p style="text-align:center;">
<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(4%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(4 \, t\right)}" title="{y} = 3 \, e^{\left(4 \, t\right)}" data-latex="{y} = 3 \, e^{\left(4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0886" title="C4 | Homogeneous second-order linear IVP | ver. 0886"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26" alt="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26" title="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26" data-latex="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" alt="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" title="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" data-latex="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7Bx%7D%20=%20-6%20%5C,%20%7Bx'%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%209%20,%20%7Bx%7D%20'(0)=%2026" alt="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26" title="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26" data-latex="-8 \, {x} = -6 \, {x'} + {x''} \hspace{2em} {x} (0)= 9 , {x} '(0)= 26">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%2016%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%2020" alt="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" title="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" data-latex="-{y''} - 16 \, {y} = 0 \hspace{2em} {y} (0)= -3 , {y} '(0)= 20">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" alt="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" title="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, 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 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" alt="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" title="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 4 \, e^{\left(4 \, t\right)} + 5 \, 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?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%205%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2135" title="C4 | Homogeneous second-order linear IVP | ver. 2135"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13" alt="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13" title="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13" data-latex="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10" alt="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10" title="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10" data-latex="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx'%7D%20=%2010%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%20-13" alt="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13" title="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13" data-latex="-3 \, {x'} = 10 \, {x} - {x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -13">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-100%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%2010" alt="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10" title="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10" data-latex="{y''} = -100 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 10">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}" alt="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}" title="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)" alt="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)" title="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)" data-latex="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}" title="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="{x} = -3 \, e^{\left(5 \, t\right)} - e^{\left(-2 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)" title="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)" data-latex="{y} = -3 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2361" title="C4 | Homogeneous second-order linear IVP | ver. 2361"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" alt="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" title="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" data-latex="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" alt="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" title="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" data-latex="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%2025%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%2020" alt="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" title="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" data-latex="{x''} + 25 \, {x} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-15%20%5C,%20%7By%7D%20+%208%20%5C,%20%7By'%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%203" alt="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" title="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" data-latex="0 = -15 \, {y} + 8 \, {y'} - {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)" alt="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)" title="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)" data-latex="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}" alt="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)" title="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)" data-latex="{x} = -3 \, \cos\left(5 \, t\right) + 4 \, \sin\left(5 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}" title="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1126" title="C4 | Homogeneous second-order linear IVP | ver. 1126"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19" alt="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19" title="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19" data-latex="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14" alt="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14" title="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14" data-latex="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-19" alt="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19" title="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19" data-latex="-12 \, {x} = -{x'} - {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -19">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2049%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%20-14" alt="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14" title="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14" data-latex="{y''} + 49 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -14">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}" alt="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}" title="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}" data-latex="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)" alt="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)" title="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)" data-latex="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}" title="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}" data-latex="{x} = -e^{\left(3 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)" title="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)" data-latex="{y} = -5 \, \cos\left(7 \, t\right) - 2 \, \sin\left(7 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4103" title="C4 | Homogeneous second-order linear IVP | ver. 4103"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8" alt="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8" title="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8" data-latex="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" alt="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" title="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" data-latex="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%206%20%5C,%20%7Bx%7D%20+%205%20%5C,%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%208" alt="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8" title="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8" data-latex="-{x''} - 6 \, {x} + 5 \, {x'} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20-%2081%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%200" alt="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" title="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" data-latex="0 = -{y''} - 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" alt="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" title="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" data-latex="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, 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 \, \cos\left(9 \, t\right)" alt="{y} = 3 \, \cos\left(9 \, t\right)" title="{y} = 3 \, \cos\left(9 \, t\right)" data-latex="{y} = 3 \, \cos\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" title="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" data-latex="{x} = 4 \, e^{\left(3 \, t\right)} - 2 \, 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?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(9 \, t\right)" title="{y} = 3 \, \cos\left(9 \, t\right)" data-latex="{y} = 3 \, \cos\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3522" title="C4 | Homogeneous second-order linear IVP | ver. 3522"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4" alt="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4" title="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4" data-latex="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5" alt="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5" title="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5" data-latex="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20+%205%20%5C,%20%7Bx%7D%20=%204%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%20-4" alt="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4" title="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4" data-latex="-{x''} + 5 \, {x} = 4 \, {x'} \hspace{2em} {x} (0)= -4 , {x} '(0)= -4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25%20%5C,%20%7By%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%20-5" alt="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5" title="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5" data-latex="25 \, {y} = -{y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= -5">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -4 \, e^{t}" alt="{x} = -4 \, e^{t}" title="{x} = -4 \, e^{t}" data-latex="{x} = -4 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" alt="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" title="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" data-latex="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-4%20%5C,%20e%5E%7Bt%7D" alt="{x} = -4 \, e^{t}" title="{x} = -4 \, e^{t}" data-latex="{x} = -4 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" title="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" data-latex="{y} = 3 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2515" title="C4 | Homogeneous second-order linear IVP | ver. 2515"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20" alt="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20" title="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20" data-latex="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0" alt="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0" title="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0" data-latex="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%20-7%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%20-20" alt="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20" title="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20" data-latex="-12 \, {x} - {x''} = -7 \, {x'} \hspace{2em} {x} (0)= -5 , {x} '(0)= -20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%2049%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%204%20,%20%7By%7D%20'(0)=%200" alt="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0" title="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0" data-latex="-{y''} - 49 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= 0">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(4 \, t\right)}" alt="{x} = -5 \, e^{\left(4 \, t\right)}" title="{x} = -5 \, e^{\left(4 \, t\right)}" data-latex="{x} = -5 \, e^{\left(4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, \cos\left(7 \, t\right)" alt="{y} = 4 \, \cos\left(7 \, t\right)" title="{y} = 4 \, \cos\left(7 \, t\right)" data-latex="{y} = 4 \, \cos\left(7 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{x} = -5 \, e^{\left(4 \, t\right)}" title="{x} = -5 \, e^{\left(4 \, t\right)}" data-latex="{x} = -5 \, e^{\left(4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)" alt="{y} = 4 \, \cos\left(7 \, t\right)" title="{y} = 4 \, \cos\left(7 \, t\right)" data-latex="{y} = 4 \, \cos\left(7 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2121" title="C4 | Homogeneous second-order linear IVP | ver. 2121"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12" alt="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12" title="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12" data-latex="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" alt="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" title="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" data-latex="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-9%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%2012" alt="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12" title="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12" data-latex="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= 12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20+%206%20%5C,%20%7By'%7D%20-%205%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%203" alt="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" title="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3" data-latex="0 = -{y''} + 6 \, {y'} - 5 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 3">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, \sin\left(3 \, t\right)" alt="{x} = 4 \, \sin\left(3 \, t\right)" title="{x} = 4 \, \sin\left(3 \, t\right)" data-latex="{x} = 4 \, \sin\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} = e^{\left(5 \, t\right)} - 2 \, e^{t}" alt="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}" title="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}" data-latex="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = 4 \, \sin\left(3 \, t\right)" title="{x} = 4 \, \sin\left(3 \, t\right)" data-latex="{x} = 4 \, \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7Bt%7D" alt="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}" title="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}" data-latex="{y} = e^{\left(5 \, t\right)} - 2 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4399" title="C4 | Homogeneous second-order linear IVP | ver. 4399"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6" alt="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6" title="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6" data-latex="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24" alt="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24" title="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24" data-latex="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%209%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%20-6" alt="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6" title="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6" data-latex="-{x''} = 9 \, {x} \hspace{2em} {x} (0)= 5 , {x} '(0)= -6">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%2020%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-24" alt="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24" title="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24" data-latex="{y''} = 20 \, {y} + {y'} \hspace{2em} {y} (0)= -3 , {y} '(0)= -24">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" alt="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" title="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" data-latex="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\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(5 \, t\right)} + e^{\left(-4 \, t\right)}" alt="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" title="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" data-latex="{x} = 5 \, \cos\left(3 \, t\right) - 2 \, \sin\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?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(5 \, t\right)} + e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9108" title="C4 | Homogeneous second-order linear IVP | ver. 9108"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12" alt="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12" title="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12" data-latex="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" alt="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" title="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" data-latex="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20-%209%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%2012" alt="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12" title="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12" data-latex="{x''} - 9 \, {x} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2081%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-36" alt="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" title="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" data-latex="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= -36">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, e^{\left(3 \, t\right)}" alt="{x} = 4 \, e^{\left(3 \, t\right)}" title="{x} = 4 \, e^{\left(3 \, t\right)}" data-latex="{x} = 4 \, 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} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" alt="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = 4 \, e^{\left(3 \, t\right)}" title="{x} = 4 \, e^{\left(3 \, t\right)}" data-latex="{x} = 4 \, 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?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6011" title="C4 | Homogeneous second-order linear IVP | ver. 6011"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18" alt="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18" title="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18" data-latex="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26" alt="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26" title="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26" data-latex="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-36%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%2018" alt="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18" title="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18" data-latex="0 = -36 \, {x} - {x''} \hspace{2em} {x} (0)= 4 , {x} '(0)= 18">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20-%7By''%7D%20+%2012%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%2026" alt="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26" title="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26" data-latex="-{y'} = -{y''} + 12 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 26">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)" alt="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)" title="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)" data-latex="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" alt="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)" title="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)" data-latex="{x} = 4 \, \cos\left(6 \, t\right) + 3 \, \sin\left(6 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 5 \, e^{\left(4 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9342" title="C4 | Homogeneous second-order linear IVP | ver. 9342"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17" alt="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17" title="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17" data-latex="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18" alt="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18" title="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18" data-latex="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-15%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20-%208%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%20-17" alt="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17" title="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17" data-latex="0 = -15 \, {x} - {x''} - 8 \, {x'} \hspace{2em} {x} (0)= 5 , {x} '(0)= -17">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2036%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%20-18" alt="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18" title="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18" data-latex="{y''} + 36 \, {y} = 0 \hspace{2em} {y} (0)= -5 , {y} '(0)= -18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}" alt="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}" title="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)" alt="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)" title="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)" data-latex="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}" title="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="{x} = 4 \, e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)" title="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)" data-latex="{y} = -5 \, \cos\left(6 \, t\right) - 3 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5624" title="C4 | Homogeneous second-order linear IVP | ver. 5624"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25" alt="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25" title="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25" data-latex="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" alt="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" title="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" data-latex="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-25%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-25" alt="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25" title="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25" data-latex="-25 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= -25">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20-%206%20%5C,%20%7By'%7D%20=%20-5%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%20-16" alt="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" title="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" data-latex="{y''} - 6 \, {y'} = -5 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" alt="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}" alt="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}" title="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}" data-latex="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{x} = 3 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}" title="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}" data-latex="{y} = -3 \, e^{\left(5 \, t\right)} - e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5908" title="C4 | Homogeneous second-order linear IVP | ver. 5908"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8" alt="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8" title="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8" data-latex="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4" alt="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4" title="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4" data-latex="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-8" alt="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8" title="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8" data-latex="-2 \, {x} - {x'} + {x''} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%20-4" alt="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4" title="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4" data-latex="-{y''} = 4 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= -4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" alt="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" alt="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" title="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" title="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)" data-latex="{y} = -5 \, \cos\left(2 \, t\right) - 2 \, \sin\left(2 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6302" title="C4 | Homogeneous second-order linear IVP | ver. 6302"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" alt="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" title="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" data-latex="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21" alt="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21" title="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21" data-latex="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-12" alt="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" title="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" data-latex="16 \, {x} = -{x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%20-9%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%20-21" alt="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21" title="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21" data-latex="-{y''} = -9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -21">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)" alt="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)" title="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)" data-latex="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" alt="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)" title="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)" data-latex="{x} = -2 \, \cos\left(4 \, t\right) - 3 \, \sin\left(4 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -4 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0305" title="C4 | Homogeneous second-order linear IVP | ver. 0305"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17" alt="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17" title="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17" data-latex="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%208" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7By'%7D%20-%20%7By''%7D%20+%2015%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-17" alt="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17" title="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17" data-latex="0 = -2 \, {y'} - {y''} + 15 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -17">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\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} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5597" title="C4 | Homogeneous second-order linear IVP | ver. 5597"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20" alt="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20" title="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20" data-latex="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16" alt="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16" title="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16" data-latex="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-20" alt="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20" title="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20" data-latex="100 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%20%7By'%7D%20=%20-12%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%2016" alt="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16" title="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16" data-latex="-{y''} - {y'} = -12 \, {y} \hspace{2em} {y} (0)= -4 , {y} '(0)= 16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" alt="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" title="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" data-latex="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)"/></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(-4 \, t\right)}" alt="{y} = -4 \, e^{\left(-4 \, t\right)}" title="{y} = -4 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" title="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)" data-latex="{x} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)">
</p>
<p style="text-align:center;">
<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(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-4 \, t\right)}" title="{y} = -4 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0392" title="C4 | Homogeneous second-order linear IVP | ver. 0392"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9" alt="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9" title="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9" data-latex="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0" alt="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0" title="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0" data-latex="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20+%209%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%20-9" alt="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9" title="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9" data-latex="-{x''} + 9 \, {x} = 0 \hspace{2em} {x} (0)= 5 , {x} '(0)= -9">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?100%20%5C,%20%7By%7D%20+%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%200" alt="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0" title="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0" data-latex="100 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 1 , {y} '(0)= 0">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="{x} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="{x} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="{x} = e^{\left(3 \, t\right)} + 4 \, 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} = \cos\left(10 \, t\right)" alt="{y} = \cos\left(10 \, t\right)" title="{y} = \cos\left(10 \, t\right)" data-latex="{y} = \cos\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="{x} = e^{\left(3 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="{x} = e^{\left(3 \, t\right)} + 4 \, 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?%7By%7D%20=%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = \cos\left(10 \, t\right)" title="{y} = \cos\left(10 \, t\right)" data-latex="{y} = \cos\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8275" title="C4 | Homogeneous second-order linear IVP | ver. 8275"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0" alt="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0" title="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0" data-latex="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26" alt="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26" title="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26" data-latex="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-64%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%200" alt="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0" title="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0" data-latex="-64 \, {x} = {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= 0">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7By%7D%20=%206%20%5C,%20%7By'%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-6%20,%20%7By%7D%20'(0)=%20-26" alt="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26" title="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26" data-latex="5 \, {y} = 6 \, {y'} - {y''} \hspace{2em} {y} (0)= -6 , {y} '(0)= -26">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(8 \, t\right)" alt="{x} = \cos\left(8 \, t\right)" title="{x} = \cos\left(8 \, t\right)" data-latex="{x} = \cos\left(8 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}" alt="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}" title="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}" data-latex="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)" alt="{x} = \cos\left(8 \, t\right)" title="{x} = \cos\left(8 \, t\right)" data-latex="{x} = \cos\left(8 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}" title="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}" data-latex="{y} = -5 \, e^{\left(5 \, t\right)} - e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2018" title="C4 | Homogeneous second-order linear IVP | ver. 2018"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1" alt="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1" title="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1" data-latex="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15" alt="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15" title="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15" data-latex="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%207%20,%20%7Bx%7D%20'(0)=%201" alt="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1" title="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1" data-latex="0 = -{x} + {x''} \hspace{2em} {x} (0)= 7 , {x} '(0)= 1">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-15" alt="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15" title="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15" data-latex="9 \, {y} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -15">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}" alt="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}" title="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}" data-latex="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)" alt="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)" title="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)" data-latex="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%204%20%5C,%20e%5E%7Bt%7D" alt="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}" title="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}" data-latex="{x} = 3 \, e^{\left(-t\right)} + 4 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)" title="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)" data-latex="{y} = -3 \, \cos\left(3 \, t\right) - 5 \, \sin\left(3 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5121" title="C4 | Homogeneous second-order linear IVP | ver. 5121"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4" alt="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4" title="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4" data-latex="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" alt="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" title="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" data-latex="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%204" alt="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4" title="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4" data-latex="{x''} = 4 \, {x} \hspace{2em} {x} (0)= -4 , {x} '(0)= 4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20+%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%204" alt="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" title="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" data-latex="4 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -e^{\left(2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" alt="{x} = -e^{\left(2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="{x} = -e^{\left(2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{x} = -e^{\left(2 \, t\right)} - 3 \, 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 \, \sin\left(2 \, t\right)" alt="{y} = 2 \, \sin\left(2 \, t\right)" title="{y} = 2 \, \sin\left(2 \, t\right)" data-latex="{y} = 2 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = -e^{\left(2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="{x} = -e^{\left(2 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{x} = -e^{\left(2 \, t\right)} - 3 \, 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?%7By%7D%20=%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = 2 \, \sin\left(2 \, t\right)" title="{y} = 2 \, \sin\left(2 \, t\right)" data-latex="{y} = 2 \, \sin\left(2 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8742" title="C4 | Homogeneous second-order linear IVP | ver. 8742"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" alt="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" title="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" data-latex="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1" alt="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1" title="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1" data-latex="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%20-6" alt="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" title="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" data-latex="4 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -4 , {x} '(0)= -6">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By%7D%20-%204%20%5C,%20%7By'%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-1" alt="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1" title="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1" data-latex="0 = -3 \, {y} - 4 \, {y'} - {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -1">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\left(2 \, t\right)" alt="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\left(2 \, t\right)" title="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\left(2 \, t\right)" data-latex="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\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} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}" alt="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-4%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\left(2 \, t\right)" title="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\left(2 \, t\right)" data-latex="{x} = -4 \, \cos\left(2 \, t\right) - 3 \, \sin\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?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="{y} = -5 \, e^{\left(-t\right)} + 2 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6019" title="C4 | Homogeneous second-order linear IVP | ver. 6019"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7" alt="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7" title="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7" data-latex="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16" alt="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16" title="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16" data-latex="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%20-7" alt="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7" title="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7" data-latex="49 \, {x} = -{x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= -7">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By'%7D%20-%204%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-9%20,%20%7By%7D%20'(0)=%20-16" alt="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16" title="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16" data-latex="0 = -3 \, {y'} - 4 \, {y} + {y''} \hspace{2em} {y} (0)= -9 , {y} '(0)= -16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)" alt="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)" title="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)" data-latex="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" alt="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)" title="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)" data-latex="{x} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{y} = -5 \, e^{\left(4 \, t\right)} - 4 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2357" title="C4 | Homogeneous second-order linear IVP | ver. 2357"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" alt="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" alt="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" title="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" data-latex="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20+%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%2020" alt="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="-{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By''%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%200" alt="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" title="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0" data-latex="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 0">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, e^{\left(-2 \, t\right)}" alt="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, e^{\left(-2 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, e^{\left(-2 \, t\right)}" data-latex="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, 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 \, \cos\left(2 \, t\right)" alt="{y} = 3 \, \cos\left(2 \, t\right)" title="{y} = 3 \, \cos\left(2 \, t\right)" data-latex="{y} = 3 \, \cos\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, e^{\left(-2 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, e^{\left(-2 \, t\right)}" data-latex="{x} = 5 \, e^{\left(2 \, t\right)} - 5 \, 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?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(2 \, t\right)" title="{y} = 3 \, \cos\left(2 \, t\right)" data-latex="{y} = 3 \, \cos\left(2 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7843" title="C4 | Homogeneous second-order linear IVP | ver. 7843"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35" alt="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35" title="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35" data-latex="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45" alt="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45" title="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45" data-latex="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%2049%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-35" alt="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35" title="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35" data-latex="-{x''} - 49 \, {x} = 0 \hspace{2em} {x} (0)= 3 , {x} '(0)= -35">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-25%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%2045" alt="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45" title="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45" data-latex="0 = -25 \, {y} + {y''} \hspace{2em} {y} (0)= -1 , {y} '(0)= 45">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" alt="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" title="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" data-latex="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)"/></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(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}" alt="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" title="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" data-latex="{x} = 3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)">
</p>
<p style="text-align:center;">
<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(5%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 4 \, e^{\left(5 \, t\right)} - 5 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0489" title="C4 | Homogeneous second-order linear IVP | ver. 0489"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7" alt="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7" title="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7" data-latex="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27" alt="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27" title="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27" data-latex="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%20-15%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-7" alt="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7" title="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7" data-latex="-{x''} = -15 \, {x} + 2 \, {x'} \hspace{2em} {x} (0)= 3 , {x} '(0)= -7">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2081%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%2027" alt="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27" title="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27" data-latex="{y''} + 81 \, {y} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 27">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" alt="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="{x} = e^{\left(3 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4001" title="C4 | Homogeneous second-order linear IVP | ver. 4001"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0" alt="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0" title="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0" data-latex="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4" alt="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4" title="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4" data-latex="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%20-6%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%200" alt="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0" title="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0" data-latex="-5 \, {x} - {x''} = -6 \, {x'} \hspace{2em} {x} (0)= 4 , {x} '(0)= 0">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By''%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%204" alt="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4" title="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4" data-latex="0 = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}" alt="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}" title="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}" data-latex="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)" alt="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)" title="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7Bt%7D" alt="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}" title="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}" data-latex="{x} = -e^{\left(5 \, t\right)} + 5 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)" title="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)" data-latex="{y} = \cos\left(2 \, t\right) + 2 \, \sin\left(2 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4518" title="C4 | Homogeneous second-order linear IVP | ver. 4518"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28" alt="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28" title="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28" data-latex="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17" alt="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17" title="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17" data-latex="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-49%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%2028" alt="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28" title="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28" data-latex="{x''} = -49 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= 28">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7By%7D%20-%20%7By''%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-6%20,%20%7By%7D%20'(0)=%2017" alt="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17" title="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17" data-latex="12 \, {y} - {y''} = {y'} \hspace{2em} {y} (0)= -6 , {y} '(0)= 17">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)" alt="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)" title="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)" data-latex="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)"/></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} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}" alt="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}" title="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)" title="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)" data-latex="{x} = 2 \, \cos\left(7 \, t\right) + 4 \, \sin\left(7 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}" title="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -e^{\left(3 \, t\right)} - 5 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1154" title="C4 | Homogeneous second-order linear IVP | ver. 1154"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27" alt="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27" title="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27" data-latex="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" alt="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" title="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" data-latex="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%206%20%5C,%20%7Bx'%7D%20=%20-5%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-7%20,%20%7Bx%7D%20'(0)=%2027" alt="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27" title="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27" data-latex="{x''} + 6 \, {x'} = -5 \, {x} \hspace{2em} {x} (0)= -7 , {x} '(0)= 27">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-81%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-36" alt="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" title="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36" data-latex="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -36">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" alt="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" alt="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" title="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)" data-latex="{y} = 5 \, \cos\left(9 \, t\right) - 4 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1761" title="C4 | Homogeneous second-order linear IVP | ver. 1761"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5" alt="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5" title="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5" data-latex="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10" alt="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10" title="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10" data-latex="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%207%20%5C,%20%7Bx'%7D%20-%20%7Bx''%7D%20-%2010%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%20-5" alt="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5" title="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5" data-latex="0 = 7 \, {x'} - {x''} - 10 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -5">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%2025%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%2010" alt="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10" title="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10" data-latex="-{y''} - 25 \, {y} = 0 \hspace{2em} {y} (0)= 3 , {y} '(0)= 10">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -e^{\left(5 \, t\right)}" alt="{x} = -e^{\left(5 \, t\right)}" title="{x} = -e^{\left(5 \, t\right)}" data-latex="{x} = -e^{\left(5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)" alt="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)" title="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{x} = -e^{\left(5 \, t\right)}" title="{x} = -e^{\left(5 \, t\right)}" data-latex="{x} = -e^{\left(5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)" title="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)" data-latex="{y} = 3 \, \cos\left(5 \, t\right) + 2 \, \sin\left(5 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4560" title="C4 | Homogeneous second-order linear IVP | ver. 4560"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13" alt="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13" title="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13" data-latex="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9" alt="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9" title="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9" data-latex="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20-%204%20%5C,%20%7Bx%7D%20=%20-3%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-13" alt="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13" title="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13" data-latex="{x''} - 4 \, {x} = -3 \, {x'} \hspace{2em} {x} (0)= 2 , {x} '(0)= -13">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-81%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%20-9" alt="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9" title="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9" data-latex="{y''} = -81 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -9">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}" alt="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}" title="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}" data-latex="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -\sin\left(9 \, t\right)" alt="{y} = -\sin\left(9 \, t\right)" title="{y} = -\sin\left(9 \, t\right)" data-latex="{y} = -\sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}" title="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}" data-latex="{x} = 3 \, e^{\left(-4 \, t\right)} - e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = -\sin\left(9 \, t\right)" title="{y} = -\sin\left(9 \, t\right)" data-latex="{y} = -\sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2037" title="C4 | Homogeneous second-order linear IVP | ver. 2037"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18" alt="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18" title="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18" data-latex="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%203" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= 3">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%2018" alt="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18" title="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18" data-latex="9 \, {y} = {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" alt="{x} = -3 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" title="{x} = -3 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" data-latex="{x} = -3 \, \cos\left(3 \, t\right) + \sin\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} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}" alt="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}" title="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" title="{x} = -3 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" data-latex="{x} = -3 \, \cos\left(3 \, t\right) + \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}" title="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} - 5 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4146" title="C4 | Homogeneous second-order linear IVP | ver. 4146"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12" alt="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12" title="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12" data-latex="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" alt="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" title="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" data-latex="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-12" alt="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12" title="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12" data-latex="-36 \, {x} = {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20-%206%20%5C,%20%7By'%7D%20=%20-8%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%20-8" alt="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" title="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" data-latex="{y''} - 6 \, {y'} = -8 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= -8">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" alt="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)"/></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(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" title="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)" data-latex="{x} = 2 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)">
</p>
<p style="text-align:center;">
<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(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9642" title="C4 | Homogeneous second-order linear IVP | ver. 9642"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35" alt="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35" title="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35" data-latex="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17" alt="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17" title="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17" data-latex="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%2049%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%20-35" alt="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35" title="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35" data-latex="-{x''} = 49 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -35">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20+%204%20%5C,%20%7By%7D%20=%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%208%20,%20%7By%7D%20'(0)=%20-17" alt="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17" title="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17" data-latex="-{y''} + 4 \, {y} = 3 \, {y'} \hspace{2em} {y} (0)= 8 , {y} '(0)= -17">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" alt="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" title="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" data-latex="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}" alt="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}" title="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}" data-latex="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" title="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)" data-latex="{x} = -3 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}" title="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}" data-latex="{y} = 5 \, e^{\left(-4 \, t\right)} + 3 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9907" title="C4 | Homogeneous second-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>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3" alt="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3" title="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3" data-latex="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30" alt="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30" title="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30" data-latex="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%2012%20%5C,%20%7Bx%7D%20=%207%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%203" alt="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3" title="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3" data-latex="-{x''} - 12 \, {x} = 7 \, {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= 3">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20-%20100%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%20-30" alt="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30" title="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30" data-latex="-{y''} - 100 \, {y} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -30">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" alt="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" title="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" data-latex="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" alt="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" title="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}" data-latex="{x} = -5 \, e^{\left(-3 \, t\right)} + 3 \, e^{\left(-4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = -\cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0385" title="C4 | Homogeneous second-order linear IVP | ver. 0385"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" alt="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" title="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" data-latex="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%203" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 3">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By%7D%20-%206%20%5C,%20%7By'%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%20-16" alt="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" title="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16" data-latex="8 \, {y} - 6 \, {y'} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= -16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" alt="{x} = 5 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" title="{x} = 5 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" data-latex="{x} = 5 \, \cos\left(3 \, t\right) + \sin\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(4 \, t\right)}" alt="{y} = -4 \, e^{\left(4 \, t\right)}" title="{y} = -4 \, e^{\left(4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = 5 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" title="{x} = 5 \, \cos\left(3 \, t\right) + \sin\left(3 \, t\right)" data-latex="{x} = 5 \, \cos\left(3 \, t\right) + \sin\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?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(4 \, t\right)}" title="{y} = -4 \, e^{\left(4 \, t\right)}" data-latex="{y} = -4 \, e^{\left(4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7053" title="C4 | Homogeneous second-order linear IVP | ver. 7053"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19" alt="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19" title="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19" data-latex="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7" alt="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7" title="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7" data-latex="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-19" alt="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19" title="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19" data-latex="6 \, {x} = {x'} + {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -19">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49%20%5C,%20%7By%7D%20+%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%207" alt="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7" title="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7" data-latex="49 \, {y} + {y''} = 0 \hspace{2em} {y} (0)= 5 , {y} '(0)= 7">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" alt="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, 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} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)" alt="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)" title="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)" data-latex="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" title="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}" data-latex="{x} = -5 \, e^{\left(2 \, t\right)} + 3 \, 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?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)" title="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)" data-latex="{y} = 5 \, \cos\left(7 \, t\right) + \sin\left(7 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9197" title="C4 | Homogeneous second-order linear IVP | ver. 9197"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" alt="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" title="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" data-latex="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15" alt="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15" title="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15" data-latex="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-4%20,%20%7Bx%7D%20'(0)=%20-6" alt="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" title="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6" data-latex="-{x} = -{x''} \hspace{2em} {x} (0)= -4 , {x} '(0)= -6">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20+%2025%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%204%20,%20%7By%7D%20'(0)=%20-15" alt="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15" title="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15" data-latex="{y''} + 25 \, {y} = 0 \hspace{2em} {y} (0)= 4 , {y} '(0)= -15">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = e^{\left(-t\right)} - 5 \, e^{t}" alt="{x} = e^{\left(-t\right)} - 5 \, e^{t}" title="{x} = e^{\left(-t\right)} - 5 \, e^{t}" data-latex="{x} = e^{\left(-t\right)} - 5 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" alt="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%205%20%5C,%20e%5E%7Bt%7D" alt="{x} = e^{\left(-t\right)} - 5 \, e^{t}" title="{x} = e^{\left(-t\right)} - 5 \, e^{t}" data-latex="{x} = e^{\left(-t\right)} - 5 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%204%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{y} = 4 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7888" title="C4 | Homogeneous second-order linear IVP | ver. 7888"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21" alt="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21" title="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21" data-latex="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0" alt="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0" title="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0" data-latex="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7Bx'%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-9%20,%20%7Bx%7D%20'(0)=%20-21" alt="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21" title="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21" data-latex="5 \, {x} + 4 \, {x'} = {x''} \hspace{2em} {x} (0)= -9 , {x} '(0)= -21">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-49%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%200" alt="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0" title="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0" data-latex="-49 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= 0">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}" alt="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, \cos\left(7 \, t\right)" alt="{y} = 2 \, \cos\left(7 \, t\right)" title="{y} = 2 \, \cos\left(7 \, t\right)" data-latex="{y} = 2 \, \cos\left(7 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20%5Ccos%5Cleft(7%20%5C,%20t%5Cright)" alt="{y} = 2 \, \cos\left(7 \, t\right)" title="{y} = 2 \, \cos\left(7 \, t\right)" data-latex="{y} = 2 \, \cos\left(7 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6990" title="C4 | Homogeneous second-order linear IVP | ver. 6990"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20" alt="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20" title="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20" data-latex="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8" alt="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8" title="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8" data-latex="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-16%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%2020" alt="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20" title="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20" data-latex="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -2 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20-%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%208" alt="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8" title="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8" data-latex="{y} - {y''} = 0 \hspace{2em} {y} (0)= 0 , {y} '(0)= 8">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" alt="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}" alt="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}" title="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}" data-latex="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%205%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{x} = -2 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%204%20%5C,%20e%5E%7Bt%7D" alt="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}" title="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}" data-latex="{y} = -4 \, e^{\left(-t\right)} + 4 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6755" title="C4 | Homogeneous second-order linear IVP | ver. 6755"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24" alt="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24" title="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24" data-latex="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25" alt="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25" title="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25" data-latex="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2064%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%2024" alt="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24" title="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24" data-latex="0 = 64 \, {x} + {x''} \hspace{2em} {x} (0)= -5 , {x} '(0)= 24">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-15%20%5C,%20%7By%7D%20=%20-8%20%5C,%20%7By'%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%2025" alt="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25" title="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25" data-latex="-15 \, {y} = -8 \, {y'} + {y''} \hspace{2em} {y} (0)= 5 , {y} '(0)= 25">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" alt="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" title="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" data-latex="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, e^{\left(5 \, t\right)}" alt="{y} = 5 \, e^{\left(5 \, t\right)}" title="{y} = 5 \, e^{\left(5 \, t\right)}" data-latex="{y} = 5 \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" title="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" data-latex="{x} = -5 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="{y} = 5 \, e^{\left(5 \, t\right)}" title="{y} = 5 \, e^{\left(5 \, t\right)}" data-latex="{y} = 5 \, e^{\left(5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5438" title="C4 | Homogeneous second-order linear IVP | ver. 5438"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" alt="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2" alt="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2" title="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2" data-latex="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%20100%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%2020" alt="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="{x''} + 100 \, {x} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%203%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%20-2" alt="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2" title="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2" data-latex="{y''} = 3 \, {y'} - 2 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -2">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \sin\left(10 \, t\right)" alt="{x} = 2 \, \sin\left(10 \, t\right)" title="{x} = 2 \, \sin\left(10 \, t\right)" data-latex="{x} = 2 \, \sin\left(10 \, t\right)"/></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)} + 2 \, e^{t}" alt="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" title="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{x} = 2 \, \sin\left(10 \, t\right)" title="{x} = 2 \, \sin\left(10 \, t\right)" data-latex="{x} = 2 \, \sin\left(10 \, t\right)">
</p>
<p style="text-align:center;">
<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%20+%202%20%5C,%20e%5E%7Bt%7D" alt="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" title="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" data-latex="{y} = -2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9091" title="C4 | Homogeneous second-order linear IVP | ver. 9091"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6" alt="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6" title="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6" data-latex="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30" alt="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30" title="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30" data-latex="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx''%7D%20-%206%20%5C,%20%7Bx'%7D%20+%208%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%20-6" alt="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6" title="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6" data-latex="0 = {x''} - 6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 0 , {x} '(0)= -6">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-100%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%205%20,%20%7By%7D%20'(0)=%20-30" alt="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30" title="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30" data-latex="{y''} = -100 \, {y} \hspace{2em} {y} (0)= 5 , {y} '(0)= -30">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, e^{\left(2 \, t\right)}" alt="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, e^{\left(2 \, t\right)}" title="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, e^{\left(2 \, t\right)}" data-latex="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, 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} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" alt="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, e^{\left(2 \, t\right)}" title="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, e^{\left(2 \, t\right)}" data-latex="{x} = -3 \, e^{\left(4 \, t\right)} + 3 \, 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?%7By%7D%20=%205%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" title="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)" data-latex="{y} = 5 \, \cos\left(10 \, t\right) - 3 \, \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8725" title="C4 | Homogeneous second-order linear IVP | ver. 8725"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5" alt="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5" title="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5" data-latex="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24" alt="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24" title="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24" data-latex="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?25%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-5" alt="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5" title="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5" data-latex="25 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -5">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By'%7D%20+%205%20%5C,%20%7By%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-4%20,%20%7By%7D%20'(0)=%2024" alt="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24" title="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24" data-latex="6 \, {y'} + 5 \, {y} = -{y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 24">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" alt="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" title="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" data-latex="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" alt="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" title="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)" data-latex="{x} = 2 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" title="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}" data-latex="{y} = e^{\left(-t\right)} - 5 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7802" title="C4 | Homogeneous second-order linear IVP | ver. 7802"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2" alt="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2" title="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2" data-latex="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21" alt="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21" title="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21" data-latex="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%20-2" alt="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2" title="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2" data-latex="-{x''} = 4 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -2">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By'%7D%20=%2020%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%2021" alt="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21" title="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21" data-latex="-9 \, {y'} = 20 \, {y} + {y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= 21">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" alt="{x} = -3 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" title="{x} = -3 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" data-latex="{x} = -3 \, \cos\left(2 \, t\right) - \sin\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(-4 \, t\right)} - e^{\left(-5 \, t\right)}" alt="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}" title="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" title="{x} = -3 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" data-latex="{x} = -3 \, \cos\left(2 \, t\right) - \sin\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?%7By%7D%20=%20-4%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}" title="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="{y} = -4 \, e^{\left(-4 \, t\right)} - e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2317" title="C4 | Homogeneous second-order linear IVP | ver. 2317"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24" alt="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24" title="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24" data-latex="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3" alt="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3" title="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3" data-latex="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?64%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%204%20,%20%7Bx%7D%20'(0)=%2024" alt="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24" title="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24" data-latex="64 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 4 , {x} '(0)= 24">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20+%203%20%5C,%20%7By'%7D%20=%20-10%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%203" alt="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3" title="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3" data-latex="-{y''} + 3 \, {y'} = -10 \, {y} \hspace{2em} {y} (0)= -5 , {y} '(0)= 3">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" alt="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" title="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" data-latex="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)"/></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} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" alt="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%204%20%5C,%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" title="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)" data-latex="{x} = 4 \, \cos\left(8 \, t\right) + 3 \, \sin\left(8 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -e^{\left(5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9081" title="C4 | Homogeneous second-order linear IVP | ver. 9081"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3" alt="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3" title="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3" data-latex="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" alt="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" title="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" data-latex="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20-6%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%20-3" alt="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3" title="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3" data-latex="-{x'} = -6 \, {x} + {x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -3">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2016%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%20-12" alt="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" title="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" data-latex="0 = 16 \, {y} + {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = e^{\left(-3 \, t\right)}" alt="{x} = e^{\left(-3 \, t\right)}" title="{x} = e^{\left(-3 \, t\right)}" data-latex="{x} = 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 \, \sin\left(4 \, t\right)" alt="{y} = -3 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{x} = e^{\left(-3 \, t\right)}" title="{x} = e^{\left(-3 \, t\right)}" data-latex="{x} = 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?%7By%7D%20=%20-3%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9603" title="C4 | Homogeneous second-order linear IVP | ver. 9603"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9" alt="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9" title="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9" data-latex="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35" alt="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35" title="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35" data-latex="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%20-9" alt="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9" title="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9" data-latex="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= -9">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2015%20%5C,%20%7By%7D%20-%20%7By''%7D%20-%202%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%2035" alt="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35" title="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35" data-latex="0 = 15 \, {y} - {y''} - 2 \, {y'} \hspace{2em} {y} (0)= 1 , {y} '(0)= 35">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" alt="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" title="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" data-latex="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" title="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)" data-latex="{x} = \cos\left(9 \, t\right) - \sin\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 5 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3202" title="C4 | Homogeneous second-order linear IVP | ver. 3202"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5" alt="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5" title="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5" data-latex="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15" alt="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15" title="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15" data-latex="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%203%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7Bx'%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%20-5" alt="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5" title="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5" data-latex="0 = 3 \, {x} + 2 \, {x'} - {x''} \hspace{2em} {x} (0)= -3 , {x} '(0)= -5">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By''%7D%20+%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%2015" alt="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15" title="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15" data-latex="0 = {y''} + 9 \, {y} \hspace{2em} {y} (0)= -1 , {y} '(0)= 15">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}" alt="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)" alt="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)" title="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)" data-latex="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}" title="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}" data-latex="{x} = -2 \, e^{\left(3 \, t\right)} - e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20+%205%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)" title="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)" data-latex="{y} = -\cos\left(3 \, t\right) + 5 \, \sin\left(3 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8607" title="C4 | Homogeneous second-order linear IVP | ver. 8607"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" alt="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" title="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" data-latex="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" alt="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" title="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" data-latex="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-2" alt="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" title="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2" data-latex="-4 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -2">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By'%7D%20-%20%7By''%7D%20=%20-3%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%204" alt="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" title="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4" data-latex="2 \, {y'} - {y''} = -3 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" alt="{x} = -2 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" title="{x} = -2 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" data-latex="{x} = -2 \, \cos\left(2 \, t\right) - \sin\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} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}" alt="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}" title="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = -2 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" title="{x} = -2 \, \cos\left(2 \, t\right) - \sin\left(2 \, t\right)" data-latex="{x} = -2 \, \cos\left(2 \, t\right) - \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}" title="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}" data-latex="{y} = e^{\left(3 \, t\right)} - e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3565" title="C4 | Homogeneous second-order linear IVP | ver. 3565"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" alt="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" title="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" data-latex="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12" alt="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12" title="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12" data-latex="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%2020" alt="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" title="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20" data-latex="-100 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -3 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7By'%7D%20=%20%7By''%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%20-12" alt="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12" title="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12" data-latex="-5 \, {y'} = {y''} + 4 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= -12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" alt="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" title="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" data-latex="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)"/></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(-4 \, t\right)}" alt="{y} = 3 \, e^{\left(-4 \, t\right)}" title="{y} = 3 \, e^{\left(-4 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" title="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" data-latex="{x} = -3 \, \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)">
</p>
<p style="text-align:center;">
<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(-4%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-4 \, t\right)}" title="{y} = 3 \, e^{\left(-4 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9522" title="C4 | Homogeneous second-order linear IVP | ver. 9522"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10" alt="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10" title="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10" data-latex="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" alt="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" title="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" data-latex="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7Bx'%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%2010" alt="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10" title="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10" data-latex="2 \, {x} - 3 \, {x'} = -{x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= 10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%2016%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%2020" alt="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" title="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20" data-latex="-{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= 20">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, e^{\left(2 \, t\right)}" alt="{x} = 5 \, e^{\left(2 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 5 \, 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 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" alt="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 5 \, e^{\left(2 \, t\right)}" title="{x} = 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 5 \, 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?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%205%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) + 5 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6253" title="C4 | Homogeneous second-order linear IVP | ver. 6253"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14" alt="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14" title="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14" data-latex="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18" alt="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18" title="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18" data-latex="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%208%20%5C,%20%7Bx'%7D%20=%2015%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-14" alt="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14" title="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14" data-latex="-{x''} - 8 \, {x'} = 15 \, {x} \hspace{2em} {x} (0)= 2 , {x} '(0)= -14">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7By''%7D%20+%2081%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%2018" alt="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18" title="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18" data-latex="0 = {y''} + 81 \, {y} \hspace{2em} {y} (0)= 3 , {y} '(0)= 18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" alt="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" title="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" data-latex="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-3 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" title="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)" data-latex="{y} = 3 \, \cos\left(9 \, t\right) + 2 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9607" title="C4 | Homogeneous second-order linear IVP | ver. 9607"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21" alt="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21" title="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21" data-latex="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" alt="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" title="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" data-latex="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?49%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%20-21" alt="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21" title="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21" data-latex="49 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= -21">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By'%7D%20-%208%20%5C,%20%7By%7D%20-%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%20-8" alt="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" title="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8" data-latex="6 \, {y'} - 8 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -1 , {y} '(0)= -8">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \sin\left(7 \, t\right)" alt="{x} = -3 \, \sin\left(7 \, t\right)" title="{x} = -3 \, \sin\left(7 \, t\right)" data-latex="{x} = -3 \, \sin\left(7 \, t\right)"/></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(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Csin%5Cleft(7%20%5C,%20t%5Cright)" alt="{x} = -3 \, \sin\left(7 \, t\right)" title="{x} = -3 \, \sin\left(7 \, t\right)" data-latex="{x} = -3 \, \sin\left(7 \, t\right)">
</p>
<p style="text-align:center;">
<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(4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 2 \, e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9616" title="C4 | Homogeneous second-order linear IVP | ver. 9616"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" alt="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20" alt="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20" title="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20" data-latex="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20-%202%20%5C,%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%2020" alt="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" title="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20" data-latex="-3 \, {x} + {x''} - 2 \, {x'} = 0 \hspace{2em} {x} (0)= 0 , {x} '(0)= 20">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-16%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-20" alt="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20" title="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20" data-latex="{y''} = -16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -20">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}" alt="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}" title="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}" data-latex="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)" alt="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}" title="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}" data-latex="{x} = 5 \, e^{\left(3 \, t\right)} - 5 \, e^{\left(-t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9735" title="C4 | Homogeneous second-order linear IVP | ver. 9735"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6" alt="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6" title="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6" data-latex="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17" alt="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17" title="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17" data-latex="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-9%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%20-6" alt="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6" title="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6" data-latex="{x''} = -9 \, {x} \hspace{2em} {x} (0)= 3 , {x} '(0)= -6">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By''%7D%20=%20-6%20%5C,%20%7By'%7D%20+%205%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%2017" alt="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17" title="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17" data-latex="-{y''} = -6 \, {y'} + 5 \, {y} \hspace{2em} {y} (0)= 1 , {y} '(0)= 17">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" alt="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" title="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" data-latex="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\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(5 \, t\right)} - 3 \, e^{t}" alt="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}" title="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}" data-latex="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" title="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\left(3 \, t\right)" data-latex="{x} = 3 \, \cos\left(3 \, t\right) - 2 \, \sin\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?%7By%7D%20=%204%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7Bt%7D" alt="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}" title="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}" data-latex="{y} = 4 \, e^{\left(5 \, t\right)} - 3 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7157" title="C4 | Homogeneous second-order linear IVP | ver. 7157"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36" alt="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36" title="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36" data-latex="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2" alt="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2" title="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2" data-latex="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20-%2081%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%2036" alt="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36" title="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36" data-latex="-{x''} - 81 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= 36">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By'%7D%20+%20%7By''%7D%20=%20-20%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%202" alt="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2" title="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2" data-latex="9 \, {y'} + {y''} = -20 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= 2">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)" alt="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)" title="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)" data-latex="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)"/></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(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" alt="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)" title="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)" data-latex="{x} = 2 \, \cos\left(9 \, t\right) + 4 \, \sin\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<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(-4%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="{y} = 2 \, e^{\left(-4 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-1545" title="C4 | Homogeneous second-order linear IVP | ver. 1545"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8" alt="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8" title="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8" data-latex="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14" alt="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14" title="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14" data-latex="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20=%20-16%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%20-8" alt="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8" title="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8" data-latex="{x''} = -16 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%20%7By''%7D%20-%2012%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%20-14" alt="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14" title="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14" data-latex="-{y'} = {y''} - 12 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -14">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)" alt="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)" title="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)" data-latex="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)" title="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)" data-latex="{x} = -3 \, \cos\left(4 \, t\right) - 2 \, \sin\left(4 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-3955" title="C4 | Homogeneous second-order linear IVP | ver. 3955"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8" alt="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8" title="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8" data-latex="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4" alt="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4" title="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4" data-latex="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx''%7D%20-%208%20%5C,%20%7Bx%7D%20-%206%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-3%20,%20%7Bx%7D%20'(0)=%208" alt="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8" title="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8" data-latex="0 = -{x''} - 8 \, {x} - 6 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 8">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20%7By%7D%20-%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%204" alt="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4" title="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4" data-latex="-16 \, {y} - {y''} = 0 \hspace{2em} {y} (0)= -2 , {y} '(0)= 4">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}" alt="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}" title="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)" alt="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)" title="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)" data-latex="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}" title="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}" data-latex="{x} = -2 \, e^{\left(-2 \, t\right)} - e^{\left(-4 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)" title="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)" data-latex="{y} = -2 \, \cos\left(4 \, t\right) + \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2415" title="C4 | Homogeneous second-order linear IVP | ver. 2415"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33" alt="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33" title="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33" data-latex="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20" alt="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20" title="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20" data-latex="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx'%7D%20+%2010%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-1%20,%20%7Bx%7D%20'(0)=%20-33" alt="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33" title="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33" data-latex="3 \, {x'} + 10 \, {x} = {x''} \hspace{2em} {x} (0)= -1 , {x} '(0)= -33">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-100%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%201%20,%20%7By%7D%20'(0)=%2020" alt="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20" title="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20" data-latex="-100 \, {y} = {y''} \hspace{2em} {y} (0)= 1 , {y} '(0)= 20">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, 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} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" alt="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" title="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" data-latex="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="{x} = -5 \, e^{\left(5 \, t\right)} + 4 \, 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?%7By%7D%20=%20%5Ccos%5Cleft(10%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" title="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)" data-latex="{y} = \cos\left(10 \, t\right) + 2 \, \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5661" title="C4 | Homogeneous second-order linear IVP | ver. 5661"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10" alt="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10" title="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10" data-latex="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" alt="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" title="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" data-latex="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx''%7D%20+%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-10" alt="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10" title="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10" data-latex="{x''} + 4 \, {x} = 0 \hspace{2em} {x} (0)= 2 , {x} '(0)= -10">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7By%7D%20+%20%7By'%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%20-13" alt="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" title="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13" data-latex="0 = 2 \, {y} + {y'} - {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" alt="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" title="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" data-latex="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\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} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}" alt="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}" title="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" title="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\left(2 \, t\right)" data-latex="{x} = 2 \, \cos\left(2 \, t\right) - 5 \, \sin\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?%7By%7D%20=%20-5%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}" title="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}" data-latex="{y} = -5 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-4846" title="C4 | Homogeneous second-order linear IVP | ver. 4846"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0" alt="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0" title="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0" data-latex="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" alt="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-81%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%200" alt="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0" title="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0" data-latex="-81 \, {x} - {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= 0">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-16%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%20-32" alt="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="0 = -16 \, {y} + {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -5 \, \cos\left(9 \, t\right)" alt="{x} = -5 \, \cos\left(9 \, t\right)" title="{x} = -5 \, \cos\left(9 \, t\right)" data-latex="{x} = -5 \, \cos\left(9 \, t\right)"/></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(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-5%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(9 \, t\right)" title="{x} = -5 \, \cos\left(9 \, t\right)" data-latex="{x} = -5 \, \cos\left(9 \, t\right)">
</p>
<p style="text-align:center;">
<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(4%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" title="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = -3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-5530" title="C4 | Homogeneous second-order linear IVP | ver. 5530"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4" alt="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4" title="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4" data-latex="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" alt="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" title="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" data-latex="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20=%20-3%20%5C,%20%7Bx'%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%20-4" alt="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4" title="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4" data-latex="2 \, {x} = -3 \, {x'} - {x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= -4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By''%7D%20=%20-9%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%20-12" alt="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" title="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12" data-latex="{y''} = -9 \, {y} \hspace{2em} {y} (0)= 0 , {y} '(0)= -12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(-2 \, t\right)}" alt="{x} = 2 \, e^{\left(-2 \, t\right)}" title="{x} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{x} = 2 \, 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 \, \sin\left(3 \, t\right)" alt="{y} = -4 \, \sin\left(3 \, t\right)" title="{y} = -4 \, \sin\left(3 \, t\right)" data-latex="{y} = -4 \, \sin\left(3 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(-2 \, t\right)}" title="{x} = 2 \, e^{\left(-2 \, t\right)}" data-latex="{x} = 2 \, 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?%7By%7D%20=%20-4%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{y} = -4 \, \sin\left(3 \, t\right)" title="{y} = -4 \, \sin\left(3 \, t\right)" data-latex="{y} = -4 \, \sin\left(3 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9248" title="C4 | Homogeneous second-order linear IVP | ver. 9248"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4" alt="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4" title="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4" data-latex="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" alt="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" title="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" data-latex="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx'%7D%20-%2010%20%5C,%20%7Bx%7D%20=%20-%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%202%20,%20%7Bx%7D%20'(0)=%204" alt="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4" title="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4" data-latex="3 \, {x'} - 10 \, {x} = -{x''} \hspace{2em} {x} (0)= 2 , {x} '(0)= 4">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-2%20,%20%7By%7D%20'(0)=%2012" alt="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" title="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12" data-latex="-16 \, {y} = {y''} \hspace{2em} {y} (0)= -2 , {y} '(0)= 12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(2 \, t\right)}" alt="{x} = 2 \, e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, 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 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" alt="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" title="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" data-latex="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, 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?%7By%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" title="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)" data-latex="{y} = -2 \, \cos\left(4 \, t\right) + 3 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-7042" title="C4 | Homogeneous second-order linear IVP | ver. 7042"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18" alt="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18" title="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18" data-latex="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-12" alt="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" title="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12" data-latex="-9 \, {x} = {x''} \hspace{2em} {x} (0)= -2 , {x} '(0)= -12">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By'%7D%20=%20%7By''%7D%20-%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%206%20,%20%7By%7D%20'(0)=%20-18" alt="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18" title="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18" data-latex="-2 \, {y'} = {y''} - 8 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= -18">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\left(3 \, t\right)" alt="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\left(3 \, t\right)" title="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\left(3 \, t\right)" data-latex="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\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} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" alt="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" title="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20-2%20%5C,%20%5Ccos%5Cleft(3%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)" alt="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\left(3 \, t\right)" title="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\left(3 \, t\right)" data-latex="{x} = -2 \, \cos\left(3 \, t\right) - 4 \, \sin\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?%7By%7D%20=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" title="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}" data-latex="{y} = e^{\left(2 \, t\right)} + 5 \, e^{\left(-4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-9833" title="C4 | Homogeneous second-order linear IVP | ver. 9833"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22" alt="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22" title="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22" data-latex="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27" alt="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27" title="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27" data-latex="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx''%7D%20=%20-6%20%5C,%20%7Bx'%7D%20+%208%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%208%20,%20%7Bx%7D%20'(0)=%2022" alt="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22" title="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22" data-latex="-{x''} = -6 \, {x'} + 8 \, {x} \hspace{2em} {x} (0)= 8 , {x} '(0)= 22">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2081%20%5C,%20%7By%7D%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%2027" alt="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27" title="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27" data-latex="0 = 81 \, {y} + {y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= 27">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" alt="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" title="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, 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 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" alt="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" title="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" data-latex="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%205%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" title="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, e^{\left(2 \, t\right)}" data-latex="{x} = 3 \, e^{\left(4 \, t\right)} + 5 \, 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?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(9%20%5C,%20t%5Cright)%20+%203%20%5C,%20%5Csin%5Cleft(9%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" title="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)" data-latex="{y} = -3 \, \cos\left(9 \, t\right) + 3 \, \sin\left(9 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-0743" title="C4 | Homogeneous second-order linear IVP | ver. 0743"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5" alt="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5" title="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5" data-latex="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20" alt="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20" title="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20" data-latex="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%2012%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20-%207%20%5C,%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%200%20,%20%7Bx%7D%20'(0)=%205" alt="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5" title="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5" data-latex="0 = 12 \, {x} + {x''} - 7 \, {x'} \hspace{2em} {x} (0)= 0 , {x} '(0)= 5">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-100%20%5C,%20%7By%7D%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%200%20,%20%7By%7D%20'(0)=%2020" alt="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20" title="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20" data-latex="0 = -100 \, {y} - {y''} \hspace{2em} {y} (0)= 0 , {y} '(0)= 20">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, e^{\left(3 \, t\right)}" alt="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, e^{\left(3 \, t\right)}" title="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, e^{\left(3 \, t\right)}" data-latex="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, 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 \, \sin\left(10 \, t\right)" alt="{y} = 2 \, \sin\left(10 \, t\right)" title="{y} = 2 \, \sin\left(10 \, t\right)" data-latex="{y} = 2 \, \sin\left(10 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, e^{\left(3 \, t\right)}" title="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, e^{\left(3 \, t\right)}" data-latex="{x} = 5 \, e^{\left(4 \, t\right)} - 5 \, 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?%7By%7D%20=%202%20%5C,%20%5Csin%5Cleft(10%20%5C,%20t%5Cright)" alt="{y} = 2 \, \sin\left(10 \, t\right)" title="{y} = 2 \, \sin\left(10 \, t\right)" data-latex="{y} = 2 \, \sin\left(10 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-8296" title="C4 | Homogeneous second-order linear IVP | ver. 8296"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13" alt="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13" title="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13" data-latex="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16" alt="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16" title="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16" data-latex="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7Bx'%7D%20=%20%7Bx''%7D%20-%2020%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%2013" alt="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13" title="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13" data-latex="-{x'} = {x''} - 20 \, {x} \hspace{2em} {x} (0)= 1 , {x} '(0)= 13">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20-%2016%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-3%20,%20%7By%7D%20'(0)=%20-16" alt="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16" title="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16" data-latex="0 = -{y''} - 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -16">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}" alt="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" alt="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} - e^{\left(-5 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20-3%20%5C,%20%5Ccos%5Cleft(4%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(4%20%5C,%20t%5Cright)" alt="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" title="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)" data-latex="{y} = -3 \, \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-6301" title="C4 | Homogeneous second-order linear IVP | ver. 6301"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11" alt="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11" title="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11" data-latex="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" alt="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" title="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" data-latex="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-2%20,%20%7Bx%7D%20'(0)=%20-11" alt="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11" title="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11" data-latex="2 \, {x} = {x''} + {x'} \hspace{2em} {x} (0)= -2 , {x} '(0)= -11">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-36%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%203%20,%20%7By%7D%20'(0)=%2012" alt="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" title="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12" data-latex="-36 \, {y} = {y''} \hspace{2em} {y} (0)= 3 , {y} '(0)= 12">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}" alt="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}" title="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}" data-latex="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" alt="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20e%5E%7Bt%7D" alt="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}" title="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}" data-latex="{x} = 3 \, e^{\left(-2 \, t\right)} - 5 \, e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20%5Ccos%5Cleft(6%20%5C,%20t%5Cright)%20+%202%20%5C,%20%5Csin%5Cleft(6%20%5C,%20t%5Cright)" alt="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" title="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)" data-latex="{y} = 3 \, \cos\left(6 \, t\right) + 2 \, \sin\left(6 \, t\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="C4-2432" title="C4 | Homogeneous second-order linear IVP | ver. 2432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>C4.</strong></p><p>Explain how to find the particular solution to each given IVP.</p><ol type="a"><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25" alt="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25" title="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25" data-latex="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25"/></p></li><li><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2" alt="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2" title="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2" data-latex="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2"/></p></li></ol></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C4.</strong>
</p>
<p>Explain how to find the particular solution to each given IVP.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-25%20%5C,%20%7Bx%7D%20-%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%205%20,%20%7Bx%7D%20'(0)=%20-25" alt="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25" title="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25" data-latex="0 = -25 \, {x} - {x''} \hspace{2em} {x} (0)= 5 , {x} '(0)= -25">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By''%7D%20+%202%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%206%20,%20%7By%7D%20'(0)=%202" alt="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2" title="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2" data-latex="0 = -{y''} + 2 \, {y'} + 3 \, {y} \hspace{2em} {y} (0)= 6 , {y} '(0)= 2">
</p>
</li>
</ol>
</div>
</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" alt="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" alt="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%205%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%205%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" title="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)" data-latex="{x} = 5 \, \cos\left(5 \, t\right) - 5 \, \sin\left(5 \, t\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>
