<?xml version='1.0' encoding='UTF-8'?>
<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
<objectbank ident="X1">
<qtimetadata>
<qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- X1</fieldentry></qtimetadatafield>
</qtimetadata>
<item ident="X1-4974" title="X1 | Linear ODE systems | ver. 4974"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5" alt="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5" title="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5" data-latex="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0" alt="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0" title="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0" data-latex="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx%7D%20+%20%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5" title="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5" data-latex="0 = {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5">
</p>
<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%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0" title="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0" data-latex="-{x} + 2 \, {y} = {y'} \hspace{2em}x(0)= 0">
</p>
</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(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, 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= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, 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?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, 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=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6783" title="X1 | Linear ODE systems | ver. 6783"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5" alt="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5" title="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5" data-latex="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8" alt="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8" title="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8" data-latex="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5" title="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5" data-latex="0 = -4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%208" alt="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8" title="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8" data-latex="9 \, {x} + {y'} = {y} \hspace{2em}x(0)= 8">
</p>
</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)} - e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - e^{\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= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} - e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - e^{\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=%209%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9808" title="X1 | Linear ODE systems | ver. 9808"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3" alt="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3" title="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3" data-latex="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3"/></p><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} - 2 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3" title="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3" data-latex="{x} = 2 \, {y} + {x'} \hspace{2em}x(0)= 3">
</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=%20-6%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = -6 \, {y} - 2 \, {x} \hspace{2em}x(0)= -3">
</p>
</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)} + 2 \, e^{\left(2 \, t\right)}" alt="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, 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= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="y= -2 \, 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?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, 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=%20-2%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= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} - e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9721" title="X1 | Linear ODE systems | ver. 9721"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3" alt="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3" title="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3" data-latex="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2" alt="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2" title="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2" data-latex="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%204%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3" title="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3" data-latex="{x'} - {x} - 4 \, {y} = 0 \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2" title="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2" data-latex="2 \, {y} = {x} - {y'} \hspace{2em}x(0)= 2">
</p>
</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(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 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?x=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= 4 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4959" title="X1 | Linear ODE systems | ver. 4959"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5" alt="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5" title="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5" data-latex="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5"/></p><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} + {x} \hspace{2em}x(0)= 0" alt="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0" title="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0" data-latex="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5" title="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5" data-latex="-{x'} - 4 \, {y} = {x} \hspace{2em}x(0)= -5">
</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%7By%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0" title="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0" data-latex="-{y'} = -2 \, {y} + {x} \hspace{2em}x(0)= 0">
</p>
</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(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, 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?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-3192" title="X1 | Linear ODE systems | ver. 3192"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2" alt="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2" title="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3" alt="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3" title="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3" data-latex="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%7By%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2" title="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{x'} = -{y} - 4 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3" title="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3" data-latex="7 \, {y} + 4 \, {x} = {y'} \hspace{2em}x(0)= -3">
</p>
</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(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8662" title="X1 | Linear ODE systems | ver. 8662"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3" alt="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3" title="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" alt="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" title="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" data-latex="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3" title="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = 2 \, {y} + {x'} - 4 \, {x} \hspace{2em}x(0)= -3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-7%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" title="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1" data-latex="0 = -7 \, {y} + 2 \, {x} + {y'} \hspace{2em}x(0)= 1">
</p>
</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(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9310" title="X1 | Linear ODE systems | ver. 9310"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3" alt="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3" title="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3" data-latex="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3" alt="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3" title="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3" title="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3" data-latex="{x'} + 2 \, {y} = 3 \, {x} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3" title="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = -{y'} + 2 \, {x} - 2 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(-t\right)}" alt="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= 2 \, e^{\left(2 \, 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= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, 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?x=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= 2 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= 2 \, e^{\left(2 \, 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=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8960" title="X1 | Linear ODE systems | ver. 8960"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2" alt="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2" title="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" alt="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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-%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2" title="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2" data-latex="{y} = -{x'} - 4 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-9%20%5C,%20%7By%7D%20-%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -9 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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)} - e^{\left(-8 \, t\right)}" alt="x= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(-5 \, t\right)} - e^{\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(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -e^{\left(-5 \, t\right)} - e^{\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=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6930" title="X1 | Linear ODE systems | ver. 6930"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><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 \, {x} = {x'} \hspace{2em}x(0)= -2" alt="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2" title="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2" data-latex="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3" alt="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3" title="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2" title="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2" data-latex="-{y} + 4 \, {x} = {x'} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7%20%5C,%20%7By%7D%20=%20-%7By'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3" title="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3" data-latex="-7 \, {y} = -{y'} - 4 \, {x} \hspace{2em}x(0)= 3">
</p>
</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(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8431" title="X1 | Linear ODE systems | ver. 8431"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3" alt="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3" title="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3" data-latex="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2" alt="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2" title="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2" data-latex="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx%7D%20-%20%7Bx'%7D%20+%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3" title="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3" data-latex="0 = {x} - {x'} + 4 \, {y} \hspace{2em}x(0)= -3">
</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-%7Bx%7D%20+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2" title="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2" data-latex="-{y'} = -{x} + 2 \, {y} \hspace{2em}x(0)= -2">
</p>
</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(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - 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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6816" title="X1 | Linear ODE systems | ver. 6816"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3" alt="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3" title="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10" alt="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10" title="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10" data-latex="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-4%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3" title="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10" title="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10" data-latex="-9 \, {x} - {y'} = {y} \hspace{2em}x(0)= 10">
</p>
</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(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(8 \, 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= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 9 \, 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(8 \, 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=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7219" title="X1 | Linear ODE systems | ver. 7219"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10" alt="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10" title="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10" data-latex="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" alt="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" title="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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+%209%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10" title="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10" data-latex="-{x'} = -3 \, {x} + 9 \, {y} \hspace{2em}x(0)= -10">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" title="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} - 2 \, {y} = {y'} \hspace{2em}x(0)= 3">
</p>
</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= -9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" alt="x= -9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= -9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -9 \, e^{\left(7 \, t\right)} - e^{\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= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" alt="y= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="y= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="y= 4 \, e^{\left(7 \, t\right)} - e^{\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?x=%20-9%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= -9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -9 \, e^{\left(7 \, t\right)} - e^{\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=%204%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="y= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="y= 4 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9856" title="X1 | Linear ODE systems | ver. 9856"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" alt="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1" alt="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="-{x'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7Bx%7D%20-%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} - {y} = {y'} \hspace{2em}x(0)= 1">
</p>
</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(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, 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?x=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1879" title="X1 | Linear ODE systems | ver. 1879"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" alt="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3" alt="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3" title="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="-4 \, {x} - 2 \, {y} = -{x'} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%207%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3" title="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="2 \, {x} + 7 \, {y} = {y'} \hspace{2em}x(0)= -3">
</p>
</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(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} - 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} + 2 \, 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=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1714" title="X1 | Linear ODE systems | ver. 1714"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4" alt="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4" title="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4" data-latex="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1" alt="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1" title="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1" data-latex="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-4" alt="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4" title="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4" data-latex="3 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -4">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7Bx%7D%20-%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1" title="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1" data-latex="12 \, {x} - {y} = {y'} \hspace{2em}x(0)= -1">
</p>
</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(8 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= -3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, 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= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, 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?x=%20-3%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, 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=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0249" title="X1 | Linear ODE systems | ver. 0249"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1" alt="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1" title="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1" alt="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1" title="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1" data-latex="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%20-2%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1" title="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1" data-latex="{x} - {x'} = -2 \, {y} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20+%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1" title="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1" data-latex="0 = -2 \, {x} - {y'} + 6 \, {y} \hspace{2em}x(0)= -1">
</p>
</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)} + 2 \, e^{\left(2 \, t\right)}" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, 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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, 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?x=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, 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=%20-2%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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8353" title="X1 | Linear ODE systems | ver. 8353"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5" alt="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5" title="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5" data-latex="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2" alt="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2" title="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2" data-latex="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5" title="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5" data-latex="-4 \, {y} = {x'} + 2 \, {x} \hspace{2em}x(0)= 5">
</p>
<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%7By'%7D%20-%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2" title="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2" data-latex="{x} = {y'} - 3 \, {y} \hspace{2em}x(0)= -2">
</p>
</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)} + 4 \, e^{\left(-t\right)}" alt="x= e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="x= 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= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="x= 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=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2418" title="X1 | Linear ODE systems | ver. 2418"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" alt="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" title="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" data-latex="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3" alt="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3" title="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7Bx%7D%20+%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" title="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" data-latex="-{x'} - 3 \, {x} + {y} = 0 \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20=%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3" title="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-8 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(-4 \, t\right)} + e^{\left(-7 \, t\right)}" alt="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(-4 \, t\right)} + e^{\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(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" alt="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} - 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(-4 \, t\right)} + e^{\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=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0354" title="X1 | Linear ODE systems | ver. 0354"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" alt="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" title="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" data-latex="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3"/></p><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 \, {x} = -{y'} \hspace{2em}x(0)= 10" alt="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10" title="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10" data-latex="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" title="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" data-latex="4 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3">
</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+%209%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10" title="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10" data-latex="-{y} + 9 \, {x} = -{y'} \hspace{2em}x(0)= 10">
</p>
</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)} + e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} + e^{\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= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="y= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} + e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} + e^{\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=%209%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1991" title="X1 | Linear ODE systems | ver. 1991"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5" alt="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5" title="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5" data-latex="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1" alt="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1" title="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1" data-latex="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5" title="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5" data-latex="-{x'} - 6 \, {y} = {x} \hspace{2em}x(0)= 5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20+%206%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%201" alt="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1" title="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1" data-latex="6 \, {x} + {y'} + 6 \, {y} = 0 \hspace{2em}x(0)= 1">
</p>
</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)} + 2 \, e^{\left(-10 \, t\right)}" alt="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-10 \, t\right)}" title="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-10 \, t\right)}" data-latex="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\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(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" alt="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\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?x=%203%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-10 \, t\right)}" title="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\left(-10 \, t\right)}" data-latex="x= 3 \, e^{\left(3 \, t\right)} + 2 \, e^{\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=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-10%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, t\right)} + 3 \, e^{\left(-10 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0108" title="X1 | Linear ODE systems | ver. 0108"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3" alt="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3" title="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3" data-latex="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3"/></p><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 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3" alt="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3" title="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3" data-latex="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3" title="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3" data-latex="-3 \, {x} + {x'} = 2 \, {y} \hspace{2em}x(0)= 3">
</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%7Bx%7D%20+%208%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3" title="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3" data-latex="{y'} = -2 \, {x} + 8 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" alt="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 2 \, 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 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" alt="y= 2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(7 \, 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?x=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 2 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="y= 2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="y= 2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8792" title="X1 | Linear ODE systems | ver. 8792"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3" alt="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3" title="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1" alt="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3" title="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20=%205%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1" title="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} + {y'} = 5 \, {y} \hspace{2em}x(0)= -1">
</p>
</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(6 \, t\right)} - 2 \, e^{t}" alt="x= -e^{\left(6 \, t\right)} - 2 \, e^{t}" title="x= -e^{\left(6 \, t\right)} - 2 \, e^{t}" data-latex="x= -e^{\left(6 \, t\right)} - 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= -2 \, e^{\left(6 \, t\right)} + e^{t}" alt="y= -2 \, e^{\left(6 \, t\right)} + e^{t}" title="y= -2 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -2 \, e^{\left(6 \, 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?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7Bt%7D" alt="x= -e^{\left(6 \, t\right)} - 2 \, e^{t}" title="x= -e^{\left(6 \, t\right)} - 2 \, e^{t}" data-latex="x= -e^{\left(6 \, t\right)} - 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=%20-2%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -2 \, e^{\left(6 \, t\right)} + e^{t}" title="y= -2 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -2 \, e^{\left(6 \, t\right)} + e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6792" title="X1 | Linear ODE systems | ver. 6792"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1" alt="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1" title="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1" data-latex="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5" alt="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5" title="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5" data-latex="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1" title="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1" data-latex="{x} - 6 \, {y} = {x'} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20-%206%20%5C,%20%7Bx%7D%20+%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5" title="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5" data-latex="0 = -{y'} - 6 \, {x} + 6 \, {y} \hspace{2em}x(0)= -5">
</p>
</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(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" alt="x= 2 \, e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(10 \, 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= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" alt="y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="y= -3 \, e^{\left(10 \, 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?x=%202%20%5C,%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" title="x= 2 \, e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}" data-latex="x= 2 \, e^{\left(10 \, 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=%20-3%20%5C,%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1115" title="X1 | Linear ODE systems | ver. 1115"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = {x'} - {x} \hspace{2em}x(0)= 2" alt="{y} = {x'} - {x} \hspace{2em}x(0)= 2" title="{y} = {x'} - {x} \hspace{2em}x(0)= 2" data-latex="{y} = {x'} - {x} \hspace{2em}x(0)= 2"/></p><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} = -4 \, {x} \hspace{2em}x(0)= 5" alt="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx'%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y} = {x'} - {x} \hspace{2em}x(0)= 2" title="{y} = {x'} - {x} \hspace{2em}x(0)= 2" data-latex="{y} = {x'} - {x} \hspace{2em}x(0)= 2">
</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-%206%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5">
</p>
</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)} + e^{\left(2 \, t\right)}" alt="x= e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="x= 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= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= 4 \, 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?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="x= 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=%204%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= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= 4 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5109" title="X1 | Linear ODE systems | ver. 5109"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} - {x} = {x'} \hspace{2em}x(0)= -2" alt="-{y} - {x} = {x'} \hspace{2em}x(0)= -2" title="-{y} - {x} = {x'} \hspace{2em}x(0)= -2" data-latex="-{y} - {x} = {x'} \hspace{2em}x(0)= -2"/></p><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} = -4 \, {x} \hspace{2em}x(0)= -5" alt="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" title="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" data-latex="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y} - {x} = {x'} \hspace{2em}x(0)= -2" title="-{y} - {x} = {x'} \hspace{2em}x(0)= -2" data-latex="-{y} - {x} = {x'} \hspace{2em}x(0)= -2">
</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-%206%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" title="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5" data-latex="-{y'} - 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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)} - e^{\left(-5 \, t\right)}" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, 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= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -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?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(-2 \, 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=%20-e%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= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7950" title="X1 | Linear ODE systems | ver. 7950"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10" alt="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10" title="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10" data-latex="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" alt="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" title="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By%7D%20=%203%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10" title="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10" data-latex="9 \, {y} = 3 \, {x} + {x'} \hspace{2em}x(0)= -10">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?8%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" title="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(-12 \, t\right)} - 9 \, e^{t}" alt="x= -e^{\left(-12 \, t\right)} - 9 \, e^{t}" title="x= -e^{\left(-12 \, t\right)} - 9 \, e^{t}" data-latex="x= -e^{\left(-12 \, t\right)} - 9 \, 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= e^{\left(-12 \, t\right)} - 4 \, e^{t}" alt="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" title="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" data-latex="y= e^{\left(-12 \, 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?x=%20-e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7Bt%7D" alt="x= -e^{\left(-12 \, t\right)} - 9 \, e^{t}" title="x= -e^{\left(-12 \, t\right)} - 9 \, e^{t}" data-latex="x= -e^{\left(-12 \, t\right)} - 9 \, 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=%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7Bt%7D" alt="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" title="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" data-latex="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7787" title="X1 | Linear ODE systems | ver. 7787"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4" alt="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4" title="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4" data-latex="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4"/></p><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 \, {x} + {y} = 0 \hspace{2em}x(0)= -1" alt="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1" title="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1" data-latex="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%203%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%204" alt="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4" title="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4" data-latex="-{x'} + 4 \, {x} - 3 \, {y} = 0 \hspace{2em}x(0)= 4">
</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+%2012%20%5C,%20%7Bx%7D%20+%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1" title="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1" data-latex="{y'} + 12 \, {x} + {y} = 0 \hspace{2em}x(0)= -1">
</p>
</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(8 \, t\right)} + e^{\left(-5 \, t\right)}" alt="x= 3 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, 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?x=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= -4 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8487" title="X1 | Linear ODE systems | ver. 8487"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3" alt="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3" title="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3"/></p><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 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3" alt="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3" title="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3" title="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="{x} = -{x'} + 2 \, {y} \hspace{2em}x(0)= 3">
</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%7Bx%7D%20=%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3" title="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3" data-latex="-{y'} - 2 \, {x} = 6 \, {y} \hspace{2em}x(0)= -3">
</p>
</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(-5 \, t\right)}" alt="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, 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= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, 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?x=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, 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=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0560" title="X1 | Linear ODE systems | ver. 0560"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3" alt="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3" title="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3" data-latex="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2" alt="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2" title="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2" data-latex="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3" title="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3" data-latex="{x} - 4 \, {y} = -{x'} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20=%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2" title="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2" data-latex="-2 \, {y} = {x} - {y'} \hspace{2em}x(0)= -2">
</p>
</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(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, 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?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6432" title="X1 | Linear ODE systems | ver. 6432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1" alt="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1" title="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1" data-latex="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4" alt="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4" title="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4" data-latex="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1" title="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1" data-latex="-{x'} + 12 \, {y} = 3 \, {x} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20=%20-3%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%204" alt="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4" title="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4" data-latex="-8 \, {y} = -3 \, {x} + {y'} \hspace{2em}x(0)= 4">
</p>
</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(-12 \, t\right)} + 3 \, e^{t}" alt="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, e^{t}" title="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, e^{t}" data-latex="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, 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 \, e^{\left(-12 \, t\right)} + e^{t}" alt="y= 3 \, e^{\left(-12 \, t\right)} + e^{t}" title="y= 3 \, e^{\left(-12 \, t\right)} + e^{t}" data-latex="y= 3 \, e^{\left(-12 \, 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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7Bt%7D" alt="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, e^{t}" title="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, e^{t}" data-latex="x= -4 \, e^{\left(-12 \, t\right)} + 3 \, 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=%203%20%5C,%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= 3 \, e^{\left(-12 \, t\right)} + e^{t}" title="y= 3 \, e^{\left(-12 \, t\right)} + e^{t}" data-latex="y= 3 \, e^{\left(-12 \, t\right)} + e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1722" title="X1 | Linear ODE systems | ver. 1722"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" alt="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" title="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" data-latex="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" alt="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" title="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" data-latex="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By%7D%20+%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5">
</p>
</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)} + e^{\left(-t\right)}" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, 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= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= 4 \, e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, 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=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0117" title="X1 | Linear ODE systems | ver. 0117"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0" alt="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0" title="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0" data-latex="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3" alt="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3" title="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0" title="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0" data-latex="-2 \, {x} = -{x'} - {y} \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20-%203%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3" title="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3" data-latex="4 \, {x} - 3 \, {y} = {y'} \hspace{2em}x(0)= -3">
</p>
</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)} + e^{t}" alt="x= -e^{\left(-2 \, t\right)} + e^{t}" title="x= -e^{\left(-2 \, t\right)} + e^{t}" data-latex="x= -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= -4 \, e^{\left(-2 \, t\right)} + e^{t}" alt="y= -4 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= -4 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -4 \, e^{\left(-2 \, 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?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -e^{\left(-2 \, t\right)} + e^{t}" title="x= -e^{\left(-2 \, t\right)} + e^{t}" data-latex="x= -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=%20-4%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -4 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= -4 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -4 \, e^{\left(-2 \, t\right)} + e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5112" title="X1 | Linear ODE systems | ver. 5112"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2" alt="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2" title="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2" data-latex="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3" alt="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3" title="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3" data-latex="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7Bx%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2" title="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2" data-latex="{x'} = -2 \, {x} + {y} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3" title="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3" data-latex="4 \, {x} = {y'} - {y} \hspace{2em}x(0)= -3">
</p>
</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)} - e^{\left(-3 \, t\right)}" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} + 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?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-4%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2342" title="X1 | Linear ODE systems | ver. 2342"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2" alt="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2" title="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2" data-latex="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3" alt="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3" title="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3" data-latex="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2" title="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2" data-latex="-{x'} + 4 \, {x} = {y} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20=%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3" title="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3" data-latex="7 \, {y} = {y'} + 4 \, {x} \hspace{2em}x(0)= 3">
</p>
</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(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8111" title="X1 | Linear ODE systems | ver. 8111"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1" alt="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1" title="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1"/></p><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 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1" alt="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-2%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1" title="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {x} = -2 \, {y} - {x'} \hspace{2em}x(0)= 1">
</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%7Bx%7D%20=%203%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1" title="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1" data-latex="-{y'} + 2 \, {x} = 3 \, {y} \hspace{2em}x(0)= -1">
</p>
</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)} + 2 \, e^{t}" alt="x= -e^{\left(-2 \, t\right)} + 2 \, e^{t}" title="x= -e^{\left(-2 \, t\right)} + 2 \, e^{t}" data-latex="x= -e^{\left(-2 \, t\right)} + 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= -2 \, e^{\left(-2 \, t\right)} + e^{t}" alt="y= -2 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= -2 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -2 \, e^{\left(-2 \, 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?x=%20-e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7Bt%7D" alt="x= -e^{\left(-2 \, t\right)} + 2 \, e^{t}" title="x= -e^{\left(-2 \, t\right)} + 2 \, e^{t}" data-latex="x= -e^{\left(-2 \, t\right)} + 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=%20-2%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -2 \, e^{\left(-2 \, t\right)} + e^{t}" title="y= -2 \, e^{\left(-2 \, t\right)} + e^{t}" data-latex="y= -2 \, e^{\left(-2 \, t\right)} + e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8604" title="X1 | Linear ODE systems | ver. 8604"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3" alt="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3" title="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3" data-latex="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3" alt="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3" title="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3" data-latex="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3" title="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3" data-latex="-2 \, {y} + 2 \, {x} + {x'} = 0 \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-7%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3" title="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3" data-latex="0 = -7 \, {y} - 2 \, {x} - {y'} \hspace{2em}x(0)= -3">
</p>
</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(-6 \, t\right)}" alt="x= 2 \, e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-3 \, t\right)} + e^{\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= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\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?x=%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-3 \, t\right)} + e^{\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=%20-e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6867" title="X1 | Linear ODE systems | ver. 6867"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1" alt="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1" title="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3" alt="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3" title="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%202%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1" title="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1" data-latex="2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3" title="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 3">
</p>
</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)} - 2 \, e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= 2 \, e^{\left(2 \, t\right)} + 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - 2 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= 2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8308" title="X1 | Linear ODE systems | ver. 8308"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0" alt="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0" title="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0" data-latex="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5" alt="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5" title="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5" data-latex="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0" title="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0" data-latex="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%205%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5" title="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5" data-latex="0 = 5 \, {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -5">
</p>
</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(6 \, t\right)} + e^{t}" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, 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= -4 \, e^{\left(6 \, t\right)} - e^{t}" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -4 \, e^{\left(6 \, 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?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= -e^{\left(6 \, t\right)} + e^{t}" title="x= -e^{\left(6 \, t\right)} + e^{t}" data-latex="x= -e^{\left(6 \, 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=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{t}" data-latex="y= -4 \, e^{\left(6 \, t\right)} - e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9598" title="X1 | Linear ODE systems | ver. 9598"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5" alt="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5" title="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5" data-latex="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5"/></p><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 \, {x} = {y} \hspace{2em}x(0)= 8" alt="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8" title="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8" data-latex="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20-4%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5" title="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5" data-latex="4 \, {x} = -4 \, {y} - {x'} \hspace{2em}x(0)= -5">
</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+%209%20%5C,%20%7Bx%7D%20=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%208" alt="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8" title="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8" data-latex="{y'} + 9 \, {x} = {y} \hspace{2em}x(0)= 8">
</p>
</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)} - e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - e^{\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= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} - e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - e^{\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=%209%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2501" title="X1 | Linear ODE systems | ver. 2501"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - {x} = -{y} \hspace{2em}x(0)= 2" alt="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2" title="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2" data-latex="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" alt="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%20-%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2" title="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2" data-latex="-{x'} - {x} = -{y} \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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)} + e^{\left(-5 \, t\right)}" alt="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, 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= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -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?x=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, 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=%20-e%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= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7622" title="X1 | Linear ODE systems | ver. 7622"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3" alt="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3" title="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3" data-latex="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1" alt="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1" title="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7Bx'%7D%20-%202%20%5C,%20%7By%7D%20+%205%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3" title="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3" data-latex="0 = -{x'} - 2 \, {y} + 5 \, {x} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7By%7D%20-%20%7By'%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1" title="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 2 \, {y} - {y'} - 2 \, {x} \hspace{2em}x(0)= 1">
</p>
</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(6 \, t\right)} + e^{t}" alt="x= 2 \, e^{\left(6 \, t\right)} + e^{t}" title="x= 2 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= 2 \, e^{\left(6 \, 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= -e^{\left(6 \, t\right)} + 2 \, e^{t}" alt="y= -e^{\left(6 \, t\right)} + 2 \, e^{t}" title="y= -e^{\left(6 \, t\right)} + 2 \, e^{t}" data-latex="y= -e^{\left(6 \, 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?x=%202%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="x= 2 \, e^{\left(6 \, t\right)} + e^{t}" title="x= 2 \, e^{\left(6 \, t\right)} + e^{t}" data-latex="x= 2 \, e^{\left(6 \, 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=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} + 2 \, e^{t}" title="y= -e^{\left(6 \, t\right)} + 2 \, e^{t}" data-latex="y= -e^{\left(6 \, t\right)} + 2 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7585" title="X1 | Linear ODE systems | ver. 7585"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = -{x'} - {y} \hspace{2em}x(0)= 0" alt="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0" title="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0" data-latex="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0"/></p><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} + 4 \, {x} \hspace{2em}x(0)= -5" alt="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5" title="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0" title="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0" data-latex="-{x} = -{x'} - {y} \hspace{2em}x(0)= 0">
</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%7By%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5" title="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-{y'} = 2 \, {y} + 4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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)} - e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" alt="y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="y= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1511" title="X1 | Linear ODE systems | ver. 1511"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5" alt="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5" title="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5" data-latex="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1" alt="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1" title="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1" data-latex="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20=%204%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5" title="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5" data-latex="-6 \, {y} = 4 \, {x} - {x'} \hspace{2em}x(0)= 5">
</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%7By'%7D%20-%206%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1" title="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1" data-latex="-{y} = {y'} - 6 \, {x} \hspace{2em}x(0)= -1">
</p>
</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(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" alt="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, 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?x=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} + 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4045" title="X1 | Linear ODE systems | ver. 4045"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2" alt="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2" title="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7" alt="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7" title="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7" data-latex="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx'%7D%20+%203%20%5C,%20%7By%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2" title="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="0 = {x'} + 3 \, {y} + 2 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7Bx%7D%20=%20-7%20%5C,%20%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7" title="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7" data-latex="12 \, {x} = -7 \, {y} - {y'} \hspace{2em}x(0)= 7">
</p>
</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^{\left(-11 \, t\right)}" alt="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" title="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" data-latex="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, 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?x=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" title="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}" data-latex="x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9764" title="X1 | Linear ODE systems | ver. 9764"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1" alt="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1" title="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1" alt="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1" title="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1" data-latex="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1" title="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1" data-latex="-2 \, {y} + 4 \, {x} = -{x'} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?9%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1" title="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1" data-latex="9 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= 1">
</p>
</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(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\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)} + 2 \, e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\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?x=%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= 2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= 2 \, e^{\left(-5 \, t\right)} - e^{\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=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} + 2 \, e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7726" title="X1 | Linear ODE systems | ver. 7726"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" alt="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" alt="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" title="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" title="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1" data-latex="-{x} = -{x'} - 2 \, {y} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" title="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3" data-latex="0 = 2 \, {x} + {y'} + 2 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(-3 \, t\right)}" alt="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, 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?x=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= -2 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4780" title="X1 | Linear ODE systems | ver. 4780"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3" alt="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3" title="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3" data-latex="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2" alt="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2" title="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2" data-latex="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%20-4%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3" title="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3" data-latex="2 \, {x} + {x'} = -4 \, {y} \hspace{2em}x(0)= -3">
</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-%7Bx%7D%20-%205%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2" title="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2" data-latex="{y'} = -{x} - 5 \, {y} \hspace{2em}x(0)= 2">
</p>
</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(-t\right)} + e^{\left(-6 \, t\right)}" alt="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -4 \, e^{\left(-t\right)} + e^{\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= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" alt="y= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} + e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= -4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -4 \, e^{\left(-t\right)} + e^{\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=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8214" title="X1 | Linear ODE systems | ver. 8214"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3" alt="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3" title="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3" data-latex="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3" alt="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3" title="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3" data-latex="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20=%202%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3" title="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3" data-latex="-2 \, {y} = 2 \, {x} + {x'} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20+%203%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3" title="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3" data-latex="2 \, {x} - {y'} + 3 \, {y} = 0 \hspace{2em}x(0)= -3">
</p>
</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)} + 2 \, e^{\left(-t\right)}" alt="x= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, t\right)} + 2 \, 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 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" alt="y= -2 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -2 \, e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + 2 \, e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, t\right)} + 2 \, 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=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -2 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}" data-latex="y= -2 \, e^{\left(2 \, t\right)} - e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7743" title="X1 | Linear ODE systems | ver. 7743"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10" alt="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10" title="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3" alt="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3" title="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10" title="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} = {x'} + {x} \hspace{2em}x(0)= -10">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3" title="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = {y'} - 4 \, {y} \hspace{2em}x(0)= -3">
</p>
</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(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 9 \, 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= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 9 \, 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=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1130" title="X1 | Linear ODE systems | ver. 1130"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2" alt="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2" title="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2" data-latex="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" alt="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7Bx%7D%20-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2" title="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2" data-latex="{x'} = 3 \, {x} - {y} \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-8%20%5C,%20%7By%7D%20+%20%7By'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" title="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5" data-latex="-8 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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(7 \, t\right)} + e^{\left(4 \, t\right)}" alt="x= e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, 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= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" alt="y= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, 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?x=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" title="x= e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}" data-latex="x= e^{\left(7 \, 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=%20-4%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0449" title="X1 | Linear ODE systems | ver. 0449"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10" alt="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10" title="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10" data-latex="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7Bx%7D%20-%209%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%2010" alt="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10" title="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10" data-latex="-3 \, {x} - 9 \, {y} = {x'} \hspace{2em}x(0)= 10">
</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-%208%20%5C,%20%7By%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - 8 \, {y} = 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(-12 \, t\right)} + 9 \, e^{t}" alt="x= e^{\left(-12 \, t\right)} + 9 \, e^{t}" title="x= e^{\left(-12 \, t\right)} + 9 \, e^{t}" data-latex="x= e^{\left(-12 \, t\right)} + 9 \, 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= e^{\left(-12 \, t\right)} - 4 \, e^{t}" alt="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" title="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" data-latex="y= e^{\left(-12 \, 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?x=%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7Bt%7D" alt="x= e^{\left(-12 \, t\right)} + 9 \, e^{t}" title="x= e^{\left(-12 \, t\right)} + 9 \, e^{t}" data-latex="x= e^{\left(-12 \, t\right)} + 9 \, 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=%20e%5E%7B%5Cleft(-12%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7Bt%7D" alt="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" title="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}" data-latex="y= e^{\left(-12 \, t\right)} - 4 \, e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8196" title="X1 | Linear ODE systems | ver. 8196"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2" alt="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2" title="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2" data-latex="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5" alt="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5" title="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2" title="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2" data-latex="{x'} - 2 \, {x} = {y} \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-7%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5" title="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= 5">
</p>
</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(6 \, t\right)} + e^{\left(3 \, t\right)}" alt="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" title="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(6 \, t\right)} + 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?x=%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" title="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="x= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6671" title="X1 | Linear ODE systems | ver. 6671"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1" alt="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1" title="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1" data-latex="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1"/></p><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 \, {x} - {y'} \hspace{2em}x(0)= 5" alt="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5" title="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5" data-latex="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1" title="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1" data-latex="6 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1">
</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=%206%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5" title="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5" data-latex="{y} = 6 \, {x} - {y'} \hspace{2em}x(0)= 5">
</p>
</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(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" alt="x= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, 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?x=%203%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="x= 3 \, e^{\left(8 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="x= 3 \, e^{\left(8 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5837" title="X1 | Linear ODE systems | ver. 5837"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3" alt="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3" title="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3" data-latex="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2" alt="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2" title="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2" data-latex="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20%7By%7D%20-%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3" title="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3" data-latex="0 = -4 \, {y} - {x} - {x'} \hspace{2em}x(0)= 3">
</p>
<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%7By'%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2" title="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2" data-latex="{x} + {y'} = 2 \, {y} \hspace{2em}x(0)= 2">
</p>
</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(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, 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?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1265" title="X1 | Linear ODE systems | ver. 1265"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1" alt="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1" title="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1" data-latex="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1"/></p><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 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1" alt="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1" title="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1" data-latex="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1" title="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1" data-latex="-{x'} + 2 \, {y} = -{x} \hspace{2em}x(0)= 1">
</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%7Bx%7D%20+%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1" title="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1" data-latex="{y'} = -2 \, {x} + 6 \, {y} \hspace{2em}x(0)= -1">
</p>
</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)} + 2 \, e^{\left(2 \, t\right)}" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, 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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, 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?x=%20-e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" title="x= -e^{\left(5 \, t\right)} + 2 \, e^{\left(2 \, t\right)}" data-latex="x= -e^{\left(5 \, 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=%20-2%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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2261" title="X1 | Linear ODE systems | ver. 2261"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2" alt="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2" title="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2" data-latex="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7" alt="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7" title="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7" data-latex="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2" title="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2" data-latex="-3 \, {y} + {x'} = -4 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7" title="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7" data-latex="12 \, {x} = {y'} - {y} \hspace{2em}x(0)= 7">
</p>
</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)} - 3 \, e^{\left(-8 \, t\right)}" alt="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 3 \, e^{\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= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" alt="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\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?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= e^{\left(5 \, t\right)} - 3 \, e^{\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=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0038" title="X1 | Linear ODE systems | ver. 0038"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2" alt="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2" title="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3" alt="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3" title="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx'%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2" title="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2" data-latex="-{y} - {x'} = -4 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20-%207%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3" title="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3" data-latex="-4 \, {x} = {y'} - 7 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6434" title="X1 | Linear ODE systems | ver. 6434"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5" alt="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5" title="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5" data-latex="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5"/></p><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} + 9 \, {x} \hspace{2em}x(0)= -8" alt="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8" title="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8" data-latex="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20-%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5" title="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5" data-latex="-{x'} + 4 \, {y} - {x} = 0 \hspace{2em}x(0)= -5">
</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%7By%7D%20+%209%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-8" alt="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8" title="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8" data-latex="{y'} = 4 \, {y} + 9 \, {x} \hspace{2em}x(0)= -8">
</p>
</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(8 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= -4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -4 \, e^{\left(8 \, 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= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" alt="y= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="y= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="y= -9 \, e^{\left(8 \, 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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -4 \, e^{\left(8 \, 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=%20-9%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" title="y= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}" data-latex="y= -9 \, e^{\left(8 \, t\right)} + e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4247" title="X1 | Linear ODE systems | ver. 4247"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2" alt="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2" title="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7" alt="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7" title="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7" data-latex="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%203%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2" title="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2" data-latex="0 = 3 \, {y} - {x'} + 2 \, {x} \hspace{2em}x(0)= -2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?7%20%5C,%20%7By%7D%20=%20%7By'%7D%20-%2012%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%207" alt="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7" title="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7" data-latex="7 \, {y} = {y'} - 12 \, {x} \hspace{2em}x(0)= 7">
</p>
</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(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" alt="x= e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(11 \, 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= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" alt="y= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, 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?x=%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(11 \, 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=%203%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" title="y= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}" data-latex="y= 3 \, e^{\left(11 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1722" title="X1 | Linear ODE systems | ver. 1722"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" alt="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" title="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" data-latex="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" alt="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx'%7D%20+%202%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" title="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2" data-latex="-{y} + {x'} + 2 \, {x} = 0 \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By%7D%20+%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" title="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5" data-latex="0 = -3 \, {y} + {y'} + 4 \, {x} \hspace{2em}x(0)= 5">
</p>
</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)} + e^{\left(-t\right)}" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, 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= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= 4 \, e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="x= e^{\left(2 \, 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=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0367" title="X1 | Linear ODE systems | ver. 0367"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10" alt="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10" title="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10" data-latex="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" alt="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" title="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20+%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10" title="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10" data-latex="{x'} = -9 \, {y} + 3 \, {x} \hspace{2em}x(0)= -10">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20+%208%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" title="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="0 = -{y'} + 8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(12 \, t\right)} - 9 \, e^{\left(-t\right)}" alt="x= -e^{\left(12 \, t\right)} - 9 \, e^{\left(-t\right)}" title="x= -e^{\left(12 \, t\right)} - 9 \, e^{\left(-t\right)}" data-latex="x= -e^{\left(12 \, t\right)} - 9 \, 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= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}" alt="y= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}" title="y= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="y= e^{\left(12 \, 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?x=%20-e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -e^{\left(12 \, t\right)} - 9 \, e^{\left(-t\right)}" title="x= -e^{\left(12 \, t\right)} - 9 \, e^{\left(-t\right)}" data-latex="x= -e^{\left(12 \, t\right)} - 9 \, 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=%20e%5E%7B%5Cleft(12%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}" title="y= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}" data-latex="y= e^{\left(12 \, t\right)} - 4 \, e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8942" title="X1 | Linear ODE systems | ver. 8942"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3" alt="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3" title="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1" alt="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1" title="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1" data-latex="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20+%20%7Bx%7D%20+%20%7Bx'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3" title="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3" data-latex="2 \, {y} + {x} + {x'} = 0 \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20+%202%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1" title="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1" data-latex="0 = -{y'} + 2 \, {y} - 2 \, {x} \hspace{2em}x(0)= -1">
</p>
</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(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="x= 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= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, 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?x=%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= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="x= 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=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -2 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1515" title="X1 | Linear ODE systems | ver. 1515"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1" alt="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1" title="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1" data-latex="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5" alt="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5" title="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5" data-latex="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20=%20-6%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1" title="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1" data-latex="2 \, {x} + {x'} = -6 \, {y} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7%20%5C,%20%7By%7D%20-%206%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5" title="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5" data-latex="-7 \, {y} - 6 \, {x} = {y'} \hspace{2em}x(0)= -5">
</p>
</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)} - 2 \, e^{\left(-11 \, t\right)}" alt="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" title="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" data-latex="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}"/></p><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(-11 \, t\right)}" alt="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, 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?x=%203%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" title="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" data-latex="x= 3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -2 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2568" title="X1 | Linear ODE systems | ver. 2568"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8" alt="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8" title="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8" data-latex="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5" alt="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5" title="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5" data-latex="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%208" alt="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8" title="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8" data-latex="{x'} = -9 \, {y} + {x} \hspace{2em}x(0)= 8">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20+%206%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5" title="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5" data-latex="4 \, {x} = -{y'} + 6 \, {y} \hspace{2em}x(0)= 5">
</p>
</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(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}" alt="x= -e^{\left(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}" title="x= -e^{\left(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(10 \, t\right)} + 9 \, 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= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(10 \, 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?x=%20-e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%209%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}" title="x= -e^{\left(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}" data-latex="x= -e^{\left(10 \, t\right)} + 9 \, 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=%20e%5E%7B%5Cleft(10%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-3365" title="X1 | Linear ODE systems | ver. 3365"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3" alt="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3" title="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3" data-latex="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3" alt="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3" title="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-3%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3" title="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3" data-latex="{x'} + 2 \, {y} = -3 \, {x} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20=%20-8%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3" title="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3" data-latex="-2 \, {x} + {y'} = -8 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(-7 \, t\right)}" alt="x= 2 \, e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= 2 \, e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= 2 \, e^{\left(-4 \, t\right)} + e^{\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(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}" alt="y= e^{\left(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + 2 \, e^{\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?x=%202%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= 2 \, e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= 2 \, e^{\left(-4 \, t\right)} + e^{\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=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + 2 \, e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1193" title="X1 | Linear ODE systems | ver. 1193"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2" alt="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2" title="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2" data-latex="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7" alt="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7" title="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7" data-latex="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20%7By%7D%20+%203%20%5C,%20%7Bx%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2" title="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2" data-latex="-3 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 2">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20=%20%7By'%7D%20+%2012%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7" title="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7" data-latex="-2 \, {y} = {y'} + 12 \, {x} \hspace{2em}x(0)= -7">
</p>
</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(7 \, t\right)} - e^{\left(-6 \, t\right)}" alt="x= 3 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= 3 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 3 \, e^{\left(7 \, t\right)} - e^{\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= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" alt="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\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?x=%203%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 3 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= 3 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 3 \, e^{\left(7 \, t\right)} - e^{\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=%20-4%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="y= -4 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4185" title="X1 | Linear ODE systems | ver. 4185"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1" alt="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1" title="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1" data-latex="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1" alt="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1" title="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20-%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%201" alt="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1" title="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1" data-latex="2 \, {y} + {x'} - 4 \, {x} = 0 \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%209%20%5C,%20%7By%7D%20-%20%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1" title="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = 9 \, {y} - {y'} + 2 \, {x} \hspace{2em}x(0)= 1">
</p>
</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(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, 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= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}" data-latex="x= -e^{\left(8 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4771" title="X1 | Linear ODE systems | ver. 4771"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" alt="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" title="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" data-latex="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2" alt="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2" title="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2" data-latex="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20-%203%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" title="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5" data-latex="-{x'} + 4 \, {y} - 3 \, {x} = 0 \hspace{2em}x(0)= -5">
</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=%20-8%20%5C,%20%7By%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2" title="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2" data-latex="{y'} = -8 \, {y} - {x} \hspace{2em}x(0)= 2">
</p>
</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)} - e^{\left(-7 \, t\right)}" alt="x= -4 \, e^{\left(-4 \, t\right)} - e^{\left(-7 \, t\right)}" title="x= -4 \, e^{\left(-4 \, t\right)} - e^{\left(-7 \, t\right)}" data-latex="x= -4 \, e^{\left(-4 \, t\right)} - e^{\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(-4 \, t\right)} + e^{\left(-7 \, t\right)}" alt="y= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(-4 \, t\right)} - e^{\left(-7 \, t\right)}" title="x= -4 \, e^{\left(-4 \, t\right)} - e^{\left(-7 \, t\right)}" data-latex="x= -4 \, e^{\left(-4 \, t\right)} - e^{\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=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1887" title="X1 | Linear ODE systems | ver. 1887"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7" alt="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7" title="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7" data-latex="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2" alt="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2" title="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2" data-latex="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-7" alt="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7" title="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7" data-latex="{x'} + 12 \, {y} = -4 \, {x} \hspace{2em}x(0)= -7">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7By'%7D%20+%20%7By%7D%20-%203%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2" title="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2" data-latex="0 = -{y'} + {y} - 3 \, {x} \hspace{2em}x(0)= 2">
</p>
</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)} - 3 \, e^{\left(-8 \, t\right)}" alt="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\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= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="y= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} - e^{\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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" title="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}" data-latex="x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\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=%203%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="y= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="y= 3 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9354" title="X1 | Linear ODE systems | ver. 9354"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10" alt="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10" title="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10"/></p><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 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3" alt="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3" title="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3" data-latex="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20+%203%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10" title="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= -10">
</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%7Bx%7D%20=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3" title="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3" data-latex="{y'} - 4 \, {x} = 2 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}" alt="x= -e^{\left(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}" title="x= -e^{\left(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} - 9 \, e^{\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(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" alt="y= -e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= -e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= -e^{\left(6 \, t\right)} + 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}" title="x= -e^{\left(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(6 \, t\right)} - 9 \, e^{\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=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" title="y= -e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}" data-latex="y= -e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4966" title="X1 | Linear ODE systems | ver. 4966"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1" alt="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1" title="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1" data-latex="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3" alt="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3" title="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3" data-latex="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20+%202%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%201" alt="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1" title="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1" data-latex="{x'} + 2 \, {y} + 2 \, {x} = 0 \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7By%7D%20=%20-%7By'%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3" title="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3" data-latex="5 \, {y} = -{y'} - 2 \, {x} \hspace{2em}x(0)= -3">
</p>
</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)} - e^{\left(-6 \, t\right)}" alt="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-t\right)} - e^{\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= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 2 \, e^{\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?x=%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-t\right)} - e^{\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=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9965" title="X1 | Linear ODE systems | ver. 9965"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5" alt="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5" title="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5" data-latex="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0" alt="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0" title="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0" data-latex="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20%7By%7D%20=%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5" title="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5" data-latex="4 \, {y} = {x} + {x'} \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx%7D%20-%20%7By'%7D%20+%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0" title="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0" data-latex="0 = {x} - {y'} + 2 \, {y} \hspace{2em}x(0)= 0">
</p>
</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(-2 \, t\right)}" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, 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?x=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" title="x= -e^{\left(3 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}" data-latex="x= -e^{\left(3 \, 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=%20-e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2634" title="X1 | Linear ODE systems | ver. 2634"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} - {x'} = -{y} \hspace{2em}x(0)= 2" alt="{x} - {x'} = -{y} \hspace{2em}x(0)= 2" title="{x} - {x'} = -{y} \hspace{2em}x(0)= 2" data-latex="{x} - {x'} = -{y} \hspace{2em}x(0)= 2"/></p><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} - 4 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%20-%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="{x} - {x'} = -{y} \hspace{2em}x(0)= 2" title="{x} - {x'} = -{y} \hspace{2em}x(0)= 2" data-latex="{x} - {x'} = -{y} \hspace{2em}x(0)= 2">
</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%7By%7D%20-%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = 2 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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)} + e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" alt="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%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="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" title="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}" data-latex="y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7547" title="X1 | Linear ODE systems | ver. 7547"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10" alt="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10" title="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10"/></p><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 \, {x} + {y} \hspace{2em}x(0)= 3" alt="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3" title="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3" data-latex="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10" title="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10" data-latex="-9 \, {y} = -4 \, {x} + {x'} \hspace{2em}x(0)= -10">
</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%7Bx%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3" title="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3" data-latex="-{y'} = 4 \, {x} + {y} \hspace{2em}x(0)= 3">
</p>
</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= -9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" alt="x= -9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -9 \, e^{\left(8 \, 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= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, 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?x=%20-9%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= -9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= -9 \, e^{\left(8 \, 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=%204%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" title="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="y= 4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9349" title="X1 | Linear ODE systems | ver. 9349"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><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 \, {x} = -{x'} \hspace{2em}x(0)= 0" alt="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0" title="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0" data-latex="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0" title="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0" data-latex="-{y} + 3 \, {x} = -{x'} \hspace{2em}x(0)= 0">
</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=%208%20%5C,%20%7By%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} = 8 \, {y} + 4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(-4 \, t\right)} + e^{\left(-7 \, t\right)}" alt="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(-4 \, t\right)} + e^{\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(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" alt="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} - 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -e^{\left(-4 \, t\right)} + e^{\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=%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" title="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}" data-latex="y= e^{\left(-4 \, t\right)} - 4 \, e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-2147" title="X1 | Linear ODE systems | ver. 2147"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><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 \, {x} - {x'} \hspace{2em}x(0)= 2" alt="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2" title="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2" data-latex="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3" alt="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2" title="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2" data-latex="-{y} = -2 \, {x} - {x'} \hspace{2em}x(0)= 2">
</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%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3" title="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3" data-latex="-{y'} - 5 \, {y} = -4 \, {x} \hspace{2em}x(0)= -3">
</p>
</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)} + e^{\left(-6 \, t\right)}" alt="x= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(-t\right)} + e^{\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= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} - 4 \, e^{\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?x=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" title="x= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(-t\right)} + e^{\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=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5234" title="X1 | Linear ODE systems | ver. 5234"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5" alt="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5" title="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1" alt="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1" title="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1" data-latex="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5" title="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5" data-latex="-6 \, {y} = -2 \, {x} - {x'} \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-6%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20+%207%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1" title="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1" data-latex="0 = -6 \, {x} + {y'} + 7 \, {y} \hspace{2em}x(0)= 1">
</p>
</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)} - 2 \, e^{\left(-11 \, t\right)}" alt="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" title="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" data-latex="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}"/></p><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(-11 \, t\right)}" alt="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, 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?x=%20-3%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" title="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}" data-latex="x= -3 \, e^{\left(2 \, t\right)} - 2 \, e^{\left(-11 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%20-2%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-11%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" title="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}" data-latex="y= -2 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5372" title="X1 | Linear ODE systems | ver. 5372"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8" alt="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8" title="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8" data-latex="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" alt="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-8" alt="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8" title="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8" data-latex="-9 \, {y} - 2 \, {x} = -{x'} \hspace{2em}x(0)= -8">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-3%20%5C,%20%7By%7D%20-%20%7By'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" title="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = -3 \, {y} - {y'} + 4 \, {x} \hspace{2em}x(0)= -5">
</p>
</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= -9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}" alt="x= -9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -9 \, e^{\left(6 \, t\right)} + e^{\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(6 \, t\right)} - e^{\left(-7 \, t\right)}" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}" data-latex="y= -4 \, e^{\left(6 \, t\right)} - e^{\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?x=%20-9%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="x= -9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}" title="x= -9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}" data-latex="x= -9 \, e^{\left(6 \, t\right)} + e^{\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=%20-4%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}" title="y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}" data-latex="y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5723" title="X1 | Linear ODE systems | ver. 5723"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1" alt="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1" title="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1" data-latex="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1" alt="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1" title="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx'%7D%20+%202%20%5C,%20%7By%7D%20-%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1" title="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1" data-latex="0 = {x'} + 2 \, {y} - {x} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7By%7D%20=%20-%7By'%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1" title="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1" data-latex="-6 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= -1">
</p>
</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)} - 2 \, e^{\left(2 \, t\right)}" alt="x= e^{\left(5 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, 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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" alt="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, 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?x=%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(5 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" title="x= e^{\left(5 \, t\right)} - 2 \, e^{\left(2 \, t\right)}" data-latex="x= e^{\left(5 \, 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=%20-2%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= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" title="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="y= -2 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4812" title="X1 | Linear ODE systems | ver. 4812"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" alt="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" title="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" data-latex="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3" alt="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3" title="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3" data-latex="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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+%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%200" alt="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" title="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0" data-latex="{x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-7%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3" title="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3" data-latex="-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3">
</p>
</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)} + e^{\left(-6 \, t\right)}" alt="x= -e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= -e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-3 \, t\right)} + e^{\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= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" title="x= -e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-3 \, t\right)} + e^{\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=%20-e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7670" title="X1 | Linear ODE systems | ver. 7670"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10" alt="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10" title="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10" data-latex="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3" alt="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3" title="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-10" alt="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10" title="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10" data-latex="{x} + 9 \, {y} = {x'} \hspace{2em}x(0)= -10">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3" title="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3" data-latex="-4 \, {x} = -{y'} - 4 \, {y} \hspace{2em}x(0)= -3">
</p>
</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= -9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= -9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -9 \, e^{\left(5 \, t\right)} - e^{\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= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" alt="y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= -4 \, e^{\left(5 \, t\right)} + e^{\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?x=%20-9%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -9 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -9 \, e^{\left(5 \, t\right)} - e^{\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=%20-4%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" title="y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}" data-latex="y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4790" title="X1 | Linear ODE systems | ver. 4790"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3" alt="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3" title="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1" alt="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1" title="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1" data-latex="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3" title="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="2 \, {y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20%7By%7D%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1" title="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1" data-latex="5 \, {y} = -2 \, {x} - {y'} \hspace{2em}x(0)= -1">
</p>
</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)} - e^{\left(-6 \, t\right)}" alt="x= -2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(-t\right)} - e^{\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= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} - 2 \, e^{\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?x=%20-2%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(-t\right)} - e^{\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=%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8329" title="X1 | Linear ODE systems | ver. 8329"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0" alt="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0" title="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0" data-latex="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" alt="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" title="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" data-latex="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20=%20-%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0" title="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0" data-latex="-2 \, {x} = -{y} - {x'} \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By%7D%20-%204%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" title="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3" data-latex="3 \, {y} - 4 \, {x} = -{y'} \hspace{2em}x(0)= 3">
</p>
</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)} - e^{t}" alt="x= e^{\left(-2 \, t\right)} - e^{t}" title="x= e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= 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= 4 \, e^{\left(-2 \, t\right)} - e^{t}" alt="y= 4 \, e^{\left(-2 \, t\right)} - e^{t}" title="y= 4 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="y= 4 \, e^{\left(-2 \, 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?x=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="x= e^{\left(-2 \, t\right)} - e^{t}" title="x= e^{\left(-2 \, t\right)} - e^{t}" data-latex="x= 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=%204%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7Bt%7D" alt="y= 4 \, e^{\left(-2 \, t\right)} - e^{t}" title="y= 4 \, e^{\left(-2 \, t\right)} - e^{t}" data-latex="y= 4 \, e^{\left(-2 \, t\right)} - e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8369" title="X1 | Linear ODE systems | ver. 8369"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5" alt="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5" title="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5" data-latex="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1" alt="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%20%7Bx'%7D%20+%206%20%5C,%20%7By%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5" title="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5" data-latex="2 \, {x} + {x'} + 6 \, {y} = 0 \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6%20%5C,%20%7Bx%7D%20+%203%20%5C,%20%7By%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1" title="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1" data-latex="-6 \, {x} + 3 \, {y} = {y'} \hspace{2em}x(0)= 1">
</p>
</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(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" alt="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\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(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\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?x=%20-2%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" title="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}" data-latex="x= -2 \, e^{\left(7 \, t\right)} - 3 \, e^{\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=%203%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= 3 \, e^{\left(7 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1990" title="X1 | Linear ODE systems | ver. 1990"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" alt="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" title="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" data-latex="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3" alt="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3" title="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" title="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3" data-latex="2 \, {y} = -4 \, {x} - {x'} \hspace{2em}x(0)= -3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3" title="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3" data-latex="-9 \, {y} - {y'} = -2 \, {x} \hspace{2em}x(0)= -3">
</p>
</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(-5 \, t\right)} - e^{\left(-8 \, t\right)}" alt="x= -2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -2 \, e^{\left(-5 \, t\right)} - e^{\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)} - 2 \, e^{\left(-8 \, t\right)}" alt="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\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?x=%20-2%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="x= -2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" title="x= -2 \, e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}" data-latex="x= -2 \, e^{\left(-5 \, t\right)} - e^{\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=%20-e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}" title="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}" data-latex="y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-9513" title="X1 | Linear ODE systems | ver. 9513"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1" alt="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3" alt="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3" title="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3" data-latex="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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=%202%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1" title="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1" data-latex="-{x'} - 2 \, {x} = 2 \, {y} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20=%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3" title="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3" data-latex="-5 \, {y} - 2 \, {x} = {y'} \hspace{2em}x(0)= -3">
</p>
</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)} - e^{\left(-6 \, t\right)}" alt="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-t\right)} - e^{\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= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" alt="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 2 \, e^{\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?x=%202%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" title="x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= 2 \, e^{\left(-t\right)} - e^{\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=%20-e%5E%7B%5Cleft(-t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" title="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}" data-latex="y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8094" title="X1 | Linear ODE systems | ver. 8094"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1" alt="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1" title="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4" alt="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4" title="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4" data-latex="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-12%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20+%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1" title="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1" data-latex="0 = -12 \, {y} - {x'} + 4 \, {x} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7Bx%7D%20=%20-%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%204" alt="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4" title="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4" data-latex="3 \, {x} = -{y} - {y'} \hspace{2em}x(0)= 4">
</p>
</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(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="x= -3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, 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= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, 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?x=%20-3%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= -3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="x= -3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="x= -3 \, e^{\left(8 \, 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=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%203%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0140" title="X1 | Linear ODE systems | ver. 0140"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3" alt="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3" title="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3" data-latex="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2" alt="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2" title="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2" data-latex="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%204%20%5C,%20%7Bx%7D%20-%204%20%5C,%20%7By%7D%20-%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3" title="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3" data-latex="0 = 4 \, {x} - 4 \, {y} - {x'} \hspace{2em}x(0)= 3">
</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%7Bx%7D%20=%20-7%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%202" alt="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2" title="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2" data-latex="-{y'} - {x} = -7 \, {y} \hspace{2em}x(0)= 2">
</p>
</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(8 \, t\right)} + 4 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, 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= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, 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=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-0189" title="X1 | Linear ODE systems | ver. 0189"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" alt="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" title="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" data-latex="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0" alt="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0" title="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0" data-latex="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20+%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" title="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3" data-latex="-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= 3">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By%7D%20=%20%7Bx%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0" title="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0" data-latex="3 \, {y} = {x} + {y'} \hspace{2em}x(0)= 0">
</p>
</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)} + 4 \, e^{\left(-t\right)}" alt="x= -e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="x= -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= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" alt="y= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, 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?x=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="x= -e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" title="x= -e^{\left(2 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="x= -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=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="y= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" title="y= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}" data-latex="y= -e^{\left(2 \, t\right)} + e^{\left(-t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5811" title="X1 | Linear ODE systems | ver. 5811"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5" alt="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5" title="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5" data-latex="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8" alt="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8" title="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8" data-latex="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20+%204%20%5C,%20%7By%7D%20=%20%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5" title="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5" data-latex="2 \, {x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-9%20%5C,%20%7Bx%7D%20=%207%20%5C,%20%7By%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-8" alt="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8" title="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8" data-latex="-9 \, {x} = 7 \, {y} - {y'} \hspace{2em}x(0)= -8">
</p>
</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(11 \, t\right)} - e^{\left(-2 \, t\right)}" alt="x= -4 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(11 \, 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= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}" alt="y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -9 \, e^{\left(11 \, 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?x=%20-4%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= -4 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= -4 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= -4 \, e^{\left(11 \, 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=%20-9%20%5C,%20e%5E%7B%5Cleft(11%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}" title="y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}" data-latex="y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1117" title="X1 | Linear ODE systems | ver. 1117"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2" alt="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2" title="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2" data-latex="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" alt="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-2%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-2" alt="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2" title="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2" data-latex="{x'} - {y} = -2 \, {x} \hspace{2em}x(0)= -2">
</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+%207%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" title="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5" data-latex="{y'} + 7 \, {y} = -4 \, {x} \hspace{2em}x(0)= 5">
</p>
</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)} - e^{\left(-6 \, t\right)}" alt="x= -e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= -e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-3 \, t\right)} - e^{\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= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-3 \, t\right)} + 4 \, e^{\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?x=%20-e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" title="x= -e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}" data-latex="x= -e^{\left(-3 \, t\right)} - e^{\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=%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-6987" title="X1 | Linear ODE systems | ver. 6987"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1" alt="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1" title="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1" data-latex="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3" alt="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3" title="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3" data-latex="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1" title="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1" data-latex="{x'} - 2 \, {y} = 4 \, {x} \hspace{2em}x(0)= -1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7Bx%7D%20-%20%7By'%7D%20=%20-7%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3" title="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3" data-latex="2 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 3">
</p>
</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(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + 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?x=%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= e^{\left(8 \, t\right)} - 2 \, 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=%202%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4852" title="X1 | Linear ODE systems | ver. 4852"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3" alt="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3" title="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3"/></p><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 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" alt="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" title="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" data-latex="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By%7D%20+%20%7Bx'%7D%20=%204%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-3" alt="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3" title="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3" data-latex="-2 \, {y} + {x'} = 4 \, {x} \hspace{2em}x(0)= -3">
</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%7Bx%7D%20=%207%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" title="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1" data-latex="{y'} - 2 \, {x} = 7 \, {y} \hspace{2em}x(0)= -1">
</p>
</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(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + 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?x=%20-e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" title="x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}" data-latex="x= -e^{\left(8 \, t\right)} - 2 \, 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=%20-2%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" title="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}" data-latex="y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-8154" title="X1 | Linear ODE systems | ver. 8154"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5" alt="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5" title="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0" alt="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0" title="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0" data-latex="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx'%7D%20-%204%20%5C,%20%7By%7D%20-%202%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5" title="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5" data-latex="0 = {x'} - 4 \, {y} - 2 \, {x} \hspace{2em}x(0)= -5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7By%7D%20=%20-%7By'%7D%20+%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0" title="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0" data-latex="-5 \, {y} = -{y'} + {x} \hspace{2em}x(0)= 0">
</p>
</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(6 \, t\right)} - 4 \, e^{t}" alt="x= -e^{\left(6 \, t\right)} - 4 \, e^{t}" title="x= -e^{\left(6 \, t\right)} - 4 \, e^{t}" data-latex="x= -e^{\left(6 \, 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= -e^{\left(6 \, t\right)} + e^{t}" alt="y= -e^{\left(6 \, t\right)} + e^{t}" title="y= -e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -e^{\left(6 \, 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?x=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7Bt%7D" alt="x= -e^{\left(6 \, t\right)} - 4 \, e^{t}" title="x= -e^{\left(6 \, t\right)} - 4 \, e^{t}" data-latex="x= -e^{\left(6 \, 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=%20-e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20+%20e%5E%7Bt%7D" alt="y= -e^{\left(6 \, t\right)} + e^{t}" title="y= -e^{\left(6 \, t\right)} + e^{t}" data-latex="y= -e^{\left(6 \, t\right)} + e^{t}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-3161" title="X1 | Linear ODE systems | ver. 3161"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" alt="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1" alt="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1" title="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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%7By%7D%20=%20-%7Bx'%7D%20%5Chspace%7B2em%7Dx(0)=%201" alt="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" title="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1" data-latex="{x} + 2 \, {y} = -{x'} \hspace{2em}x(0)= 1">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7Bx%7D%20+%206%20%5C,%20%7By%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-1" alt="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1" title="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1" data-latex="-2 \, {x} + 6 \, {y} = -{y'} \hspace{2em}x(0)= -1">
</p>
</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(-5 \, t\right)}" alt="x= 2 \, e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, 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= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" alt="y= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, 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?x=%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= 2 \, e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= 2 \, e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= 2 \, e^{\left(-2 \, 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=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%202%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" title="y= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}" data-latex="y= e^{\left(-2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-1443" title="X1 | Linear ODE systems | ver. 1443"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0" alt="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0" title="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0" data-latex="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5" alt="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5" title="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20%7Bx%7D%20-%20%7Bx'%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0" title="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0" data-latex="0 = -2 \, {x} - {x'} + {y} \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20=%20-%7By'%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5" title="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5" data-latex="-4 \, {x} = -{y'} + {y} \hspace{2em}x(0)= 5">
</p>
</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)} - e^{\left(-3 \, t\right)}" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + 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?x=%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" title="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}" data-latex="x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y=%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" title="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}" data-latex="y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-5438" title="X1 | Linear ODE systems | 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>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y} + {x'} = -{x} \hspace{2em}x(0)= 0" alt="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0" title="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0" data-latex="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0"/></p><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} = -4 \, {x} \hspace{2em}x(0)= 3" alt="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3" title="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3" data-latex="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</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%7Bx'%7D%20=%20-%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0" title="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0" data-latex="-{y} + {x'} = -{x} \hspace{2em}x(0)= 0">
</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+%206%20%5C,%20%7By%7D%20=%20-4%20%5C,%20%7Bx%7D%20%5Chspace%7B2em%7Dx(0)=%203" alt="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3" title="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3" data-latex="{y'} + 6 \, {y} = -4 \, {x} \hspace{2em}x(0)= 3">
</p>
</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)} - e^{\left(-5 \, t\right)}" alt="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, 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= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" alt="y= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -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?x=%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" title="x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}" data-latex="x= e^{\left(-2 \, 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=%20-e%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= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" title="y= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}" data-latex="y= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-4460" title="X1 | Linear ODE systems | ver. 4460"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5" alt="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5" title="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5" data-latex="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8" alt="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8" title="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8" data-latex="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<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-%204%20%5C,%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%205" alt="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5" title="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5" data-latex="-{x'} = -3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-9%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7By%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7Dx(0)=%20-8" alt="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8" title="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8" data-latex="0 = -9 \, {x} + 2 \, {y} + {y'} \hspace{2em}x(0)= -8">
</p>
</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(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" alt="x= e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 4 \, e^{\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= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}" alt="y= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(7 \, t\right)} - 9 \, e^{\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?x=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" title="x= e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}" data-latex="x= e^{\left(7 \, t\right)} + 4 \, e^{\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=%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%209%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="y= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}" title="y= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}" data-latex="y= e^{\left(7 \, t\right)} - 9 \, e^{\left(-6 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X1-7263" title="X1 | Linear ODE systems | ver. 7263"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X1.</strong></p><p>Find the solution to the given system of IVPs.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0" alt="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0" title="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0" data-latex="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5" alt="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5" title="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5" data-latex="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X1.</strong>
</p>
<p>Find the solution to the given system of IVPs.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7Bx%7D%20+%20%7Bx'%7D%20+%20%7By%7D%20%5Chspace%7B2em%7Dx(0)=%200" alt="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0" title="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0" data-latex="0 = {x} + {x'} + {y} \hspace{2em}x(0)= 0">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx%7D%20+%202%20%5C,%20%7By%7D%20-%20%7By'%7D%20=%200%20%5Chspace%7B2em%7Dx(0)=%20-5" alt="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5" title="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5" data-latex="-4 \, {x} + 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -5">
</p>
</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)} - e^{\left(-2 \, t\right)}" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, 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= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" alt="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -4 \, e^{\left(3 \, 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?x=%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="x= e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="x= e^{\left(3 \, 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=%20-4%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" title="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}" data-latex="y= -4 \, e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>
