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<qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- F4</fieldentry></qtimetadatafield>
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<item ident="F4-6141" title="F4 | Implicit solutions for exact IVPs | ver. 6141"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" alt="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" title="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" data-latex="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y%20=%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}">
</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,%20t%5E%7B3%7D%20-%202%20%5C,%20y%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y" alt="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" title="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" data-latex="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><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} + 3 \, t^{2} y + 2 \, t y = 0" alt="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" title="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" data-latex="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y%20=%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B4%7D%20+%203%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%20=%200" alt="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" title="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" data-latex="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8685" title="F4 | Implicit solutions for exact IVPs | ver. 8685"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" alt="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" title="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" data-latex="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, 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?6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y" alt="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y" title="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y" data-latex="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%20-%2010%20%5C,%20t" alt="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" title="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" data-latex="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, 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?6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%20%7By'%7D%20-%20y%20=%20-8%20%5C,%20t%20y" alt="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y" title="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y" data-latex="6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" alt="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" title="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" data-latex="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4" alt="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4" title="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4" data-latex="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%20-%2010%20%5C,%20t" alt="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" title="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t" data-latex="0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%5E%7B2%7D%20=%204" alt="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4" title="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4" data-latex="5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2307" title="F4 | Implicit solutions for exact IVPs | ver. 2307"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} {y'} - 12 \, t^{2} = -t {y'} - y" alt="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" title="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" data-latex="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y"/></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} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y" alt="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y" title="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y" data-latex="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20%7By'%7D%20-%2012%20%5C,%20t%5E%7B2%7D%20=%20-t%20%7By'%7D%20-%20y" alt="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" title="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" data-latex="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y">
</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,%20y%5E%7B2%7D%20%7By'%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20%7By'%7D%20+%2012%20%5C,%20t%5E%7B2%7D%20-%20y" alt="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y" title="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y" data-latex="-9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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} {y'} - 12 \, t^{2} = -t {y'} - y" alt="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" title="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" data-latex="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)" alt="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)" title="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)" data-latex="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20%7By'%7D%20-%2012%20%5C,%20t%5E%7B2%7D%20=%20-t%20%7By'%7D%20-%20y" alt="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" title="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y" data-latex="-9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B3%7D%20-%203%20%5C,%20y%5E%7B3%7D%20+%20t%20y%20=%20%5Cleft(-2%5Cright)" alt="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)" title="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)" data-latex="-4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5213" title="F4 | Implicit solutions for exact IVPs | ver. 5213"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}" alt="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}" title="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}" data-latex="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" alt="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" title="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" data-latex="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20+%206%20%5C,%20t%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20t%20%7By'%7D" alt="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}" title="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}" data-latex="6 \, t y^{2} + 6 \, t = 2 \, t^{2} {y'} - 2 \, t y {y'} - 2 \, t {y'}">
</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,%20t%20y%5E%7B2%7D%20-%206%20%5C,%20t%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y" alt="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" title="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" data-latex="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" alt="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" title="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" data-latex="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)" alt="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)" title="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)" data-latex="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%206%20%5C,%20t%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y" alt="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" title="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y" data-latex="-6 \, t y^{2} - 6 \, t = 6 \, t^{2} y {y'} + 2 \, t {y'} + 2 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%202%20%5C,%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)" title="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)" data-latex="-3 \, t^{2} y^{2} - 3 \, t^{2} - 2 \, t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6667" title="F4 | Implicit solutions for exact IVPs | ver. 6667"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" alt="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" title="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%20+%204%20%5C,%20t%20%7By'%7D" alt="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" title="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" alt="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" title="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" data-latex="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" title="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" data-latex="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0849" title="F4 | Implicit solutions for exact IVPs | ver. 0849"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t" alt="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t" title="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t" data-latex="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, 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?16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" alt="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" title="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" data-latex="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20=%2010%20%5C,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t" alt="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t" title="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t" data-latex="16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, 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?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20=%208%20%5C,%20t%20y" alt="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" title="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" data-latex="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" alt="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" title="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" data-latex="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)" alt="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)" title="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)" data-latex="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20=%208%20%5C,%20t%20y" alt="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" title="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y" data-latex="16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%5E%7B2%7D%20=%20%5Cleft(-5%5Cright)" alt="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)" title="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)" data-latex="4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0493" title="F4 | Implicit solutions for exact IVPs | ver. 0493"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" alt="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" title="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" data-latex="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, 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?-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}" alt="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}" title="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}" data-latex="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20%7By'%7D%20-%204%20%5C,%20y%20=%20-6%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20-%2010%20%5C,%20t" alt="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" title="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" data-latex="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, 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?-4%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20y%5E%7B2%7D%20+%206%20%5C,%20y%20%7By'%7D%20-%204%20%5C,%20y%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}" title="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}" data-latex="-4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" alt="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" title="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" data-latex="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12" alt="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12" title="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12" data-latex="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20%7By'%7D%20-%204%20%5C,%20y%20=%20-6%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20-%2010%20%5C,%20t" alt="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" title="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t" data-latex="-4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20-%204%20%5C,%20t%20y%20=%2012" alt="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12" title="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12" data-latex="3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5847" title="F4 | Implicit solutions for exact IVPs | ver. 5847"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y" alt="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y" title="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y" data-latex="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" alt="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" title="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" data-latex="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20+%205%20%5C,%20y" alt="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y" title="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y" data-latex="0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y">
</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,%20t%20y%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20-%205%20%5C,%20y" alt="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" title="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" data-latex="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" alt="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" title="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" data-latex="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4" alt="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4" title="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4" data-latex="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20-%205%20%5C,%20y" alt="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" title="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y" data-latex="6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20-%203%20%5C,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20t%20y%20=%204" alt="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4" title="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4" data-latex="-2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6005" title="F4 | Implicit solutions for exact IVPs | ver. 6005"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" alt="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" title="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" data-latex="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}" alt="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}" title="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20-%20y%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20y%5E%7B2%7D%20%7By'%7D" alt="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" title="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" data-latex="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20-%20y%20=%206%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}" title="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} + 6 \, t y - y = 6 \, y^{2} {y'} - 5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" alt="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" title="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" data-latex="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} + 2 \, y^{3} + t y = 6" alt="-5 \, t y^{2} + 2 \, y^{3} + t y = 6" title="-5 \, t y^{2} + 2 \, y^{3} + t y = 6" data-latex="-5 \, t y^{2} + 2 \, y^{3} + t y = 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20-%20y%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20y%5E%7B2%7D%20%7By'%7D" alt="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" title="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}" data-latex="5 \, y^{2} - t {y'} - y = -10 \, t y {y'} + 6 \, y^{2} {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20y%5E%7B3%7D%20+%20t%20y%20=%206" alt="-5 \, t y^{2} + 2 \, y^{3} + t y = 6" title="-5 \, t y^{2} + 2 \, y^{3} + t y = 6" data-latex="-5 \, t y^{2} + 2 \, y^{3} + t y = 6">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9233" title="F4 | Implicit solutions for exact IVPs | ver. 9233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}" alt="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}" title="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}" data-latex="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}"/></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} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" alt="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" title="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" data-latex="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D" alt="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}" title="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}" data-latex="-4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'}">
</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,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%206%20%5C,%20t%20y" alt="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" title="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" data-latex="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" alt="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" title="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" data-latex="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)" alt="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)" title="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)" data-latex="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%206%20%5C,%20t%20y" alt="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" title="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y" data-latex="-8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20y%5E%7B4%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20=%20%5Cleft(-3%5Cright)" alt="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)" title="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)" data-latex="2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9506" title="F4 | Implicit solutions for exact IVPs | ver. 9506"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y" alt="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y" title="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y" data-latex="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" alt="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" title="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" data-latex="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20=%20t%5E%7B2%7D%20%7By'%7D%20-%2015%20%5C,%20t%5E%7B2%7D%20+%20y" alt="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y" title="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y" data-latex="-4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y">
</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,%20y%5E%7B3%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20+%2015%20%5C,%20t%5E%7B2%7D%20=%203%20%5C,%20y%5E%7B2%7D" alt="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" title="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" data-latex="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" alt="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" title="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" data-latex="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><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} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)" alt="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)" title="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)" data-latex="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20+%2015%20%5C,%20t%5E%7B2%7D%20=%203%20%5C,%20y%5E%7B2%7D" alt="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" title="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}" data-latex="-4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%5E%7B4%7D%20-%205%20%5C,%20t%5E%7B3%7D%20+%203%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-1%5Cright)" alt="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)" title="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)" data-latex="y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0446" title="F4 | Implicit solutions for exact IVPs | ver. 0446"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}" alt="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}" title="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}" data-latex="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" alt="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" title="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20-%205%20%5C,%20y%20=%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D" alt="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}" title="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}" data-latex="-6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%20-5%20%5C,%20y%5E%7B2%7D" alt="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" title="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" alt="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" title="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1" alt="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1" title="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1" data-latex="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%20-5%20%5C,%20y%5E%7B2%7D" alt="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" title="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}" data-latex="-6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%5E%7B2%7D%20=%201" alt="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1" title="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1" data-latex="3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6299" title="F4 | Implicit solutions for exact IVPs | ver. 6299"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" alt="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" title="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" data-latex="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{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?6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}" alt="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}" title="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}" data-latex="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B3%7D" alt="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" title="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" data-latex="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{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?6%20%5C,%20t%20y%20-%202%20%5C,%20t%20%7By'%7D%20=%206%20%5C,%20t%20y%5E%7B2%7D%20+%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D" alt="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}" title="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}" data-latex="6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" alt="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" title="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" data-latex="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5" alt="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5" title="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5" data-latex="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B3%7D" alt="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" title="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}" data-latex="6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B4%7D%20+%203%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20=%205" alt="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5" title="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5" data-latex="-t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9747" title="F4 | Implicit solutions for exact IVPs | ver. 9747"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y" alt="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y" title="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y" data-latex="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y"/></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 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" alt="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" title="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" data-latex="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2016%20%5C,%20t%5E%7B3%7D%20-%204%20%5C,%20t%20y%20%7By'%7D%20+%205%20%5C,%20y%20=%20-4%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y" title="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y" data-latex="-4 \, t^{2} y {y'} - 16 \, t^{3} - 4 \, t y {y'} + 5 \, y = -4 \, y^{3} {y'} - 8 \, t y">
</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,%20t%20%7By'%7D%20+%205%20%5C,%20y%20=%20-4%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%2016%20%5C,%20t%5E%7B3%7D" alt="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" title="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" data-latex="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" alt="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" title="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" data-latex="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{4} - y^{4} - 5 \, t y = 8" alt="4 \, t^{4} - y^{4} - 5 \, t y = 8" title="4 \, t^{4} - y^{4} - 5 \, t y = 8" data-latex="4 \, t^{4} - y^{4} - 5 \, t y = 8"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20%7By'%7D%20+%205%20%5C,%20y%20=%20-4%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%2016%20%5C,%20t%5E%7B3%7D" alt="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" title="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}" data-latex="5 \, t {y'} + 5 \, y = -4 \, y^{3} {y'} + 16 \, t^{3}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20-%20y%5E%7B4%7D%20-%205%20%5C,%20t%20y%20=%208" alt="4 \, t^{4} - y^{4} - 5 \, t y = 8" title="4 \, t^{4} - y^{4} - 5 \, t y = 8" data-latex="4 \, t^{4} - y^{4} - 5 \, t y = 8">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3333" title="F4 | Implicit solutions for exact IVPs | ver. 3333"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" alt="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" title="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" data-latex="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{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 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}" alt="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}" title="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}" data-latex="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%2020%20%5C,%20t%5E%7B3%7D%20-%2010%20%5C,%20t%20y%5E%7B2%7D" alt="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" title="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" data-latex="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{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=%2010%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20+%20t%20%7By'%7D" alt="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}" title="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}" data-latex="0 = 10 \, t y^{2} - 2 \, t y {y'} - 8 \, t y + t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" alt="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" title="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" data-latex="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4" alt="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4" title="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4" data-latex="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%2020%20%5C,%20t%5E%7B3%7D%20-%2010%20%5C,%20t%20y%5E%7B2%7D" alt="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" title="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}" data-latex="10 \, t^{2} y {y'} - 4 \, t^{2} {y'} - 8 \, t y = 20 \, t^{3} - 10 \, t y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%205%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20=%204" alt="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4" title="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4" data-latex="-5 \, t^{4} + 5 \, t^{2} y^{2} - 4 \, t^{2} y = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3066" title="F4 | Implicit solutions for exact IVPs | ver. 3066"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" alt="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" title="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" data-latex="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y"/></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 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y" alt="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y" title="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y" data-latex="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B3%7D%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%5E%7B2%7D%20+%20t%20%7By'%7D%20+%20y" alt="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" title="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" data-latex="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y">
</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,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D%20=%20-y" alt="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y" title="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y" data-latex="-8 \, t y^{2} + 5 \, t^{2} {y'} + 15 \, y^{2} {y'} + 5 \, y^{2} = -y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" alt="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" title="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" data-latex="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)" alt="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)" title="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)" data-latex="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B3%7D%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%5E%7B2%7D%20+%20t%20%7By'%7D%20+%20y" alt="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" title="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y" data-latex="-4 \, t^{3} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + t {y'} + y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B4%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)" title="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)" data-latex="t^{4} - 4 \, t^{2} y^{2} + t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4178" title="F4 | Implicit solutions for exact IVPs | ver. 4178"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}" alt="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}" title="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}" data-latex="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}"/></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 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20+%206%20%5C,%20t%20y%20=%204%20%5C,%20t%20y%20%7By'%7D%20+%205%20%5C,%20t%20%7By'%7D" alt="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}" title="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}" data-latex="2 \, t y^{2} + 6 \, t y = 4 \, t y {y'} + 5 \, t {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)" alt="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)" title="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)" data-latex="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + 3 \, t^{2} {y'} + 6 \, t y = 4 \, t y {y'} + 2 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-6%5Cright)" alt="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)" title="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)" data-latex="-t^{2} y^{2} - 3 \, t^{2} y + 2 \, t y^{2} = \left(-6\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7073" title="F4 | Implicit solutions for exact IVPs | ver. 7073"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" alt="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" title="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" data-latex="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{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 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y" alt="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y" title="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y" data-latex="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%20-8%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D" alt="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" title="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" data-latex="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{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,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%208%20%5C,%20t%20y%20=%20-9%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%203%20%5C,%20y" alt="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y" title="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y" data-latex="-8 \, t y^{2} + 10 \, t y {y'} + 8 \, t y = -9 \, y^{2} {y'} - 3 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" alt="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" title="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" data-latex="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)" alt="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)" title="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)" data-latex="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%20-8%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D" alt="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" title="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}" data-latex="8 \, t^{2} y {y'} - 8 \, t y = -8 \, t y^{2} + 4 \, t^{2} {y'} + 10 \, t y {y'} + 5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20+%205%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-5%5Cright)" alt="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)" title="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)" data-latex="-4 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, t y^{2} = \left(-5\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7296" title="F4 | Implicit solutions for exact IVPs | ver. 7296"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y" alt="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y" title="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y" data-latex="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" title="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D%20=%202%20%5C,%20t%20y" alt="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y" title="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y" data-latex="-6 \, t y^{2} - 2 \, t y {y'} + 4 \, t {y'} = 2 \, t y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%5E%7B3%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" title="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" title="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4" alt="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4" title="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4" data-latex="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%5E%7B3%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" title="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} = -8 \, t^{3} - t^{2} {y'} - 2 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%203%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20y%20=%204" alt="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4" title="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4" data-latex="2 \, t^{4} + 3 \, t^{2} y^{2} + t^{2} y = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6615" title="F4 | Implicit solutions for exact IVPs | ver. 6615"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" alt="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" title="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" data-latex="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}"/></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 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}" alt="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}" title="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}" data-latex="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20t%5E%7B3%7D%20-%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20-%204%20%5C,%20y%20=%202%20%5C,%20t%20y%20%7By'%7D" alt="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" title="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" data-latex="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20-%20y%5E%7B2%7D%20-%204%20%5C,%20y%20=%204%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D" alt="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}" title="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}" data-latex="-5 \, t^{2} {y'} - y^{2} - 4 \, y = 4 \, t^{2} y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" alt="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" title="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" data-latex="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)" alt="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)" title="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)" data-latex="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20t%5E%7B3%7D%20-%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20-%204%20%5C,%20y%20=%202%20%5C,%20t%20y%20%7By'%7D" alt="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" title="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}" data-latex="-20 \, t^{3} - y^{2} - 4 \, t {y'} - 4 \, y = 2 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20-%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20=%20%5Cleft(-2%5Cright)" alt="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)" title="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)" data-latex="-5 \, t^{4} - t y^{2} - 4 \, t y = \left(-2\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6476" title="F4 | Implicit solutions for exact IVPs | ver. 6476"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}" alt="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}" title="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}" data-latex="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}"/></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 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" alt="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" title="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" data-latex="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20-%203%20%5C,%20t%20%7By'%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D" alt="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}" title="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}" data-latex="-4 \, t^{2} {y'} + 3 \, t^{2} - y^{2} - 3 \, t {y'} = -6 \, t^{2} y {y'}">
</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,%20t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D%20+%203%20%5C,%20y" alt="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" title="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" data-latex="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" alt="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" title="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" data-latex="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{3} - t y^{2} - 3 \, t y = 3" alt="t^{3} - t y^{2} - 3 \, t y = 3" title="t^{3} - t y^{2} - 3 \, t y = 3" data-latex="t^{3} - t y^{2} - 3 \, t y = 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D%20+%203%20%5C,%20y" alt="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" title="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y" data-latex="-2 \, t y {y'} - y^{2} = -3 \, t^{2} + 3 \, t {y'} + 3 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B3%7D%20-%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%20y%20=%203" alt="t^{3} - t y^{2} - 3 \, t y = 3" title="t^{3} - t y^{2} - 3 \, t y = 3" data-latex="t^{3} - t y^{2} - 3 \, t y = 3">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3104" title="F4 | Implicit solutions for exact IVPs | ver. 3104"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" alt="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" title="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" data-latex="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y" alt="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y" title="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y" data-latex="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20=%202%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%20+%20t%20%7By'%7D%20+%20y" alt="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" title="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" data-latex="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20=%20-2%20%5C,%20t%20y%5E%7B2%7D%20-%20y" alt="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y" title="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y" data-latex="-t^{2} {y'} - 10 \, t y {y'} = -2 \, t y^{2} - y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" alt="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" title="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" data-latex="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y^{2} - t^{2} y + t y = \left(-1\right)" alt="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)" title="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)" data-latex="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20=%202%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%20+%20t%20%7By'%7D%20+%20y" alt="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" title="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y" data-latex="-2 \, t^{2} y {y'} = 2 \, t y^{2} - t^{2} {y'} - 2 \, t y + t {y'} + y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20y%20+%20t%20y%20=%20%5Cleft(-1%5Cright)" alt="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)" title="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)" data-latex="t^{2} y^{2} - t^{2} y + t y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9567" title="F4 | Implicit solutions for exact IVPs | ver. 9567"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{2}" alt="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{2}" title="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{2}" data-latex="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{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^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" alt="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" title="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" data-latex="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20-%204%20%5C,%20t%20%7By'%7D%20=%204%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D" alt="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{2}" title="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{2}" data-latex="8 \, t y - 4 \, t {y'} = 4 \, t^{2} y {y'} - 2 \, y^{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,%20y%5E%7B2%7D%20+%202%20%5C,%20y%20%7By'%7D%20-%204%20%5C,%20y%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D" alt="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" title="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" data-latex="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" alt="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" title="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" data-latex="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)" alt="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)" title="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)" data-latex="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20+%202%20%5C,%20y%20%7By'%7D%20-%204%20%5C,%20y%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D" alt="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" title="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}" data-latex="2 \, y^{2} + 2 \, y {y'} - 4 \, y = -4 \, t y {y'} + 4 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20-%20y%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)" title="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)" data-latex="-2 \, t y^{2} + 4 \, t y - y^{2} = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7662" title="F4 | Implicit solutions for exact IVPs | ver. 7662"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y" alt="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y" title="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y" data-latex="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" alt="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" title="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" data-latex="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%205%20%5C,%20y" alt="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y" title="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y" data-latex="6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y">
</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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20t%20%7By'%7D%20+%205%20%5C,%20y%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y" alt="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" title="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" data-latex="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" alt="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" title="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" data-latex="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)" alt="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)" title="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)" data-latex="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20t%20%7By'%7D%20+%205%20%5C,%20y%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y" alt="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" title="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y" data-latex="-6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20+%205%20%5C,%20t%20y%20=%20%5Cleft(-12%5Cright)" alt="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)" title="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)" data-latex="-3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4470" title="F4 | Implicit solutions for exact IVPs | ver. 4470"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" alt="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" title="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" data-latex="-t {y'} + 2 \, y {y'} + 10 \, t - y = 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?10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}" alt="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}" title="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}" data-latex="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20+%202%20%5C,%20y%20%7By'%7D%20+%2010%20%5C,%20t%20-%20y%20=%200" alt="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" title="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" data-latex="-t {y'} + 2 \, y {y'} + 10 \, t - y = 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?10%20%5C,%20t%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20%7By'%7D%20+%20t%20%7By'%7D" alt="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}" title="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}" data-latex="10 \, t = -4 \, t y^{2} - 2 \, t^{2} {y'} + 6 \, t y {y'} + t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" alt="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" title="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" data-latex="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} - t y + y^{2} = 7" alt="5 \, t^{2} - t y + y^{2} = 7" title="5 \, t^{2} - t y + y^{2} = 7" data-latex="5 \, t^{2} - t y + y^{2} = 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20+%202%20%5C,%20y%20%7By'%7D%20+%2010%20%5C,%20t%20-%20y%20=%200" alt="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" title="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0" data-latex="-t {y'} + 2 \, y {y'} + 10 \, t - y = 0">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20-%20t%20y%20+%20y%5E%7B2%7D%20=%207" alt="5 \, t^{2} - t y + y^{2} = 7" title="5 \, t^{2} - t y + y^{2} = 7" data-latex="5 \, t^{2} - t y + y^{2} = 7">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2248" title="F4 | Implicit solutions for exact IVPs | ver. 2248"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, t" alt="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, t" title="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, t" data-latex="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, 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?4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" alt="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" title="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" data-latex="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%20-6%20%5C,%20t%20y%20%7By'%7D%20+%203%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20t" alt="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, t" title="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, t" data-latex="-8 \, t^{2} y {y'} - 8 \, t y = -6 \, t y {y'} + 3 \, t {y'} + 4 \, 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?4%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%20%7By'%7D%20+%203%20%5C,%20y%20=%20-12%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" title="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" data-latex="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" alt="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" title="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" data-latex="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)" alt="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)" title="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)" data-latex="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%20%7By'%7D%20+%203%20%5C,%20y%20=%20-12%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" title="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y" data-latex="4 \, t^{2} {y'} + 3 \, t {y'} + 3 \, y = -12 \, y^{3} {y'} - 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)" title="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)" data-latex="3 \, y^{4} + 4 \, t^{2} y + 3 \, t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3660" title="F4 | Implicit solutions for exact IVPs | ver. 3660"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" alt="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" title="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" data-latex="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, 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?-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y" alt="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y" title="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y" data-latex="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20-%20y%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20-%208%20%5C,%20t" alt="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" title="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" data-latex="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, 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?-t%5E%7B2%7D%20%7By'%7D%20=%20-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D%20-%20y" alt="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y" title="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y" data-latex="-t^{2} {y'} = -10 \, t^{2} y {y'} + 5 \, y^{2} - y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" alt="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" title="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" data-latex="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} - 4 \, t^{2} + t y = 0" alt="-5 \, t y^{2} - 4 \, t^{2} + t y = 0" title="-5 \, t y^{2} - 4 \, t^{2} + t y = 0" data-latex="-5 \, t y^{2} - 4 \, t^{2} + t y = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20-%20y%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20-%208%20%5C,%20t" alt="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" title="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t" data-latex="-t {y'} - y = -10 \, t y {y'} - 5 \, y^{2} - 8 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20+%20t%20y%20=%200" alt="-5 \, t y^{2} - 4 \, t^{2} + t y = 0" title="-5 \, t y^{2} - 4 \, t^{2} + t y = 0" data-latex="-5 \, t y^{2} - 4 \, t^{2} + t y = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2101" title="F4 | Implicit solutions for exact IVPs | ver. 2101"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}" alt="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}" title="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}" data-latex="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D%20=%20-6%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20%7By'%7D" alt="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}" title="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}" data-latex="4 \, t^{2} {y'} + 5 \, y^{2} = -6 \, t y^{2} + 2 \, t {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D%20-%202%20%5C,%20y%20=%202%20%5C,%20t%20%7By'%7D" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)" alt="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)" title="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)" data-latex="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D%20-%202%20%5C,%20y%20=%202%20%5C,%20t%20%7By'%7D" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 2 \, y {y'} - 2 \, y = 2 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20+%20y%5E%7B2%7D%20=%20%5Cleft(-4%5Cright)" alt="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)" title="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)" data-latex="-3 \, t^{2} y^{2} + 2 \, t y + y^{2} = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7075" title="F4 | Implicit solutions for exact IVPs | ver. 7075"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" alt="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" title="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y" alt="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y" title="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y" data-latex="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20+%206%20%5C,%20t%20=%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" title="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20y%5E%7B2%7D%20-%204%20%5C,%20y" alt="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y" title="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y" data-latex="-10 \, t^{2} y {y'} - 3 \, t^{2} {y'} = -4 \, y^{2} - 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" alt="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" title="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)" alt="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)" title="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)" data-latex="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20+%206%20%5C,%20t%20=%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" title="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-4 \, y^{2} + 6 \, t = 8 \, t y {y'} + 4 \, t {y'} + 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20y%20=%20%5Cleft(-11%5Cright)" alt="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)" title="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)" data-latex="4 \, t y^{2} - 3 \, t^{2} + 4 \, t y = \left(-11\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3824" title="F4 | Implicit solutions for exact IVPs | ver. 3824"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" alt="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" title="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" data-latex="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}"/></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 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y" alt="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y" title="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y" data-latex="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20-%204%20%5C,%20y%20=%204%20%5C,%20t%20%7By'%7D" alt="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" title="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" data-latex="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20=%208%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20-%204%20%5C,%20y" alt="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y" title="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y" data-latex="5 \, t^{2} {y'} + 4 \, t y {y'} + 9 \, y^{2} {y'} = 8 \, t y^{2} - 8 \, t - 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" alt="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" title="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" data-latex="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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} - 4 \, t^{2} - 4 \, t y = 3" alt="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3" title="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3" data-latex="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20-%204%20%5C,%20y%20=%204%20%5C,%20t%20%7By'%7D" alt="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" title="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}" data-latex="-9 \, y^{2} {y'} - 8 \, t - 4 \, y = 4 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B3%7D%20-%204%20%5C,%20t%5E%7B2%7D%20-%204%20%5C,%20t%20y%20=%203" alt="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3" title="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3" data-latex="-3 \, y^{3} - 4 \, t^{2} - 4 \, t y = 3">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5399" title="F4 | Implicit solutions for exact IVPs | ver. 5399"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}" alt="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}" title="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}" data-latex="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}"/></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 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" alt="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" title="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" data-latex="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20+%204%20%5C,%20y%20=%20-8%20%5C,%20t%20y%5E%7B2%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%2010%20%5C,%20y%20%7By'%7D" alt="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}" title="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}" data-latex="-8 \, t y + 4 \, y = -8 \, t y^{2} + 8 \, t y {y'} - 10 \, y {y'}">
</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,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D" alt="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" title="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" data-latex="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" alt="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" title="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" data-latex="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4" alt="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4" title="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4" data-latex="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D" alt="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" title="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}" data-latex="-8 \, t y - 4 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 4 \, t^{2} {y'} + 8 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%204" alt="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4" title="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4" data-latex="4 \, t^{2} y^{2} - 4 \, t^{2} y - 4 \, t y^{2} = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2336" title="F4 | Implicit solutions for exact IVPs | ver. 2336"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" alt="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" title="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" data-latex="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y"/></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} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y" alt="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y" title="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y" data-latex="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%208%20%5C,%20t%20y" alt="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" title="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" data-latex="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y">
</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,%20y%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20-%204%20%5C,%20y%20=%2010%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y" title="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y" data-latex="9 \, y^{2} {y'} + 3 \, t^{2} - 4 \, y = 10 \, t^{2} y {y'} - 4 \, t y {y'} - 8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" alt="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" title="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" data-latex="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)" alt="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)" title="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)" data-latex="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%208%20%5C,%20t%20y" alt="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" title="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y" data-latex="0 = -4 \, t^{2} {y'} - 9 \, y^{2} {y'} - 3 \, t^{2} - 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B3%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20y%5E%7B3%7D%20=%20%5Cleft(-6%5Cright)" alt="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)" title="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)" data-latex="t^{3} + 4 \, t^{2} y + 3 \, y^{3} = \left(-6\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7528" title="F4 | Implicit solutions for exact IVPs | ver. 7528"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y" alt="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y" title="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y" data-latex="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" alt="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" title="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" data-latex="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B3%7D%20-%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%2010%20%5C,%20t%20y%20%7By'%7D%20+%20y" alt="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y" title="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y" data-latex="10 \, t^{2} y {y'} - 4 \, t^{3} - 5 \, t^{2} {y'} = 10 \, t y {y'} + y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%20y%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B3%7D%20+%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" title="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" data-latex="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" alt="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" title="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" data-latex="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11" alt="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11" title="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11" data-latex="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%20y%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B3%7D%20+%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" title="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}" data-latex="10 \, t y {y'} = -4 \, t^{3} + 15 \, y^{2} {y'} - 5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B4%7D%20+%205%20%5C,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B3%7D%20=%2011" alt="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11" title="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11" data-latex="t^{4} + 5 \, t y^{2} - 5 \, y^{3} = 11">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3585" title="F4 | Implicit solutions for exact IVPs | ver. 3585"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" alt="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" title="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" data-latex="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}" alt="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}" title="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}" data-latex="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20%7By'%7D" alt="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" title="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" data-latex="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%5E%7B2%7D%20=%2010%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20t%20%7By'%7D%20+%2010%20%5C,%20y%20%7By'%7D" alt="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}" title="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}" data-latex="-4 \, t^{2} {y'} + 6 \, t^{2} = 10 \, t y^{2} - 3 \, y^{2} + 2 \, t {y'} + 10 \, y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" alt="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" title="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" data-latex="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2" alt="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2" title="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2" data-latex="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20%7By'%7D" alt="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" title="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}" data-latex="10 \, t^{2} y {y'} + 8 \, t y - 3 \, y^{2} = -10 \, t y^{2} - 4 \, t^{2} {y'} + 6 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%5E%7B2%7D%20=%202" alt="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2" title="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2" data-latex="-5 \, t^{2} y^{2} - 4 \, t^{2} y + 3 \, t y^{2} = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8835" title="F4 | Implicit solutions for exact IVPs | ver. 8835"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{2}" alt="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{2}" title="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{2}" data-latex="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{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 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" alt="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" title="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" data-latex="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20=%20-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{2}" title="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{2}" data-latex="4 \, t y + 2 \, t {y'} = -10 \, t^{2} y {y'} - 5 \, y^{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,%20t%20y%20=%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t" alt="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" title="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" data-latex="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" alt="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" title="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" data-latex="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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} + 2 \, t^{2} y + 4 \, t^{2} = 0" alt="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0" title="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0" data-latex="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20=%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t" alt="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" title="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t" data-latex="4 \, t y = 8 \, y^{3} {y'} - 2 \, t^{2} {y'} - 8 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20+%204%20%5C,%20t%5E%7B2%7D%20=%200" alt="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0" title="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0" data-latex="-2 \, y^{4} + 2 \, t^{2} y + 4 \, t^{2} = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2020" title="F4 | Implicit solutions for exact IVPs | ver. 2020"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" alt="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" title="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" data-latex="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, 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?-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}" alt="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}" title="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}" data-latex="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%205%20%5C,%20y%5E%7B2%7D%20-%2010%20%5C,%20t" alt="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" title="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" data-latex="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, 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?-6%20%5C,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D" alt="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}" title="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}" data-latex="-6 \, t y + 2 \, t {y'} = 2 \, t^{2} y {y'} + 5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" alt="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" title="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" data-latex="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3" alt="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3" title="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3" data-latex="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%205%20%5C,%20y%5E%7B2%7D%20-%2010%20%5C,%20t" alt="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" title="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t" data-latex="0 = -3 \, t^{2} {y'} - 10 \, t y {y'} - 6 \, t y - 5 \, y^{2} - 10 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20+%205%20%5C,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20=%203" alt="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3" title="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3" data-latex="3 \, t^{2} y + 5 \, t y^{2} + 5 \, t^{2} = 3">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4416" title="F4 | Implicit solutions for exact IVPs | ver. 4416"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" alt="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" title="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" data-latex="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{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 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}" alt="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}" title="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}" data-latex="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20=%2010%20%5C,%20t%20y%20%7By'%7D%20-%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D" alt="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" title="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" data-latex="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20=%20-5%20%5C,%20y%5E%7B2%7D%20-%205%20%5C,%20t%20%7By'%7D" alt="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}" title="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}" data-latex="2 \, t y^{2} - 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 4 \, t = -5 \, y^{2} - 5 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" alt="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" title="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" data-latex="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11" alt="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11" title="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11" data-latex="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20=%2010%20%5C,%20t%20y%20%7By'%7D%20-%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D" alt="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" title="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}" data-latex="-2 \, t^{2} y {y'} - 2 \, t y^{2} = 10 \, t y {y'} - 15 \, y^{2} {y'} + 5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%5E%7B2%7D%20+%205%20%5C,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B3%7D%20=%2011" alt="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11" title="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11" data-latex="t^{2} y^{2} + 5 \, t y^{2} - 5 \, y^{3} = 11">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3730" title="F4 | Implicit solutions for exact IVPs | ver. 3730"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" alt="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" title="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" data-latex="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}"/></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 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}" alt="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}" title="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}" data-latex="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D" alt="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" title="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" data-latex="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}">
</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,%20t%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20=%202%20%5C,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}" title="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}" data-latex="2 \, t y {y'} + 3 \, t^{2} - 4 \, t {y'} = 2 \, t y^{2} + 5 \, t^{2} {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" alt="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" title="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" data-latex="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)" alt="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)" title="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)" data-latex="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D" alt="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" title="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}" data-latex="0 = -2 \, t^{2} y {y'} - 2 \, t y^{2} + 2 \, t y {y'} + y^{2} + 10 \, y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)" title="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)" data-latex="t^{2} y^{2} - t y^{2} - 5 \, y^{2} = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2695" title="F4 | Implicit solutions for exact IVPs | ver. 2695"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}" alt="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}" title="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}" data-latex="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" alt="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" title="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" data-latex="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%203%20%5C,%20t%20%7By'%7D%20=%206%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20y%20+%202%20%5C,%20y%20%7By'%7D" alt="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}" title="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}" data-latex="-6 \, t y^{2} + 4 \, t y {y'} + 3 \, t {y'} = 6 \, t^{2} + 4 \, t y + 2 \, y {y'}">
</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,%20t%20y%20=%20-2%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D" alt="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" title="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" data-latex="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" alt="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" title="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" data-latex="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{3} + 2 \, t^{2} y + y^{2} = 1" alt="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1" title="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1" data-latex="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20=%20-2%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D" alt="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" title="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}" data-latex="4 \, t y = -2 \, t^{2} {y'} - 6 \, t^{2} - 2 \, y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B3%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20+%20y%5E%7B2%7D%20=%201" alt="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1" title="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1" data-latex="2 \, t^{3} + 2 \, t^{2} y + y^{2} = 1">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7369" title="F4 | Implicit solutions for exact IVPs | ver. 7369"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y" alt="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y" title="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y" data-latex="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y"/></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 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" alt="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" title="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" data-latex="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20y" alt="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y" title="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y" data-latex="3 \, y^{2} = -8 \, t y^{2} + 4 \, t^{2} {y'} - 15 \, y^{2} {y'} + 3 \, y">
</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,%20t%20y%5E%7B2%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20=%208%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D" alt="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" title="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" data-latex="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" alt="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" title="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" data-latex="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6" alt="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6" title="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6" data-latex="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20=%208%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D" alt="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" title="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}" data-latex="-8 \, t y^{2} - 6 \, t y {y'} = 8 \, t^{2} y {y'} + 15 \, y^{2} {y'} + 3 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20y%5E%7B3%7D%20=%206" alt="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6" title="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6" data-latex="4 \, t^{2} y^{2} + 3 \, t y^{2} + 5 \, y^{3} = 6">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9577" title="F4 | Implicit solutions for exact IVPs | ver. 9577"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" alt="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" title="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" data-latex="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y"/></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 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0" alt="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0" title="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0" data-latex="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20t%5E%7B3%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20=%205%20%5C,%20y" alt="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" title="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" data-latex="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2020%20%5C,%20t%5E%7B3%7D%20+%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%205%20%5C,%20y%20=%200" alt="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0" title="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0" data-latex="-2 \, t^{2} y {y'} + 20 \, t^{3} + t^{2} {y'} + 4 \, t y {y'} + 5 \, y = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" alt="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" title="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" data-latex="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)" alt="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)" title="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)" data-latex="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20t%5E%7B3%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20=%205%20%5C,%20y" alt="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" title="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y" data-latex="-20 \, t^{3} - t^{2} {y'} - 2 \, t y - 5 \, t {y'} = 5 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20-%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%20=%20%5Cleft(-1%5Cright)" alt="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)" title="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)" data-latex="-5 \, t^{4} - t^{2} y - 5 \, t y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7878" title="F4 | Implicit solutions for exact IVPs | ver. 7878"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}" alt="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}" title="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}" data-latex="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" alt="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" title="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" data-latex="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20+%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20=%208%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D" alt="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}" title="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}" data-latex="-6 \, t y + y^{2} - 4 \, t {y'} = 8 \, t^{2} y {y'}">
</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,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" title="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" data-latex="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" alt="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" title="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" data-latex="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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} - t^{3} - t y^{2} = 1" alt="3 \, y^{4} - t^{3} - t y^{2} = 1" title="3 \, y^{4} - t^{3} - t y^{2} = 1" data-latex="3 \, y^{4} - t^{3} - t y^{2} = 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" title="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}" data-latex="-12 \, y^{3} {y'} + 2 \, t y {y'} = -3 \, t^{2} - y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%20t%5E%7B3%7D%20-%20t%20y%5E%7B2%7D%20=%201" alt="3 \, y^{4} - t^{3} - t y^{2} = 1" title="3 \, y^{4} - t^{3} - t y^{2} = 1" data-latex="3 \, y^{4} - t^{3} - t y^{2} = 1">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2737" title="F4 | Implicit solutions for exact IVPs | ver. 2737"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" alt="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" title="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" data-latex="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y" alt="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y" title="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y" data-latex="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B3%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%20=%20-4%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20t%20%7By'%7D" alt="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" title="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" data-latex="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-2%20%5C,%20y" alt="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y" title="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y" data-latex="-10 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 4 \, t y - 4 \, y^{2} = -2 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" alt="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" title="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" data-latex="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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} + 4 \, t y^{2} - 2 \, t y = 9" alt="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9" title="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9" data-latex="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%20=%20-4%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20t%20%7By'%7D" alt="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" title="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}" data-latex="12 \, y^{3} {y'} + 8 \, t y {y'} - 2 \, y = -4 \, y^{2} + 2 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%20=%209" alt="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9" title="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9" data-latex="3 \, y^{4} + 4 \, t y^{2} - 2 \, t y = 9">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6619" title="F4 | Implicit solutions for exact IVPs | ver. 6619"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{2}" alt="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{2}" title="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{2}" data-latex="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{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?-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" alt="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" title="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" data-latex="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%205%20%5C,%20y%5E%7B2%7D%20-%205%20%5C,%20t%20%7By'%7D%20=%20-t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D" alt="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{2}" title="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{2}" data-latex="-10 \, t^{2} y {y'} + 5 \, y^{2} - 5 \, t {y'} = -t^{2} {y'} + 3 \, t^{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?-3%20%5C,%20t%5E%7B2%7D%20-%205%20%5C,%20t%20%7By'%7D%20+%208%20%5C,%20y%20%7By'%7D%20-%205%20%5C,%20y%20=%200" alt="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" title="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" data-latex="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" alt="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" title="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" data-latex="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)" alt="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)" title="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)" data-latex="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20-%205%20%5C,%20t%20%7By'%7D%20+%208%20%5C,%20y%20%7By'%7D%20-%205%20%5C,%20y%20=%200" alt="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" title="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0" data-latex="-3 \, t^{2} - 5 \, t {y'} + 8 \, y {y'} - 5 \, y = 0">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B3%7D%20+%205%20%5C,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D%20=%20%5Cleft(-10%5Cright)" alt="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)" title="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)" data-latex="t^{3} + 5 \, t y - 4 \, y^{2} = \left(-10\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0601" title="F4 | Implicit solutions for exact IVPs | ver. 0601"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" alt="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" title="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" data-latex="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}" alt="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}" title="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B3%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20=%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y" alt="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" title="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" data-latex="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%5E%7B3%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D" alt="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}" title="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 8 \, t^{3} = -3 \, t^{2} {y'} - 3 \, y^{2} + 3 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" alt="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" title="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" data-latex="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)" alt="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)" title="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)" data-latex="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20=%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y" alt="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" title="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y" data-latex="8 \, y^{3} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 3 \, t^{2} {y'} + 6 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%203%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-2%5Cright)" alt="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)" title="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)" data-latex="-2 \, y^{4} + 3 \, t^{2} y + 3 \, t y^{2} = \left(-2\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6814" title="F4 | Implicit solutions for exact IVPs | ver. 6814"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}" alt="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}" title="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}" data-latex="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" alt="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" title="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" data-latex="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%20=%20t%5E%7B2%7D%20%7By'%7D" alt="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}" title="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}" data-latex="6 \, t y^{2} - 2 \, t y {y'} + 4 \, y = t^{2} {y'}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20=%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" title="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" data-latex="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" alt="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" title="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" data-latex="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y + 4 \, y^{3} - 4 \, t y = 9" alt="t^{2} y + 4 \, y^{3} - 4 \, t y = 9" title="t^{2} y + 4 \, y^{3} - 4 \, t y = 9" data-latex="t^{2} y + 4 \, y^{3} - 4 \, t y = 9"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20=%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" title="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y" data-latex="t^{2} {y'} + 12 \, y^{2} {y'} + 2 \, t y = 4 \, t {y'} + 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%20+%204%20%5C,%20y%5E%7B3%7D%20-%204%20%5C,%20t%20y%20=%209" alt="t^{2} y + 4 \, y^{3} - 4 \, t y = 9" title="t^{2} y + 4 \, y^{3} - 4 \, t y = 9" data-latex="t^{2} y + 4 \, y^{3} - 4 \, t y = 9">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8448" title="F4 | Implicit solutions for exact IVPs | ver. 8448"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" alt="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" title="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" data-latex="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y"/></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} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y" alt="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y" title="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y" data-latex="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%205%20%5C,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20-%20y" alt="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" title="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" data-latex="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y">
</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,%20y%5E%7B2%7D%20+%206%20%5C,%20y%20%7By'%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%20y" alt="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y" title="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y" data-latex="5 \, y^{2} + 6 \, y {y'} = 4 \, t y^{2} - 3 \, t^{2} {y'} + y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" alt="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" title="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" data-latex="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)" alt="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)" title="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)" data-latex="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%205%20%5C,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20-%20y" alt="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" title="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y" data-latex="0 = 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y + 5 \, y^{2} - t {y'} - y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%5E%7B2%7D%20+%20t%20y%20=%20%5Cleft(-3%5Cright)" alt="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)" title="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)" data-latex="-3 \, t^{2} y - 5 \, t y^{2} + t y = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8001" title="F4 | Implicit solutions for exact IVPs | ver. 8001"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y" alt="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y" title="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y" data-latex="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" alt="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" title="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" data-latex="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20%20%5C,%20t%5E%7B3%7D%20-%208%20%5C,%20t%20y%20%7By'%7D%20+%20t%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y" alt="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y" title="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y" data-latex="20 \, t^{3} - 8 \, t y {y'} + t {y'} = 2 \, t^{2} y {y'} + 8 \, t y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%2020%20%5C,%20t%5E%7B3%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D%20=%200" alt="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" title="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" data-latex="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" alt="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" title="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" data-latex="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13" alt="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13" title="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13" data-latex="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%2020%20%5C,%20t%5E%7B3%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D%20=%200" alt="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" title="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0" data-latex="-16 \, y^{3} {y'} - 20 \, t^{3} + 8 \, t y {y'} + 4 \, y^{2} = 0">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%204%20%5C,%20y%5E%7B4%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%2013" alt="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13" title="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13" data-latex="5 \, t^{4} + 4 \, y^{4} - 4 \, t y^{2} = 13">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5407" title="F4 | Implicit solutions for exact IVPs | ver. 5407"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 0" alt="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 0" title="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 0" data-latex="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 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 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" alt="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" title="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" data-latex="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B3%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20-%206%20%5C,%20t%20=%200" alt="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 0" title="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 0" data-latex="-8 \, y^{3} {y'} - 10 \, t y^{2} - 4 \, t^{2} {y'} - 2 \, y^{2} - t {y'} - 6 \, t = 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,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20=%20-8%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D" alt="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" title="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" data-latex="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" alt="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" title="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" data-latex="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)" alt="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)" title="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)" data-latex="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20=%20-8%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D" alt="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" title="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}" data-latex="4 \, t y {y'} + 6 \, t = -8 \, y^{3} {y'} - 2 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)" title="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)" data-latex="-2 \, y^{4} - 2 \, t y^{2} - 3 \, t^{2} = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2772" title="F4 | Implicit solutions for exact IVPs | ver. 2772"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}" alt="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}" title="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}" data-latex="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}"/></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 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" alt="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" title="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" data-latex="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%204%20%5C,%20t%20%7By'%7D" alt="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}" title="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}" data-latex="3 \, t^{2} = 4 \, t^{2} y {y'} - 8 \, t y {y'} - 4 \, t y - 4 \, t {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20y%20+%204%20%5C,%20y%5E%7B2%7D" alt="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" title="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" data-latex="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" alt="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" title="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" data-latex="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)" alt="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)" title="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)" data-latex="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20y%20+%204%20%5C,%20y%5E%7B2%7D" alt="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" title="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}" data-latex="0 = 2 \, t^{2} {y'} + 8 \, t y {y'} + 3 \, t^{2} + 4 \, t y + 4 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B3%7D%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)" title="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)" data-latex="-t^{3} - 2 \, t^{2} y - 4 \, t y^{2} = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3214" title="F4 | Implicit solutions for exact IVPs | ver. 3214"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" alt="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" title="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" data-latex="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0" alt="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0" title="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0" data-latex="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20=%20t%20%7By'%7D%20+%20y" alt="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" title="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" data-latex="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y">
</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,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20=%200" alt="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0" title="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0" data-latex="4 \, t y^{2} + 3 \, t^{2} {y'} - 4 \, y^{2} - t {y'} = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" alt="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" title="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" data-latex="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7" alt="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7" title="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7" data-latex="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20=%20t%20%7By'%7D%20+%20y" alt="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" title="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y" data-latex="4 \, t^{2} y {y'} + 16 \, y^{3} {y'} + 4 \, t y^{2} = t {y'} + y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20y%5E%7B4%7D%20-%20t%20y%20=%207" alt="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7" title="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7" data-latex="2 \, t^{2} y^{2} + 4 \, y^{4} - t y = 7">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2585" title="F4 | Implicit solutions for exact IVPs | ver. 2585"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}" alt="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}" title="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}" data-latex="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" alt="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" title="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" data-latex="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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=%20-20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20-%2010%20%5C,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D" alt="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}" title="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}" data-latex="0 = -20 \, y^{3} {y'} - 2 \, t y^{2} - 10 \, t y - 4 \, y^{2} - 4 \, t {y'}">
</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,%20t%20%7By'%7D%20=%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%2010%20%5C,%20t%20y%20+%204%20%5C,%20y%5E%7B2%7D%20+%204%20%5C,%20y" alt="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" title="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" data-latex="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" alt="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" title="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" data-latex="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5" alt="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5" title="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5" data-latex="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20%7By'%7D%20=%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%2010%20%5C,%20t%20y%20+%204%20%5C,%20y%5E%7B2%7D%20+%204%20%5C,%20y" alt="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" title="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y" data-latex="-4 \, t {y'} = 5 \, t^{2} {y'} + 8 \, t y {y'} + 10 \, t y + 4 \, y^{2} + 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20=%205" alt="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5" title="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5" data-latex="-5 \, t^{2} y - 4 \, t y^{2} - 4 \, t y = 5">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2597" title="F4 | Implicit solutions for exact IVPs | ver. 2597"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" alt="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" title="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" data-latex="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{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?20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t" alt="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t" title="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t" data-latex="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%20y%5E%7B2%7D" alt="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" title="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" data-latex="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{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?20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-5%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20t" alt="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t" title="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t" data-latex="20 \, y^{3} {y'} - 10 \, t y^{2} - 2 \, t^{2} {y'} - y^{2} = -5 \, t {y'} + 4 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" alt="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" title="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" data-latex="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2" alt="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2" title="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2" data-latex="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%20y%5E%7B2%7D" alt="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" title="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}" data-latex="2 \, t y {y'} + 4 \, t = 20 \, y^{3} {y'} - y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20=%202" alt="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2" title="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2" data-latex="5 \, y^{4} - t y^{2} - 2 \, t^{2} = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9370" title="F4 | Implicit solutions for exact IVPs | ver. 9370"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" alt="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" title="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" data-latex="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}"/></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 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y" alt="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y" title="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y" data-latex="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%20y%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%20t%20%7By'%7D" alt="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" title="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" data-latex="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20=%208%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%20y" alt="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y" title="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y" data-latex="2 \, t^{2} {y'} - 10 \, t y {y'} + 6 \, t = 8 \, t^{2} y {y'} + y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" alt="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" title="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" data-latex="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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} - 2 \, t^{2} y + t y = \left(-3\right)" alt="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)" title="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)" data-latex="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%20y%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%20t%20%7By'%7D" alt="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" title="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}" data-latex="-16 \, y^{3} {y'} - 4 \, t y + y = 2 \, t^{2} {y'} - t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%202%20%5C,%20t%5E%7B2%7D%20y%20+%20t%20y%20=%20%5Cleft(-3%5Cright)" alt="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)" title="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)" data-latex="-4 \, y^{4} - 2 \, t^{2} y + t y = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8011" title="F4 | Implicit solutions for exact IVPs | ver. 8011"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" alt="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" title="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" data-latex="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}"/></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 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t" alt="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t" title="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t" data-latex="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%20=%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D%20-%202%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y%20%7By'%7D" alt="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" title="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" data-latex="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20y%20=%20t%5E%7B2%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t" alt="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t" title="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t" data-latex="-8 \, t^{2} y {y'} - 2 \, y = t^{2} {y'} - 8 \, t y {y'} + 6 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" alt="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" title="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" data-latex="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)" alt="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)" title="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)" data-latex="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%20=%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D%20-%202%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y%20%7By'%7D" alt="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" title="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}" data-latex="2 \, y = 8 \, t y {y'} + 4 \, y^{2} - 2 \, t {y'} + 4 \, y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%20%5Cleft(-8%5Cright)" alt="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)" title="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)" data-latex="-4 \, t y^{2} + 2 \, t y - 2 \, y^{2} = \left(-8\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4653" title="F4 | Implicit solutions for exact IVPs | ver. 4653"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y" alt="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y" title="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y" data-latex="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y"/></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 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%20y%20+%205%20%5C,%20y%5E%7B2%7D%20-%20y" alt="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y" title="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y" data-latex="16 \, y^{3} {y'} = -2 \, t^{2} y {y'} + 8 \, t y + 5 \, y^{2} - y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20y%20+%20t%20%7By'%7D%20+%20y%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)" alt="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)" title="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)" data-latex="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20y%20+%20t%20%7By'%7D%20+%20y%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" title="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} - 8 \, t y + t {y'} + y = 4 \, t^{2} {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20+%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)" title="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)" data-latex="t^{2} y^{2} - 4 \, t^{2} y + t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5765" title="F4 | Implicit solutions for exact IVPs | ver. 5765"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" alt="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" title="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" data-latex="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}" alt="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}" title="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}" data-latex="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20%7By'%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B3%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y" alt="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" title="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" data-latex="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20%7By'%7D%20-%20y%20=%20-4%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B3%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20%7By'%7D" alt="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}" title="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}" data-latex="t^{2} {y'} - y = -4 \, y^{3} {y'} - 4 \, t^{3} + 2 \, t y^{2} - 4 \, t y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" alt="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" title="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" data-latex="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)" alt="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)" title="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)" data-latex="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20%7By'%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B3%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y" alt="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" title="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y" data-latex="-t^{2} {y'} = -2 \, t^{2} y {y'} + 4 \, t^{3} - 2 \, t y^{2} + 2 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B4%7D%20-%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20y%20=%20%5Cleft(-1%5Cright)" alt="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)" title="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)" data-latex="t^{4} - t^{2} y^{2} + t^{2} y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2736" title="F4 | Implicit solutions for exact IVPs | ver. 2736"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" alt="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}" alt="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}" title="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}" data-latex="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%206%20%5C,%20t%5E%7B2%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y">
</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,%20t%20y%20%7By'%7D%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%20+%20t%20%7By'%7D" alt="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}" title="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}" data-latex="-4 \, t y {y'} = -4 \, t y^{2} + 2 \, t y + t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" alt="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)" alt="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)" title="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)" data-latex="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%206%20%5C,%20t%5E%7B2%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-4 \, t^{2} y {y'} - 6 \, t^{2} = 4 \, t y^{2} - t^{2} {y'} - 2 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B3%7D%20+%20t%5E%7B2%7D%20y%20=%20%5Cleft(-5%5Cright)" alt="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)" title="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)" data-latex="-2 \, t^{2} y^{2} - 2 \, t^{3} + t^{2} y = \left(-5\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2129" title="F4 | Implicit solutions for exact IVPs | ver. 2129"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" alt="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" title="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" data-latex="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{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 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y" alt="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y" title="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y" data-latex="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2016%20%5C,%20t%5E%7B3%7D%20=%208%20%5C,%20t%20y%5E%7B2%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D" alt="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" title="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" data-latex="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{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,%20t%5E%7B2%7D%20%7By'%7D%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20y%5E%7B2%7D%20-%203%20%5C,%20y" alt="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y" title="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y" data-latex="-4 \, t^{2} {y'} = 16 \, y^{3} {y'} - 8 \, t y^{2} - 4 \, y^{2} - 3 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" alt="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" title="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" data-latex="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)" alt="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)" title="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)" data-latex="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2016%20%5C,%20t%5E%7B3%7D%20=%208%20%5C,%20t%20y%5E%7B2%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D" alt="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" title="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}" data-latex="-8 \, t^{2} y {y'} + 16 \, t^{3} = 8 \, t y^{2} + 8 \, t y {y'} + 4 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-4%5Cright)" alt="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)" title="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)" data-latex="-4 \, t^{4} + 4 \, t^{2} y^{2} + 4 \, t y^{2} = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0587" title="F4 | Implicit solutions for exact IVPs | ver. 0587"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y" alt="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y" title="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y" data-latex="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" alt="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" title="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" data-latex="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20=%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%5E%7B3%7D%20-%206%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20y" alt="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y" title="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y" data-latex="3 \, t^{2} {y'} - 10 \, t y {y'} = 12 \, y^{3} {y'} - 4 \, t^{3} - 6 \, t y^{2} + 4 \, y">
</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,%20t%5E%7B3%7D%20+%205%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20%7By'%7D%20-%204%20%5C,%20y" alt="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" title="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" data-latex="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" alt="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" title="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" data-latex="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{4} + 5 \, t y^{2} + 4 \, t y = 0" alt="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0" title="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0" data-latex="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B3%7D%20+%205%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20%7By'%7D%20-%204%20%5C,%20y" alt="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" title="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y" data-latex="-4 \, t^{3} + 5 \, y^{2} = -10 \, t y {y'} - 4 \, t {y'} - 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B4%7D%20+%205%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20=%200" alt="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0" title="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0" data-latex="-t^{4} + 5 \, t y^{2} + 4 \, t y = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6300" title="F4 | Implicit solutions for exact IVPs | ver. 6300"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y" alt="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y" title="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y" data-latex="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" title="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20y%5E%7B2%7D%20+%20t%20%7By'%7D%20-%204%20%5C,%20y%20%7By'%7D%20+%208%20%5C,%20t%20=%202%20%5C,%20t%20y" alt="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y" title="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y" data-latex="-4 \, t^{2} y {y'} + 4 \, y^{2} + t {y'} - 4 \, y {y'} + 8 \, t = 2 \, t y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" title="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" title="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7" alt="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7" title="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7" data-latex="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" title="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} - 4 \, y^{2} = -t^{2} {y'} + 8 \, t y {y'} - 2 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%207" alt="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7" title="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7" data-latex="2 \, t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 7">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0439" title="F4 | Implicit solutions for exact IVPs | ver. 0439"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" alt="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, 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?3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" alt="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%202%20%5C,%20t" alt="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, 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?3%20%5C,%20y%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20-%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%202%20%5C,%20t" alt="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="3 \, y = -6 \, t^{2} y {y'} + 4 \, t y {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" alt="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y - 3 \, y^{3} + t^{2} = 6" alt="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6" title="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6" data-latex="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%202%20%5C,%20t" alt="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" title="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t" data-latex="0 = -2 \, t^{2} {y'} - 9 \, y^{2} {y'} - 4 \, t y + 2 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20-%203%20%5C,%20y%5E%7B3%7D%20+%20t%5E%7B2%7D%20=%206" alt="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6" title="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6" data-latex="-2 \, t^{2} y - 3 \, y^{3} + t^{2} = 6">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6062" title="F4 | Implicit solutions for exact IVPs | ver. 6062"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" alt="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" title="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" data-latex="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0" alt="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0" title="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0" data-latex="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%203%20%5C,%20y%20=%20-10%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D" alt="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" title="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" data-latex="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-10%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%20y%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D%20+%203%20%5C,%20y%20=%200" alt="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0" title="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0" data-latex="-10 \, t y^{2} + 2 \, t^{2} {y'} - y^{2} - 2 \, y {y'} + 3 \, y = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" alt="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" title="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" data-latex="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)" alt="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)" title="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)" data-latex="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%203%20%5C,%20y%20=%20-10%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D" alt="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" title="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}" data-latex="10 \, t^{2} y {y'} - 2 \, t^{2} {y'} - 4 \, t y - 3 \, y = -10 \, t y^{2} + 3 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%20=%20%5Cleft(-6%5Cright)" alt="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)" title="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)" data-latex="-5 \, t^{2} y^{2} + 2 \, t^{2} y + 3 \, t y = \left(-6\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8301" title="F4 | Implicit solutions for exact IVPs | ver. 8301"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" alt="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" title="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" data-latex="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t" alt="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t" title="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t" data-latex="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%208%20%5C,%20t%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20%7By'%7D" alt="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" title="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" data-latex="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20=%208%20%5C,%20t%20y%20+%205%20%5C,%20t%20%7By'%7D%20+%208%20%5C,%20t" alt="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t" title="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t" data-latex="-4 \, t^{2} y {y'} + 6 \, y^{2} {y'} - 2 \, y^{2} = 8 \, t y + 5 \, t {y'} + 8 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" alt="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" title="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" data-latex="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)" alt="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)" title="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)" data-latex="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%208%20%5C,%20t%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20%7By'%7D" alt="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" title="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}" data-latex="4 \, t^{2} y {y'} + 2 \, y^{2} + 8 \, t = -4 \, t y^{2} - 4 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20=%20%5Cleft(-8%5Cright)" alt="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)" title="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)" data-latex="-2 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t^{2} = \left(-8\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8669" title="F4 | Implicit solutions for exact IVPs | ver. 8669"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y" alt="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y" title="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y" data-latex="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y"/></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 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" alt="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" title="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%205%20%5C,%20y" alt="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y" title="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y" data-latex="6 \, t y - 3 \, y^{2} = -8 \, t^{2} y {y'} - 8 \, y^{3} {y'} - 5 \, y">
</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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20=%20-15%20%5C,%20t%5E%7B2%7D%20+%205%20%5C,%20y" alt="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" title="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" alt="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" title="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)" alt="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)" title="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)" data-latex="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20=%20-15%20%5C,%20t%5E%7B2%7D%20+%205%20%5C,%20y" alt="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" title="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 5 \, t {y'} = -15 \, t^{2} + 5 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B3%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%20=%20%5Cleft(-3%5Cright)" alt="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)" title="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)" data-latex="5 \, t^{3} - 3 \, t^{2} y - 5 \, t y = \left(-3\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8869" title="F4 | Implicit solutions for exact IVPs | ver. 8869"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}" alt="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}" title="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}" data-latex="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}"/></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 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" alt="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" title="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" data-latex="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20=%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%20-%203%20%5C,%20t%20%7By'%7D" alt="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}" title="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}" data-latex="8 \, t y^{2} = 4 \, t y {y'} + 2 \, t y - 3 \, t {y'}">
</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,%20t%20y%20=%20-t%5E%7B2%7D%20%7By'%7D%20+%2015%20%5C,%20t%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D" alt="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" title="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" data-latex="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" alt="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" title="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" data-latex="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)" alt="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)" title="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)" data-latex="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20=%20-t%5E%7B2%7D%20%7By'%7D%20+%2015%20%5C,%20t%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D" alt="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" title="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}" data-latex="2 \, t y = -t^{2} {y'} + 15 \, t^{2} + 10 \, y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B3%7D%20+%20t%5E%7B2%7D%20y%20-%205%20%5C,%20y%5E%7B2%7D%20=%20%5Cleft(-11%5Cright)" alt="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)" title="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)" data-latex="-5 \, t^{3} + t^{2} y - 5 \, y^{2} = \left(-11\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9033" title="F4 | Implicit solutions for exact IVPs | ver. 9033"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y" alt="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y" title="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y" data-latex="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" title="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20%7By'%7D%20+%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20y%20=%204%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20y" alt="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y" title="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y" data-latex="-6 \, t y {y'} + 9 \, y^{2} {y'} - 4 \, y = 4 \, t y^{2} - 8 \, t y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20+%209%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y%20=%200" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" title="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" title="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)" alt="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)" title="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)" data-latex="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20+%209%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y%20=%200" alt="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" title="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0" data-latex="4 \, t^{2} y {y'} + 4 \, t y^{2} + 9 \, t^{2} + 4 \, t {y'} + 4 \, y = 0">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B3%7D%20-%204%20%5C,%20t%20y%20=%20%5Cleft(-1%5Cright)" alt="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)" title="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)" data-latex="-2 \, t^{2} y^{2} - 3 \, t^{3} - 4 \, t y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8066" title="F4 | Implicit solutions for exact IVPs | ver. 8066"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" alt="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" title="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" data-latex="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{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 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y" alt="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y" title="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y" data-latex="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20%7By'%7D%20=%20-16%20%5C,%20t%5E%7B3%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D" alt="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" title="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" data-latex="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{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,%20t%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20y" alt="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y" title="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y" data-latex="-2 \, t^{2} {y'} + 2 \, y^{2} = -10 \, t^{2} y {y'} - 2 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" alt="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" title="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" data-latex="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)" alt="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)" title="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)" data-latex="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20%7By'%7D%20=%20-16%20%5C,%20t%5E%7B3%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D" alt="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" title="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}" data-latex="9 \, y^{2} {y'} = -16 \, t^{3} + 4 \, t y {y'} + 2 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20y%5E%7B3%7D%20=%20%5Cleft(-1%5Cright)" alt="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)" title="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)" data-latex="4 \, t^{4} - 2 \, t y^{2} + 3 \, y^{3} = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1080" title="F4 | Implicit solutions for exact IVPs | ver. 1080"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y" alt="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y" title="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y" data-latex="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" alt="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" title="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" data-latex="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%20+%205%20%5C,%20y" alt="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y" title="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y" data-latex="2 \, t^{2} y {y'} + 3 \, t^{2} {y'} = -4 \, t y {y'} + 4 \, t + 5 \, y">
</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,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t" alt="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" title="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" data-latex="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" alt="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" title="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" data-latex="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0" alt="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0" title="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0" data-latex="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t" alt="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" title="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t" data-latex="4 \, y^{3} {y'} + 2 \, t y^{2} = -2 \, t^{2} y {y'} + 4 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20y%5E%7B2%7D%20-%20y%5E%7B4%7D%20+%202%20%5C,%20t%5E%7B2%7D%20=%200" alt="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0" title="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0" data-latex="-t^{2} y^{2} - y^{4} + 2 \, t^{2} = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0327" title="F4 | Implicit solutions for exact IVPs | ver. 0327"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" alt="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" title="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" data-latex="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{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 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y" alt="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y" title="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y" data-latex="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20%7By'%7D%20=%204%20%5C,%20t%5E%7B3%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D" alt="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" title="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" data-latex="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{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,%20t%5E%7B3%7D%20-%208%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%20%7By'%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2010%20%5C,%20t%20y" alt="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y" title="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y" data-latex="4 \, t^{3} - 8 \, t y {y'} + 2 \, t {y'} = -6 \, t^{2} y {y'} - 10 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" alt="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" title="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" data-latex="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10" alt="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10" title="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10" data-latex="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20=%204%20%5C,%20t%5E%7B3%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%2010%20%5C,%20t%20y%20-%204%20%5C,%20y%5E%7B2%7D" alt="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" title="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}" data-latex="8 \, t y {y'} = 4 \, t^{3} + 5 \, t^{2} {y'} + 10 \, t y - 4 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B4%7D%20+%205%20%5C,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%2010" alt="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10" title="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10" data-latex="t^{4} + 5 \, t^{2} y - 4 \, t y^{2} = 10">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4296" title="F4 | Implicit solutions for exact IVPs | ver. 4296"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" alt="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" title="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" data-latex="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}" alt="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}" title="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}" data-latex="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%20y%5E%7B2%7D%20-%20y%20=%202%20%5C,%20t%20y%20%7By'%7D%20+%20t%20%7By'%7D" alt="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" title="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" data-latex="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20-%20y%5E%7B2%7D%20-%20y%20=%20-10%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D" alt="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}" title="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}" data-latex="16 \, y^{3} {y'} - 4 \, t y - y^{2} - y = -10 \, t^{2} y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" alt="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" title="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" data-latex="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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} - t y^{2} - t y = 4" alt="4 \, y^{4} - t y^{2} - t y = 4" title="4 \, y^{4} - t y^{2} - t y = 4" data-latex="4 \, y^{4} - t y^{2} - t y = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?16%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%20y%5E%7B2%7D%20-%20y%20=%202%20%5C,%20t%20y%20%7By'%7D%20+%20t%20%7By'%7D" alt="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" title="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}" data-latex="16 \, y^{3} {y'} - y^{2} - y = 2 \, t y {y'} + t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%20t%20y%5E%7B2%7D%20-%20t%20y%20=%204" alt="4 \, y^{4} - t y^{2} - t y = 4" title="4 \, y^{4} - t y^{2} - t y = 4" data-latex="4 \, y^{4} - t y^{2} - t y = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2517" title="F4 | Implicit solutions for exact IVPs | ver. 2517"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y" alt="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y" title="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y" data-latex="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" alt="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" title="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2010%20%5C,%20t%20y%20=%202%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20y%20%7By'%7D%20+%204%20%5C,%20y" alt="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y" title="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y" data-latex="-6 \, t^{2} y {y'} - 10 \, t y = 2 \, y^{2} + 2 \, y {y'} + 4 \, y">
</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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" title="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" alt="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" title="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)" alt="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)" title="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)" data-latex="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%204%20%5C,%20t%20%7By'%7D%20+%204%20%5C,%20y" alt="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" title="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y" data-latex="-6 \, t y^{2} - 4 \, t y {y'} - 2 \, y^{2} = 6 \, t^{2} y {y'} + 4 \, t {y'} + 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y%20=%20%5Cleft(-1%5Cright)" alt="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)" title="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)" data-latex="-3 \, t^{2} y^{2} - 2 \, t y^{2} - 4 \, t y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-9523" title="F4 | Implicit solutions for exact IVPs | ver. 9523"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y" alt="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y" title="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y" data-latex="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" alt="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" title="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" data-latex="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20=%20-12%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20t%20-%202%20%5C,%20y" alt="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y" title="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y" data-latex="-6 \, t^{2} y {y'} = -12 \, y^{3} {y'} + 4 \, t^{2} {y'} - 5 \, y^{2} + 2 \, t - 2 \, y">
</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,%20t%20y%5E%7B2%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t" alt="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" title="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" data-latex="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" alt="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" title="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" data-latex="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9" alt="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9" title="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9" data-latex="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t" alt="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" title="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t" data-latex="6 \, t y^{2} - 10 \, t y {y'} - 5 \, y^{2} = -6 \, t^{2} y {y'} - 2 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%205%20%5C,%20t%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20=%209" alt="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9" title="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9" data-latex="3 \, t^{2} y^{2} - 5 \, t y^{2} + t^{2} = 9">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1736" title="F4 | Implicit solutions for exact IVPs | ver. 1736"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" alt="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}" alt="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}" title="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}" data-latex="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20%7By'%7D%20+%2015%20%5C,%20t%5E%7B2%7D%20+%20t%20%7By'%7D%20=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20y%5E%7B2%7D" alt="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}" title="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}" data-latex="t^{2} {y'} + 15 \, t^{2} + t {y'} = 20 \, y^{3} {y'} + 2 \, t y^{2} + 4 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" alt="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4" alt="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4" title="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4" data-latex="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20t%20y" alt="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" title="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y" data-latex="-8 \, t y {y'} - 4 \, y^{2} = 2 \, t^{2} y {y'} + 2 \, t y^{2} - t^{2} {y'} - 2 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20y%20-%204%20%5C,%20t%20y%5E%7B2%7D%20=%204" alt="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4" title="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4" data-latex="-t^{2} y^{2} + t^{2} y - 4 \, t y^{2} = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6233" title="F4 | Implicit solutions for exact IVPs | ver. 6233"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" alt="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" title="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" data-latex="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y" alt="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y" title="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y" data-latex="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%20t%20%7By'%7D%20-%206%20%5C,%20y%20%7By'%7D%20+%20y" alt="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" title="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" data-latex="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20y" alt="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y" title="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y" data-latex="-t {y'} = 4 \, t y^{2} - 6 \, t y {y'} - 4 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" alt="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" title="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" data-latex="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)" alt="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)" title="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)" data-latex="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20+%20t%20%7By'%7D%20-%206%20%5C,%20y%20%7By'%7D%20+%20y" alt="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" title="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y" data-latex="0 = -2 \, t^{2} {y'} - 4 \, t y + t {y'} - 6 \, y {y'} + y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20+%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20=%20%5Cleft(-6%5Cright)" alt="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)" title="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)" data-latex="-2 \, t^{2} y + t y - 3 \, y^{2} = \left(-6\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1327" title="F4 | Implicit solutions for exact IVPs | ver. 1327"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" alt="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" title="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" data-latex="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y"/></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 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}" alt="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}" title="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}" data-latex="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20%7By'%7D%20=%20-8%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%204%20%5C,%20y" alt="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" title="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" data-latex="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y">
</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,%20t%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D" alt="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}" title="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}" data-latex="-2 \, t y^{2} = 4 \, t^{2} {y'} - 2 \, y^{2} - 4 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" alt="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" title="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" data-latex="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)" alt="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)" title="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)" data-latex="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20-%204%20%5C,%20t%20%7By'%7D%20=%20-8%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%204%20%5C,%20y" alt="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" title="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y" data-latex="-4 \, t y {y'} - 4 \, t {y'} = -8 \, y^{3} {y'} + 2 \, y^{2} + 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)" title="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)" data-latex="-2 \, y^{4} + 2 \, t y^{2} + 4 \, t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4788" title="F4 | Implicit solutions for exact IVPs | ver. 4788"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y" alt="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y" title="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y" data-latex="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20=%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20+%203%20%5C,%20y" alt="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y" title="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y" data-latex="-4 \, t y^{2} = 5 \, t^{2} {y'} - 10 \, t y {y'} + 3 \, y">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20=%20-5%20%5C,%20y%5E%7B2%7D" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0" alt="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0" title="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0" data-latex="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%209%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20=%20-5%20%5C,%20y%5E%7B2%7D" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 10 \, t y {y'} + 9 \, y^{2} {y'} = -5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%205%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20y%5E%7B3%7D%20=%200" alt="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0" title="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0" data-latex="-2 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{3} = 0">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1476" title="F4 | Implicit solutions for exact IVPs | ver. 1476"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" alt="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" title="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" data-latex="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}" alt="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}" title="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}" data-latex="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20y" alt="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" title="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" data-latex="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y">
</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,%20y%5E%7B2%7D%20=%204%20%5C,%20t%20y%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20t%20%7By'%7D" alt="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}" title="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}" data-latex="3 \, y^{2} = 4 \, t y^{2} + 4 \, t^{2} {y'} + 3 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" alt="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" title="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" data-latex="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4" alt="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4" title="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4" data-latex="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%20y%5E%7B2%7D%20-%208%20%5C,%20t%20y" alt="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" title="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y" data-latex="4 \, t^{2} y {y'} - 8 \, y^{3} {y'} + 4 \, t^{2} {y'} = -4 \, t y^{2} - 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20y%5E%7B4%7D%20-%204%20%5C,%20t%5E%7B2%7D%20y%20=%204" alt="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4" title="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4" data-latex="-2 \, t^{2} y^{2} + 2 \, y^{4} - 4 \, t^{2} y = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1079" title="F4 | Implicit solutions for exact IVPs | ver. 1079"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{2}" alt="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{2}" title="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{2}" data-latex="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{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?20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" alt="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" title="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" data-latex="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20%7By'%7D%20=%20-20%20%5C,%20t%5E%7B3%7D%20-%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D" alt="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{2}" title="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{2}" data-latex="6 \, t^{2} y {y'} - 4 \, t {y'} = -20 \, t^{3} - t^{2} {y'} - 2 \, y^{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?20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%20=%20-2%20%5C,%20y%5E%7B2%7D" alt="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" title="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" data-latex="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" alt="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" title="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" data-latex="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4" alt="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4" title="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4" data-latex="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%20=%20-2%20%5C,%20y%5E%7B2%7D" alt="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" title="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}" data-latex="20 \, y^{3} {y'} + t^{2} {y'} + 4 \, t y {y'} + 2 \, t y = -2 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%20t%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%5E%7B2%7D%20=%204" alt="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4" title="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4" data-latex="5 \, y^{4} + t^{2} y + 2 \, t y^{2} = 4">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1487" title="F4 | Implicit solutions for exact IVPs | ver. 1487"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" alt="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" title="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" data-latex="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{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 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y" alt="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y" title="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y" data-latex="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%208%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%209%20%5C,%20t%5E%7B2%7D" alt="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" title="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" data-latex="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{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,%20t%20y%20%7By'%7D%20-%209%20%5C,%20t%5E%7B2%7D%20=%208%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20+%20y" alt="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y" title="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y" data-latex="-8 \, t y {y'} - 9 \, t^{2} = 8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 6 \, t y + y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" alt="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" title="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" data-latex="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)" alt="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)" title="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)" data-latex="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%20=%208%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%209%20%5C,%20t%5E%7B2%7D" alt="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" title="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}" data-latex="-8 \, t^{2} y {y'} + 6 \, t y = 8 \, t y^{2} - 3 \, t^{2} {y'} + 9 \, t^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%203%20%5C,%20t%5E%7B3%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20=%20%5Cleft(-2%5Cright)" alt="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)" title="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)" data-latex="4 \, t^{2} y^{2} + 3 \, t^{3} - 3 \, t^{2} y = \left(-2\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-0575" title="F4 | Implicit solutions for exact IVPs | ver. 0575"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" alt="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" title="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" data-latex="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y" alt="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y" title="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y" data-latex="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B3%7D%20+%208%20%5C,%20t%20y" alt="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" title="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" data-latex="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y">
</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,%20t%20y%20%7By'%7D%20-%208%20%5C,%20t%20y%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%5E%7B3%7D%20-%203%20%5C,%20y" alt="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y" title="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y" data-latex="-6 \, t y {y'} - 8 \, t y = -4 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 4 \, t^{3} - 3 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" alt="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" title="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" data-latex="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2" alt="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2" title="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2" data-latex="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%203%20%5C,%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B3%7D%20+%208%20%5C,%20t%20y" alt="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" title="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y" data-latex="-4 \, t^{2} {y'} - 6 \, t y {y'} - 3 \, y^{2} = 4 \, t^{3} + 8 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B4%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%5E%7B2%7D%20=%202" alt="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2" title="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2" data-latex="t^{4} + 4 \, t^{2} y + 3 \, t y^{2} = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5983" title="F4 | Implicit solutions for exact IVPs | ver. 5983"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{2}" alt="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{2}" title="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{2}" data-latex="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{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 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" alt="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" title="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" data-latex="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20%7By'%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%2016%20%5C,%20t%5E%7B3%7D%20-%206%20%5C,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D" alt="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{2}" title="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{2}" data-latex="-t {y'} = -6 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 16 \, t^{3} - 6 \, t y - 3 \, y^{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,%20y%5E%7B3%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y" alt="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" title="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" data-latex="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" alt="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" title="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" data-latex="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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} - 3 \, t^{2} y - 3 \, t y^{2} = 3" alt="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3" title="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3" data-latex="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B3%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y" alt="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" title="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y" data-latex="-12 \, y^{3} {y'} + 3 \, y^{2} = -3 \, t^{2} {y'} - 6 \, t y {y'} - 6 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%203%20%5C,%20t%20y%5E%7B2%7D%20=%203" alt="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3" title="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3" data-latex="3 \, y^{4} - 3 \, t^{2} y - 3 \, t y^{2} = 3">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6448" title="F4 | Implicit solutions for exact IVPs | ver. 6448"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y" alt="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y" title="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y" data-latex="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" alt="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" title="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" data-latex="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20+%203%20%5C,%20t%20%7By'%7D%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y" title="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y" data-latex="2 \, y^{2} + 3 \, t {y'} = 6 \, t^{2} y {y'} - 8 \, t y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%208%20%5C,%20t%20y%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" title="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" data-latex="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" alt="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" title="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" data-latex="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2" alt="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2" title="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2" data-latex="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-20%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%208%20%5C,%20t%20y%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D" alt="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" title="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}" data-latex="-20 \, y^{3} {y'} - 3 \, t^{2} - 8 \, t y = 4 \, t^{2} {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20+%20t%5E%7B3%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20=%202" alt="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2" title="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2" data-latex="5 \, y^{4} + t^{3} + 4 \, t^{2} y = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7821" title="F4 | Implicit solutions for exact IVPs | ver. 7821"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}" alt="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}" title="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}" data-latex="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}"/></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 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" alt="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" title="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" data-latex="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%208%20%5C,%20t%5E%7B3%7D%20+%20y%5E%7B2%7D%20-%203%20%5C,%20t%20%7By'%7D" alt="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}" title="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}" data-latex="2 \, t^{2} {y'} = -4 \, t^{2} y {y'} + 8 \, t^{3} + y^{2} - 3 \, t {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%5E%7B3%7D" alt="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" title="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" data-latex="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" alt="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" title="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" data-latex="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1" alt="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1" title="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1" data-latex="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20t%20y%20=%2020%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%208%20%5C,%20t%5E%7B3%7D" alt="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" title="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}" data-latex="-2 \, t^{2} {y'} - 4 \, t y = 20 \, y^{3} {y'} - 8 \, t^{3}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%205%20%5C,%20y%5E%7B4%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20=%201" alt="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1" title="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1" data-latex="-2 \, t^{4} + 5 \, y^{4} + 2 \, t^{2} y = 1">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6714" title="F4 | Implicit solutions for exact IVPs | ver. 6714"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" alt="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" title="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" data-latex="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}" alt="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}" title="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}" data-latex="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2016%20%5C,%20t%5E%7B3%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20=%20y" alt="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" title="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" data-latex="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-16%20%5C,%20t%5E%7B3%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%20y%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20y%20%7By'%7D" alt="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}" title="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}" data-latex="-16 \, t^{3} - 4 \, t^{2} {y'} - y = -4 \, t^{2} y {y'} - 2 \, t y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" alt="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" title="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" data-latex="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)" alt="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)" title="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)" data-latex="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2016%20%5C,%20t%5E%7B3%7D%20+%204%20%5C,%20t%20y%5E%7B2%7D%20-%20t%20%7By'%7D%20=%20y" alt="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" title="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y" data-latex="4 \, t^{2} y {y'} - 16 \, t^{3} + 4 \, t y^{2} - t {y'} = y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B4%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%20t%20y%20=%20%5Cleft(-1%5Cright)" alt="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)" title="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)" data-latex="-4 \, t^{4} + 2 \, t^{2} y^{2} - t y = \left(-1\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-3502" title="F4 | Implicit solutions for exact IVPs | ver. 3502"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}" alt="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}" title="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}" data-latex="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" alt="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" title="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" data-latex="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20-%2015%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20=%20-8%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20+%202%20%5C,%20t%20%7By'%7D" alt="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}" title="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}" data-latex="6 \, t y^{2} - 15 \, y^{2} {y'} = -8 \, t y - 2 \, y^{2} + 2 \, t {y'}">
</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,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%20y%20+%204%20%5C,%20t" alt="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" title="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" data-latex="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" alt="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" title="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" data-latex="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)" alt="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)" title="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)" data-latex="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%20y%20+%204%20%5C,%20t" alt="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" title="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t" data-latex="4 \, t^{2} {y'} + 4 \, t y {y'} + 2 \, y^{2} = -8 \, t y + 4 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20=%20%5Cleft(-4%5Cright)" alt="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)" title="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)" data-latex="4 \, t^{2} y + 2 \, t y^{2} - 2 \, t^{2} = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6785" title="F4 | Implicit solutions for exact IVPs | ver. 6785"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, 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?t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y" alt="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y" title="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y" data-latex="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%20y%20=%20-t%20%7By'%7D%20-%204%20%5C,%20t" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, 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?t%20%7By'%7D%20+%204%20%5C,%20t%20=%208%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%202%20%5C,%20t%20y%5E%7B2%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20-%208%20%5C,%20t%20y" alt="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y" title="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y" data-latex="t {y'} + 4 \, t = 8 \, y^{3} {y'} - 2 \, t y^{2} + 10 \, t y {y'} - 8 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} y^{2} + 2 \, t^{2} + t y = 2" alt="t^{2} y^{2} + 2 \, t^{2} + t y = 2" title="t^{2} y^{2} + 2 \, t^{2} + t y = 2" data-latex="t^{2} y^{2} + 2 \, t^{2} + t y = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%20y%20=%20-t%20%7By'%7D%20-%204%20%5C,%20t" alt="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" title="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t" data-latex="2 \, t^{2} y {y'} + 2 \, t y^{2} + y = -t {y'} - 4 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20+%20t%20y%20=%202" alt="t^{2} y^{2} + 2 \, t^{2} + t y = 2" title="t^{2} y^{2} + 2 \, t^{2} + t y = 2" data-latex="t^{2} y^{2} + 2 \, t^{2} + t y = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1650" title="F4 | Implicit solutions for exact IVPs | ver. 1650"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{2}" alt="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{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 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" alt="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" title="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" data-latex="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%20=%203%20%5C,%20y%5E%7B2%7D" alt="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{2}" title="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{2}" data-latex="-4 \, t^{2} y {y'} - 4 \, t^{2} {y'} + 5 \, y = 3 \, y^{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,%20t%5E%7B2%7D%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%20=%208%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D" alt="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" title="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" data-latex="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" alt="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" title="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" data-latex="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5" alt="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5" title="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5" data-latex="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D%20+%205%20%5C,%20y%20=%208%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D" alt="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" title="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}" data-latex="-4 \, t^{2} {y'} + 12 \, y^{2} {y'} + 5 \, y = 8 \, t y - 5 \, t {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20-%204%20%5C,%20y%5E%7B3%7D%20-%205%20%5C,%20t%20y%20=%205" alt="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5" title="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5" data-latex="4 \, t^{2} y - 4 \, y^{3} - 5 \, t y = 5">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6447" title="F4 | Implicit solutions for exact IVPs | ver. 6447"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" alt="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" title="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" data-latex="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}" alt="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}" title="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}" data-latex="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B3%7D%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20+%205%20%5C,%20t%20%7By'%7D%20+%205%20%5C,%20y" alt="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" title="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" data-latex="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y">
</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,%20t%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%206%20%5C,%20y%20%7By'%7D%20-%205%20%5C,%20y%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B3%7D" alt="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}" title="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}" data-latex="-4 \, t^{2} {y'} + 2 \, y^{2} + 6 \, y {y'} - 5 \, y = 2 \, t^{2} y {y'} - 4 \, t^{3}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" alt="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" title="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" data-latex="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)" alt="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)" title="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)" data-latex="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B3%7D%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20+%205%20%5C,%20t%20%7By'%7D%20+%205%20%5C,%20y" alt="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" title="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y" data-latex="4 \, t^{3} = 4 \, t^{2} {y'} + 8 \, t y + 5 \, t {y'} + 5 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B4%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20+%205%20%5C,%20t%20y%20=%20%5Cleft(-10%5Cright)" alt="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)" title="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)" data-latex="-t^{4} + 4 \, t^{2} y + 5 \, t y = \left(-10\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5507" title="F4 | Implicit solutions for exact IVPs | ver. 5507"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" alt="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" title="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" data-latex="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y"/></p><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} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y" alt="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y" title="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y" data-latex="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20+%204%20%5C,%20t%20=%206%20%5C,%20t%20y" alt="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" title="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" data-latex="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B2%7D%20-%204%20%5C,%20t%20=%202%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%20y" alt="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y" title="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y" data-latex="-y^{2} - 4 \, t = 2 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 3 \, t^{2} {y'} + y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" alt="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" title="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" data-latex="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6" alt="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6" title="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6" data-latex="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20%7By'%7D%20+%202%20%5C,%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20+%204%20%5C,%20t%20=%206%20%5C,%20t%20y" alt="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" title="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y" data-latex="-3 \, t^{2} {y'} + 2 \, t y {y'} + y^{2} + 4 \, t = 6 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20+%20t%20y%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20=%206" alt="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6" title="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6" data-latex="-3 \, t^{2} y + t y^{2} + 2 \, t^{2} = 6">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1357" title="F4 | Implicit solutions for exact IVPs | ver. 1357"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y" alt="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y" title="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y" data-latex="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y"/></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 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" alt="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" title="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" data-latex="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20=%20-10%20%5C,%20t%20y%20-%203%20%5C,%20y" alt="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y" title="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y" data-latex="-8 \, t^{2} y {y'} + 12 \, y^{3} {y'} - 4 \, y^{2} = -10 \, t y - 3 \, y">
</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,%20t%20y%20%7By'%7D%20-%203%20%5C,%20t%20%7By'%7D%20=%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20+%203%20%5C,%20y" alt="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" title="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" data-latex="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" alt="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" title="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" data-latex="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)" alt="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)" title="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)" data-latex="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20-%203%20%5C,%20t%20%7By'%7D%20=%2012%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20-%204%20%5C,%20y%5E%7B2%7D%20+%203%20%5C,%20y" alt="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" title="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y" data-latex="8 \, t y {y'} - 3 \, t {y'} = 12 \, y^{3} {y'} - 4 \, y^{2} + 3 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20y%5E%7B4%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20t%20y%20=%20%5Cleft(-4%5Cright)" alt="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)" title="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)" data-latex="3 \, y^{4} - 4 \, t y^{2} + 3 \, t y = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-7823" title="F4 | Implicit solutions for exact IVPs | ver. 7823"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}" alt="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}" title="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}" data-latex="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D%20=%20-4%20%5C,%20t%20y%20%7By'%7D" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}">
</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,%20t%20y%5E%7B2%7D%20-%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20-%205%20%5C,%20y%20=%2010%20%5C,%20y%20%7By'%7D" alt="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}" title="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}" data-latex="4 \, t y^{2} - 3 \, t^{2} {y'} - 2 \, y^{2} - 5 \, y = 10 \, y {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5" alt="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5" title="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5" data-latex="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20t%20y%5E%7B2%7D%20+%202%20%5C,%20y%5E%7B2%7D%20+%2010%20%5C,%20y%20%7By'%7D%20=%20-4%20%5C,%20t%20y%20%7By'%7D" alt="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" title="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}" data-latex="-4 \, t^{2} y {y'} - 4 \, t y^{2} + 2 \, y^{2} + 10 \, y {y'} = -4 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20y%5E%7B2%7D%20=%205" alt="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5" title="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5" data-latex="-2 \, t^{2} y^{2} + 2 \, t y^{2} + 5 \, y^{2} = 5">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-5950" title="F4 | Implicit solutions for exact IVPs | ver. 5950"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}" alt="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}" title="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}" data-latex="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%20y%20+%205%20%5C,%20t%20%7By'%7D" alt="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}" title="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}" data-latex="-6 \, t y^{2} = -8 \, t y {y'} + 2 \, t y + 5 \, t {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20=%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)" alt="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)" title="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)" data-latex="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%206%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20=%2012%20%5C,%20y%5E%7B2%7D%20%7By'%7D" alt="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" title="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}" data-latex="6 \, t^{2} y {y'} + 6 \, t y^{2} - 4 \, t = 12 \, y^{2} {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%204%20%5C,%20y%5E%7B3%7D%20+%202%20%5C,%20t%5E%7B2%7D%20=%20%5Cleft(-5%5Cright)" alt="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)" title="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)" data-latex="-3 \, t^{2} y^{2} + 4 \, y^{3} + 2 \, t^{2} = \left(-5\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-2330" title="F4 | Implicit solutions for exact IVPs | ver. 2330"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}" alt="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}" title="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}" data-latex="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" alt="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" title="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" data-latex="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20=%206%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2010%20%5C,%20t%20y%20+%203%20%5C,%20t%20%7By'%7D" alt="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}" title="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}" data-latex="-5 \, y^{2} = 6 \, t^{2} y {y'} - 10 \, t y + 3 \, t {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20y%20%7By'%7D%20=%20-6%20%5C,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" title="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" data-latex="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" alt="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" title="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" data-latex="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1" alt="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1" title="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1" data-latex="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%2010%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20y%20%7By'%7D%20=%20-6%20%5C,%20t%20y%5E%7B2%7D%20-%205%20%5C,%20y%5E%7B2%7D" alt="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" title="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}" data-latex="6 \, t^{2} y {y'} + 10 \, t y {y'} + 6 \, y {y'} = -6 \, t y^{2} - 5 \, y^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%205%20%5C,%20t%20y%5E%7B2%7D%20+%203%20%5C,%20y%5E%7B2%7D%20=%201" alt="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1" title="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1" data-latex="3 \, t^{2} y^{2} + 5 \, t y^{2} + 3 \, y^{2} = 1">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6731" title="F4 | Implicit solutions for exact IVPs | ver. 6731"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" alt="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" title="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" data-latex="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y" alt="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y" title="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y" data-latex="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20y%20%7By'%7D%20=%20-6%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y" alt="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" title="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" data-latex="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y">
</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,%20t%20y%5E%7B2%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20-%204%20%5C,%20y%20%7By'%7D%20-%205%20%5C,%20y%20=%20-4%20%5C,%20t%20y" alt="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y" title="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y" data-latex="6 \, t y^{2} + 8 \, t y {y'} - 4 \, y {y'} - 5 \, y = -4 \, t y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" alt="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" title="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" data-latex="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3" alt="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3" title="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3" data-latex="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%204%20%5C,%20y%20%7By'%7D%20=%20-6%20%5C,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%20y" alt="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" title="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y" data-latex="6 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 4 \, y {y'} = -6 \, t y^{2} - 4 \, t y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%203" alt="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3" title="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3" data-latex="3 \, t^{2} y^{2} + 2 \, t^{2} y - 2 \, y^{2} = 3">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-4911" title="F4 | Implicit solutions for exact IVPs | ver. 4911"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" alt="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" title="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" data-latex="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}"/></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 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}" alt="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}" title="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}" data-latex="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20y%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D%20+%204%20%5C,%20t%20=%206%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" title="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" data-latex="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}">
</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,%20t%5E%7B2%7D%20y%20%7By'%7D%20=%206%20%5C,%20t%20y%20+%203%20%5C,%20y%5E%7B2%7D%20-%203%20%5C,%20t%20%7By'%7D" alt="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}" title="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}" data-latex="-8 \, t^{2} y {y'} = 6 \, t y + 3 \, y^{2} - 3 \, t {y'}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" alt="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" title="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" data-latex="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} - 2 \, t^{2} + y^{2} = 2" alt="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2" title="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2" data-latex="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20y%5E%7B2%7D%20-%202%20%5C,%20y%20%7By'%7D%20+%204%20%5C,%20t%20=%206%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" title="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}" data-latex="-3 \, y^{2} - 2 \, y {y'} + 4 \, t = 6 \, t y {y'}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20-%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%202" alt="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2" title="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2" data-latex="3 \, t y^{2} - 2 \, t^{2} + y^{2} = 2">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6441" title="F4 | Implicit solutions for exact IVPs | ver. 6441"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, t" alt="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, t" title="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, t" data-latex="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, 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?0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" alt="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" title="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" data-latex="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 1 )= -1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20%7By'%7D%20=%20-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20+%208%20%5C,%20t" alt="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, t" title="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, t" data-latex="4 \, t {y'} = -6 \, t^{2} y {y'} + 3 \, t^{2} {y'} - 2 \, y^{2} + 8 \, 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?0%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20+%208%20%5C,%20t%20-%204%20%5C,%20y" alt="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" title="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" data-latex="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%201%20)=%20-1" alt="y( 1 )= -1" title="y( 1 )= -1" data-latex="y( 1 )= -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>The following ODE is exact.</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 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" alt="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" title="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" data-latex="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)" alt="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)" title="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)" data-latex="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20-%202%20%5C,%20y%5E%7B2%7D%20-%204%20%5C,%20t%20%7By'%7D%20+%208%20%5C,%20t%20-%204%20%5C,%20y" alt="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" title="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y" data-latex="0 = -4 \, t y {y'} - 2 \, y^{2} - 4 \, t {y'} + 8 \, t - 4 \, y">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%20y%5E%7B2%7D%20-%204%20%5C,%20t%5E%7B2%7D%20+%204%20%5C,%20t%20y%20=%20%5Cleft(-6%5Cright)" alt="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)" title="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)" data-latex="2 \, t y^{2} - 4 \, t^{2} + 4 \, t y = \left(-6\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-1727" title="F4 | Implicit solutions for exact IVPs | ver. 1727"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}" alt="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}" title="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}" data-latex="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}"/></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 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" alt="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" title="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" data-latex="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%205%20%5C,%20y%5E%7B2%7D%20=%20-8%20%5C,%20t%20y%20-%204%20%5C,%20t%20%7By'%7D" alt="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}" title="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}" data-latex="-2 \, t^{2} y {y'} - 5 \, y^{2} = -8 \, t y - 4 \, t {y'}">
</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,%20t%20y%20+%205%20%5C,%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%2012%20%5C,%20t%5E%7B2%7D" alt="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" title="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" data-latex="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</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 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" alt="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" title="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" data-latex="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}"/></p><p> Its implicit solution satisfying the initial value is: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13" alt="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13" title="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13" data-latex="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20+%205%20%5C,%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%2010%20%5C,%20t%20y%20%7By'%7D%20-%2012%20%5C,%20t%5E%7B2%7D" alt="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" title="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}" data-latex="-8 \, t y + 5 \, y^{2} = 4 \, t^{2} {y'} - 10 \, t y {y'} - 12 \, t^{2}">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B3%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%5E%7B2%7D%20=%2013" alt="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13" title="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13" data-latex="-4 \, t^{3} + 4 \, t^{2} y - 5 \, t y^{2} = 13">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-6200" title="F4 | Implicit solutions for exact IVPs | ver. 6200"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </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 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}" alt="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}" title="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}" data-latex="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}"/></p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" alt="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" title="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" data-latex="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%202%20%5C,%20t%20=%20-2%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%203%20%5C,%20y%5E%7B2%7D%20+%20t%20%7By'%7D" alt="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}" title="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}" data-latex="8 \, t^{2} y {y'} - 2 \, t = -2 \, t^{2} {y'} + 3 \, y^{2} + t {y'}">
</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,%20t%20y%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20-%202%20%5C,%20t" alt="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" title="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" data-latex="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t">
</p>
<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 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>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" alt="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" title="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" data-latex="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)" alt="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)" title="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)" data-latex="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20-%203%20%5C,%20y%5E%7B2%7D%20-%202%20%5C,%20t" alt="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" title="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t" data-latex="6 \, t y {y'} = 2 \, t^{2} {y'} + 4 \, t y - 3 \, y^{2} - 2 \, t">
</p>
<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%20+%203%20%5C,%20t%20y%5E%7B2%7D%20+%20t%5E%7B2%7D%20=%20%5Cleft(-4%5Cright)" alt="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)" title="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)" data-latex="-2 \, t^{2} y + 3 \, t y^{2} + t^{2} = \left(-4\right)">
</p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="F4-8126" title="F4 | Implicit solutions for exact IVPs | ver. 8126"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F4.</strong></p><p> Determine which of the following ODEs is exact. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t" alt="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t" title="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, 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?4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" alt="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" title="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y"/></p><p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -1 )= 1" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"/>. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>F4.</strong>
</p>
<p> Determine which of the following ODEs is exact. </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,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%204%20%5C,%20y%20%7By'%7D%20+%205%20%5C,%20y%20=%202%20%5C,%20t%20y%20%7By'%7D%20+%208%20%5C,%20t" alt="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t" title="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} + 4 \, y {y'} + 5 \, y = 2 \, t y {y'} + 8 \, t">
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<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,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2010%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20-%205%20%5C,%20y" alt="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" title="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y">
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<p> Then find an implicit solution for this exact ODE satisfying the initial value <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1">. </p>
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</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p>The following ODE is exact.</p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" alt="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" title="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y"/></p><p> Its implicit solution satisfying the initial value is: </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 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)" alt="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)" title="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)" data-latex="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p>The following ODE is exact.</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,%20t%20y%5E%7B2%7D%20+%205%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%2010%20%5C,%20t%20y%20-%205%20%5C,%20t%20%7By'%7D%20-%205%20%5C,%20y" alt="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" title="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y" data-latex="4 \, t y^{2} + 5 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 10 \, t y - 5 \, t {y'} - 5 \, y">
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<p> Its implicit solution satisfying the initial value is: </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,%20t%5E%7B2%7D%20y%5E%7B2%7D%20-%205%20%5C,%20t%5E%7B2%7D%20y%20-%205%20%5C,%20t%20y%20=%20%5Cleft(-2%5Cright)" alt="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)" title="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)" data-latex="-2 \, t^{2} y^{2} - 5 \, t^{2} y - 5 \, t y = \left(-2\right)">
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