<item ident="F4-9233" title="F4 | Implicit solutions for exact IVPs | ver. 9233">
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<presentation>
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<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>
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<response_str ident="response1" rcardinality="Single">
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<response_label ident="answer1" rshuffle="No"/>
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<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>
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<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>
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