<item ident="F4-9506" title="F4 | Implicit solutions for exact IVPs | ver. 9506">
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<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>
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<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>
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<response_label ident="answer1" rshuffle="No"/>
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<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>
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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?-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)">
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