0001.qti

<item ident="F4-8685" title="F4 | Implicit solutions for exact IVPs | ver. 8685">
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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?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>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F4.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Determine which of the following ODEs is exact. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;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"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;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"&gt;
  &lt;/p&gt;
  &lt;p&gt; Then find an implicit solution for this exact ODE satisfying the initial value &lt;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"&gt;. &lt;/p&gt;
&lt;/div&gt;

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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?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>
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        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt;The following ODE is exact.&lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;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"&gt;
  &lt;/p&gt;
  &lt;p&gt; Its implicit solution satisfying the initial value is: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;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"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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