0002.qti

<item ident="F4-2307" title="F4 | Implicit solutions for exact IVPs | ver. 2307">
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  <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>
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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?-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"&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?-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"&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;

</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">
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      <material>
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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?-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">&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?-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"&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?-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)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext>
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