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<item ident="F4-5847" title="F4 | Implicit solutions for exact IVPs | ver. 5847">
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        <fieldlabel>question_type</fieldlabel>
        <fieldentry>essay_question</fieldentry>
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  </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">&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=%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"&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%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"&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">
    <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">&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?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"&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?-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"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext>
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</item>