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<item ident="F4-6667" title="F4 | Implicit solutions for exact IVPs | ver. 6667">
  <itemmetadata>
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      <qtimetadatafield>
        <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?-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, 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?-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" alt="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" title="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, 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">&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?-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, 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?-6%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20+%202%20%5C,%20t%20=%20-4%20%5C,%20t%20y%20%7By'%7D%20-%206%20%5C,%20t%20y%20+%204%20%5C,%20t%20%7By'%7D" alt="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" title="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'}" data-latex="-6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, 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;

</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?-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, 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 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" alt="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" title="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" data-latex="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\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?-3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20-%206%20%5C,%20t%20y%20-%202%20%5C,%20y%5E%7B2%7D%20=%2016%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%204%20%5C,%20t%20y%20%7By'%7D" alt="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" title="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'}" data-latex="-3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, 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,%20y%5E%7B4%7D%20-%203%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5C,%20t%20y%5E%7B2%7D%20=%20%5Cleft(-3%5Cright)" alt="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" title="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)" data-latex="-4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext>
      </material>
    </flow_mat>
  </itemfeedback>
</item>