0000.qti

<item ident="F4-6141" title="F4 | Implicit solutions for exact IVPs | ver. 6141">
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  <presentation>
    <material>
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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?3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {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?12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" alt="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" title="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" data-latex="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, 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>
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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?3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y%20=%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {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?12%20%5C,%20t%5E%7B3%7D%20-%202%20%5C,%20y%20=%20-8%20%5C,%20t%5E%7B2%7D%20y%20%7By'%7D%20-%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D%20+%208%20%5C,%20t%20y%20%7By'%7D%20+%206%20%5C,%20t%20y" alt="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" title="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y" data-latex="12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, 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(%20-1%20)=%201" alt="y( -1 )= 1" title="y( -1 )= 1" data-latex="y( -1 )= 1"&gt;. &lt;/p&gt;
&lt;/div&gt;

</mattext>
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        <response_label ident="answer1" rshuffle="No"/>
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  </presentation>
  <itemfeedback ident="general_fb">
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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?3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {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?-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" alt="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" title="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" data-latex="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0"/>
            </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?3%20%5C,%20t%5E%7B2%7D%20%7By'%7D%20+%206%20%5C,%20t%20y%20+%202%20%5C,%20t%20%7By'%7D%20+%202%20%5C,%20y%20=%204%20%5C,%20y%5E%7B3%7D%20%7By'%7D" alt="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" title="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'}" data-latex="3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {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?-y%5E%7B4%7D%20+%203%20%5C,%20t%5E%7B2%7D%20y%20+%202%20%5C,%20t%20y%20=%200" alt="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" title="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0" data-latex="-y^{4} + 3 \, t^{2} y + 2 \, t y = 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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