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<item ident="N2-6130" title="N2 | Euler's method for approximating IVP solutions | ver. 6130">
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        <div class="exercise-statement">
          <p>
            <strong>N2.</strong>
          </p>
          <p> Use Euler's method with <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h= 0.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"/> to approximate <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"/> given the following system of IVPs. </p>
          <p style="text-align:center;">
            <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" alt="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" title="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2"/>
          </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'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" alt="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" title="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" data-latex="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0"/>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;N2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Use Euler's method with &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?h=%200.10" alt="h= 0.10" title="h= 0.10" data-latex="h= 0.10"&gt; to approximate &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)" alt="x( -0.80 )" title="x( -0.80 )" data-latex="x( -0.80 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)" alt="y( -0.80 )" title="y( -0.80 )" data-latex="y( -0.80 )"&gt; given the following system of IVPs. &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?x'=%20-2%20%5C,%20t%5E%7B2%7D%20x%20-%202%20%5C,%20t%5E%7B2%7D%20y%20-%202%20%5Chspace%7B2em%7D%20x(%20-1%20)=%202" alt="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" title="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2" data-latex="x'= -2 \, t^{2} x - 2 \, t^{2} y - 2 \hspace{2em} x( -1 )= 2"&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?y'=%20-t%5E%7B2%7D%20x%5E%7B2%7D%20-%20x%20y%5E%7B2%7D%20%5Chspace%7B2em%7D%20y(%20-1%20)=%200" alt="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" title="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0" data-latex="y'= -t^{2} x^{2} - x y^{2} \hspace{2em} y( -1 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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          <div class="exercise-answer">
            <h4>Partial Answer:</h4>
            <ul>
              <li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.90 )\approx 1.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.400" alt="y( -0.90 )\approx -0.400" title="y( -0.90 )\approx -0.400" data-latex="y( -0.90 )\approx -0.400"/></li>
              <li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 1.04" alt="x( -0.80 )\approx 1.04" title="x( -0.80 )\approx 1.04" data-latex="x( -0.80 )\approx 1.04"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.581" alt="y( -0.80 )\approx -0.581" title="y( -0.80 )\approx -0.581" data-latex="y( -0.80 )\approx -0.581"/></li>
            </ul>
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        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ul&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.90%20)%5Capprox%201.40" alt="x( -0.90 )\approx 1.40" title="x( -0.90 )\approx 1.40" data-latex="x( -0.90 )\approx 1.40"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.90%20)%5Capprox%20-0.400" alt="y( -0.90 )\approx -0.400" title="y( -0.90 )\approx -0.400" data-latex="y( -0.90 )\approx -0.400"&gt;
&lt;/li&gt;
    &lt;li&gt;
&lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x(%20-0.80%20)%5Capprox%201.04" alt="x( -0.80 )\approx 1.04" title="x( -0.80 )\approx 1.04" data-latex="x( -0.80 )\approx 1.04"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%20-0.80%20)%5Capprox%20-0.581" alt="y( -0.80 )\approx -0.581" title="y( -0.80 )\approx -0.581" data-latex="y( -0.80 )\approx -0.581"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

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