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<item ident="N2-2652" title="N2 | Euler's method for approximating IVP solutions | ver. 2652">
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          <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'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" alt="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" title="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0"/>
          </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'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" alt="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1"/>
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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'=%20t%5E%7B2%7D%20x%5E%7B2%7D%20+%204%20%5C,%20t%5E%7B2%7D%20y%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%20-1%20)=%200" alt="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" title="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0" data-latex="x'= t^{2} x^{2} + 4 \, t^{2} y^{2} + 1 \hspace{2em} x( -1 )= 0"&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'=%202%20%5C,%20x%20y%5E%7B2%7D%20+%204%20%5C,%20t%20y%20-%203%20%5Chspace%7B2em%7D%20y(%20-1%20)=%20-1" alt="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" title="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1" data-latex="y'= 2 \, x y^{2} + 4 \, t y - 3 \hspace{2em} y( -1 )= -1"&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 0.500" alt="x( -0.90 )\approx 0.500" title="x( -0.90 )\approx 0.500" data-latex="x( -0.90 )\approx 0.500"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.90 )\approx -0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"/></li>
              <li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( -0.80 )\approx 0.883" alt="x( -0.80 )\approx 0.883" title="x( -0.80 )\approx 0.883" data-latex="x( -0.80 )\approx 0.883"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( -0.80 )\approx -0.795" alt="y( -0.80 )\approx -0.795" title="y( -0.80 )\approx -0.795" data-latex="y( -0.80 )\approx -0.795"/></li>
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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%200.500" alt="x( -0.90 )\approx 0.500" title="x( -0.90 )\approx 0.500" data-latex="x( -0.90 )\approx 0.500"&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%20-0.900" alt="y( -0.90 )\approx -0.900" title="y( -0.90 )\approx -0.900" data-latex="y( -0.90 )\approx -0.900"&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%200.883" alt="x( -0.80 )\approx 0.883" title="x( -0.80 )\approx 0.883" data-latex="x( -0.80 )\approx 0.883"&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%20-0.795" alt="y( -0.80 )\approx -0.795" title="y( -0.80 )\approx -0.795" data-latex="y( -0.80 )\approx -0.795"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

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