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<item ident="N2-4511" title="N2 | Euler's method for approximating IVP solutions | ver. 4511">
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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( 0.20 )" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"/> 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 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" alt="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" title="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 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'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" alt="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" title="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" data-latex="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 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(%200.20%20)" alt="x( 0.20 )" title="x( 0.20 )" data-latex="x( 0.20 )"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)" alt="y( 0.20 )" title="y( 0.20 )" data-latex="y( 0.20 )"&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'=%202%20%5C,%20x%5E%7B2%7D%20y%20+%204%20%5C,%20t%20x%20-%201%20%5Chspace%7B2em%7D%20x(%200%20)=%200" alt="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" title="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 0" data-latex="x'= 2 \, x^{2} y + 4 \, t x - 1 \hspace{2em} x( 0 )= 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'=%20-x%5E%7B2%7D%20y%5E%7B2%7D%20+%202%20%5C,%20t%20y%5E%7B2%7D%20-%203%20%5Chspace%7B2em%7D%20y(%200%20)=%200" alt="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" title="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0" data-latex="y'= -x^{2} y^{2} + 2 \, t y^{2} - 3 \hspace{2em} y( 0 )= 0"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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    <response_str ident="response1" rcardinality="Single">
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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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            <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( 0.10 )\approx -0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.10 )\approx -0.300" alt="y( 0.10 )\approx -0.300" title="y( 0.10 )\approx -0.300" data-latex="y( 0.10 )\approx -0.300"/></li>
              <li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?x( 0.20 )\approx -0.205" alt="x( 0.20 )\approx -0.205" title="x( 0.20 )\approx -0.205" data-latex="x( 0.20 )\approx -0.205"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y( 0.20 )\approx -0.598" alt="y( 0.20 )\approx -0.598" title="y( 0.20 )\approx -0.598" data-latex="y( 0.20 )\approx -0.598"/></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(%200.10%20)%5Capprox%20-0.100" alt="x( 0.10 )\approx -0.100" title="x( 0.10 )\approx -0.100" data-latex="x( 0.10 )\approx -0.100"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.10%20)%5Capprox%20-0.300" alt="y( 0.10 )\approx -0.300" title="y( 0.10 )\approx -0.300" data-latex="y( 0.10 )\approx -0.300"&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(%200.20%20)%5Capprox%20-0.205" alt="x( 0.20 )\approx -0.205" title="x( 0.20 )\approx -0.205" data-latex="x( 0.20 )\approx -0.205"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y(%200.20%20)%5Capprox%20-0.598" alt="y( 0.20 )\approx -0.598" title="y( 0.20 )\approx -0.598" data-latex="y( 0.20 )\approx -0.598"&gt;
&lt;/li&gt;
  &lt;/ul&gt;
&lt;/div&gt;

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