0000.qti

<item ident="C4-9124" title="C4 | Homogeneous second-order linear IVP | ver. 9124">
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  <presentation>
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        <div class="exercise-statement">
          <p>
            <strong>C4.</strong>
          </p>
          <p>Explain how to find the particular solution to each given IVP.</p>
          <ol type="a">
            <li>
              <p style="text-align:center;">
                <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8"/>
              </p>
            </li>
            <li>
              <p style="text-align:center;">
                <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" alt="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" title="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" data-latex="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2"/>
              </p>
            </li>
          </ol>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C4.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the particular solution to each given IVP.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&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?-%7Bx''%7D%20-%204%20%5C,%20%7Bx%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%201%20,%20%7Bx%7D%20'(0)=%208" alt="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" title="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8" data-latex="-{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&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?-%7By''%7D%20-%2012%20%5C,%20%7By%7D%20=%207%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-1%20,%20%7By%7D%20'(0)=%202" alt="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" title="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2" data-latex="-{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&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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            <h4>Partial Answer:</h4>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)"/>
            </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 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" alt="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}"/>
            </p>
          </div>
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        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&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?%7Bx%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)" alt="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" title="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)" data-latex="{x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right)"&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?%7By%7D%20=%20-2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" title="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}" data-latex="{y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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