0002.qti

<item ident="C4-5644" title="C4 | Homogeneous second-order linear IVP | ver. 5644">
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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?25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" alt="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" title="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" data-latex="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15"/>
              </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?-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" alt="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" title="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" data-latex="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3"/>
              </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?25%20%5C,%20%7Bx%7D%20+%20%7Bx''%7D%20=%200%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%20-5%20,%20%7Bx%7D%20'(0)=%20-15" alt="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" title="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15" data-latex="25 \, {x} + {x''} = 0 \hspace{2em} {x} (0)= -5 , {x} '(0)= -15"&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?-3%20%5C,%20%7By%7D%20-%202%20%5C,%20%7By'%7D%20=%20-%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%20-5%20,%20%7By%7D%20'(0)=%20-3" alt="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" title="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3" data-latex="-3 \, {y} - 2 \, {y'} = -{y''} \hspace{2em} {y} (0)= -5 , {y} '(0)= -3"&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} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" alt="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, 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)} - 3 \, e^{\left(-t\right)}" alt="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}"/>
            </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 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=%20-5%20%5C,%20%5Ccos%5Cleft(5%20%5C,%20t%5Cright)%20-%203%20%5C,%20%5Csin%5Cleft(5%20%5C,%20t%5Cright)" alt="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" title="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, t\right)" data-latex="{x} = -5 \, \cos\left(5 \, t\right) - 3 \, \sin\left(5 \, 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-%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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