0010.qti

<item ident="C4-7786" title="C4 | Homogeneous second-order linear IVP | ver. 7786">
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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?6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" alt="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" title="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" data-latex="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10"/>
              </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?-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" alt="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32"/>
              </p>
            </li>
          </ol>
        </div>
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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?6%20%5C,%20%7Bx'%7D%20-%208%20%5C,%20%7Bx%7D%20=%20%7Bx''%7D%20%5Chspace%7B2em%7D%20%7Bx%7D%20(0)=%203%20,%20%7Bx%7D%20'(0)=%2010" alt="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" title="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10" data-latex="6 \, {x'} - 8 \, {x} = {x''} \hspace{2em} {x} (0)= 3 , {x} '(0)= 10"&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?-64%20%5C,%20%7By%7D%20=%20%7By''%7D%20%5Chspace%7B2em%7D%20%7By%7D%20(0)=%202%20,%20%7By%7D%20'(0)=%20-32" alt="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" title="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32" data-latex="-64 \, {y} = {y''} \hspace{2em} {y} (0)= 2 , {y} '(0)= -32"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext>
    </material>
    <response_str ident="response1" rcardinality="Single">
      <render_fib>
        <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} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" alt="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\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 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" alt="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" title="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" data-latex="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)"/>
            </p>
          </div>
        </mattextxml>
        <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=%202%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" title="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\left(2 \, t\right)}" data-latex="{x} = 2 \, e^{\left(4 \, t\right)} + e^{\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=%202%20%5C,%20%5Ccos%5Cleft(8%20%5C,%20t%5Cright)%20-%204%20%5C,%20%5Csin%5Cleft(8%20%5C,%20t%5Cright)" alt="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" title="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)" data-latex="{y} = 2 \, \cos\left(8 \, t\right) - 4 \, \sin\left(8 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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