0000.qti

<item ident="C5-9755" title="C5 | Non-homogeneous second-order linear ODE | ver. 9755">
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  <presentation>
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        <div class="exercise-statement">
          <p>
            <strong>C5.</strong>
          </p>
          <p>Explain how to find the general solution to the given ODE.</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 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" alt="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" title="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" data-latex="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}"/>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to the given ODE.&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%7By'%7D%20-%207%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%204%20%5C,%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%20=%203%20%5C,%20%7By%7D" alt="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" title="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}" data-latex="{y''} - 2 \, {y'} - 7 \, \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) = 3 \, {y}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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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?{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)"/>
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  &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?%7By%7D%20=%20k_%7B1%7D%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)" alt="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" title="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)" data-latex="{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} - \cos\left(2 \, t\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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