0002.qti

<item ident="C5-1377" title="C5 | Non-homogeneous second-order linear ODE | ver. 1377">
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          <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''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" alt="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" title="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" data-latex="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}"/>
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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=%20-16%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20%7By%7D%20-%2012%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" title="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}" data-latex="{y''} = -16 \, t e^{\left(3 \, t\right)} + {y} - 12 \, e^{\left(3 \, t\right)}"&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} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}"/>
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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=%20-2%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7Bt%7D" alt="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" title="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}" data-latex="{y} = -2 \, t e^{\left(3 \, t\right)} + k_{2} e^{\left(-t\right)} + k_{1} e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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