0007.qti

<item ident="C5-5831" title="C5 | Non-homogeneous second-order linear ODE | ver. 5831">
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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?18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" alt="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}"/>
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  &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?18%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%7By''%7D%20+%20%7By'%7D%20-%2015%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20-12%20%5C,%20%7By%7D" alt="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" title="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}" data-latex="18 \, t e^{\left(3 \, t\right)} - {y''} + {y'} - 15 \, e^{\left(3 \, t\right)} = -12 \, {y}"&gt;
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            <h4>Partial Answer:</h4>
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              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" alt="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, 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_%7B2%7D%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20t%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%20k_%7B1%7D%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" title="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}" data-latex="{y} = k_{2} e^{\left(4 \, t\right)} - 3 \, t e^{\left(3 \, t\right)} + k_{1} e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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