0009.ptx

<exercise masterit-seed="3702" masterit-slug="C3" masterit-name="Homogeneous second-order linear ODE">
  <statement>
    <p>Explain how to find the general solution to each given ODE using
        exponential functions.</p>
    <p>For each exponential solution using complex numbers, also provide
        an alternate general solution using only real numbers.</p>
    <ol>
      <li>
        <me> 3 \, {x''} + 78 \, {x} + 6 \, {x'} = 0 </me>
      </li>
      <li>
        <me> -18 \, {y} = 2 \, {y''} + 12 \, {y'} </me>
      </li>
    </ol>
  </statement>
  <answer>
    <me> {x} = c_{1} e^{\left(\left(5 i - 1\right) \, t\right)} + c_{2} e^{\left(-\left(5 i + 1\right) \, t\right)} </me>
    <me> {x} = {\left(d_{1} \cos\left(5 \, t\right) + d_{2} \sin\left(5 \, t\right)\right)} e^{\left(-t\right)} </me>
    <me> {y} = k_{1} t e^{\left(-3 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} </me>
  </answer>
</exercise>