0005.ptx

<exercise masterit-seed="3912" 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> 2 \, {y''} = -4 \, {y'} - 2 \, {y} </me>
      </li>
      <li>
        <me> 24 \, {x} = -12 \, {x'} - 3 \, {x''} </me>
      </li>
    </ol>
  </statement>
  <answer>
    <me> {x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)} </me>
    <me> {x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)} </me>
    <me> {y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)} </me>
  </answer>
</exercise>