\begin{exerciseStatement}
Explain how to find the general solution to each given ODE using exponential functions.
For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.
\begin{enumerate}[(a)]
\item \[ -2 \, {x''} - 128 \, {x} = -32 \, {x'} \]
\item \[ -3 \, {y''} - 12 \, {y'} = 24 \, {y} \]
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}
\[ {y} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)} \]\[ {y} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)} \]\[ {x} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)} \]
\end{exerciseAnswer}
