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The characteristic equation is

$\lambda ^2 + 4\lambda + \pi ^2 + 4 = 0.$

Let's find the roots of the characteristic equation.

\begin{align*} \lambda ^2 + 4\lambda + \pi ^2 + 4 = 0 &\iff \lambda _{1/2} = \frac{-4 \pm \sqrt{4^2 -4(\pi ^2 + 4 )}}{2} \\ & \iff \lambda _{1/2} =\frac{-4 \pm 2\sqrt{\cancel{4} -\pi ^2 - \cancel{4} }}{2} = -2 \pm i\pi \\ & \iff \lambda _1 =-2 + i\pi, \quad \lambda _2 = -2 - i\pi \end{align*}

So, it has the complex conjugate roots:

$\lambda _1 = -2 - i\pi (=a + ib)\quad \wedge \quad \lambda _2 = -2 - i\pi (=a-ib)$

A basis is:

\begin{align*} \begin{cases} & \phi _1 (x) = e^{ax} \cos (bx) = e^{-2 x} \cos (\pi x) \\ &\phi _2 (x) =e^{ax} \sin (bx) = e^{-2 x} \sin (\pi x) \end{cases} \\ \end{align*}

The general solution is:

\begin{align*} y & = c_1 \phi _1 (x) + c_2 \phi _2 (x) \\ & = c_1e^{-2 x} \cos (\pi x) + c_2e^{-2 x} \sin (\pi x) \\ & = \boxed{{\color{#4257b2}{ e^{-2 x} (c_1\cos (\pi x) + c_2\sin (\pi x)) }}} \end{align*}

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