Solution
= Solution
The <characteristic equation> of the <homogeneous linear differential equation> is
$$
m^2-4m+13=0,\qquad m=2\pm3i.
$$
The real <general solution> is $y=e^{2x}(A\cos3x+B\sin3x)$. The first boundary value gives $A=\pi$. At $x=-\pi/2$, $\cos(-3\pi/2)=0$ and $\sin(-3\pi/2)=1$, so $1=e^{-\pi}B$. Hence
$$
\boxed{y(x)=e^{2x}\bigl(\pi\cos3x+e^\pi\sin3x\bigr)}.
$$