System of differential equations Can you help me solve this system of differential equations
$ x'= 4x - 2y $
$ y'= 3x - y - 2e^{3t} $
Initial conditions are  $ x(0) = y(0) = 0 $

$\mathbf x' + \begin{bmatrix} -4&2\\-3&1\end{bmatrix} \mathbf x = \begin{bmatrix}0\\-2e^{3t}\end{bmatrix}$
$e^{At} \mathbf x' + A e^{At} \mathbf x = e^{At}g(t)$
$A = P^{-1}DP$
$A = \begin{bmatrix}1&2\\1&3\end{bmatrix}\begin{bmatrix}-2&0\\0&-1\end{bmatrix}\begin{bmatrix}3&-2\\-1&1\end{bmatrix}\\
e^{At} = P^{-1} e^{Dt}P$
$\mathbf x =$$ e^{-At}C + e^{-At} \int e^{At}g(t) dt\\
e^{-At}C + P^{-1}e^{-Dt} \int e^{Dt}Pg(t) dt$
$e^{Dt}Pg(t) =\begin{bmatrix}4e^t\\-2e^{2t}\end{bmatrix}$ 
$\mathbf x =e^{-At}C + P^{-1}\begin{bmatrix}4\\-1\end{bmatrix} e^{3t}\\
P\mathbf x(0) =C + \begin{bmatrix}4\\-1\end{bmatrix}\\
C = \begin{bmatrix}-4\\1\end{bmatrix}$
$\mathbf x = \begin{bmatrix}-4e^{2t}+2e^{t}+2e^{3t}\\-4e^{2t}+3e^t + e^{3t}\end{bmatrix}$
