Given the general solution to the corresponding homogeneous system is $\Psi(t)=\begin{bmatrix} \mathit{e}^{-3t} &\mathit{e}^{-t} \\ -\mathit{e}^{-3t} &\mathit{e}^{-t} \end{bmatrix}C$, find the general solution of the system $x'=\begin{bmatrix} -2 &1 \\ -1 &-2 \end{bmatrix}x+\begin{bmatrix} 2\mathit{e}^{-t}\\ 3t \end{bmatrix}$, where $C$ is a nonsingular constant matrix.
Do I first try to find the system that $\Psi$ satisfies or do I solve the nonhomogeneous system? I am not sure what the first step is. I even tried to introduce $\Psi$ into the nonhomogeneous system to see if I can determine $C$, but I don't think this is the right direction.