General integral of a linear system of ODEs I want to find the solution of 
\begin{cases} 
x'= -5x-y+e^t \\
y'= 2x-3y
\end{cases}
$$A = \begin{bmatrix} -5 & -1 \\ 2 & -3 \end{bmatrix}$$
I calculated the exponential matrix $e^{tA}$ and found the solution of the homogeneous system
$$\begin{bmatrix} x\\ y\end{bmatrix} = e^{tA} \begin{bmatrix} k_1\\ k_2\end{bmatrix}$$
where $k_1,k_2 \in \mathbb R$. How do I find a particular solution?

Laplace-transforming the system of ODEs,
$$\begin{bmatrix}s+5 & 1\\ -2 & s+3\end{bmatrix} \begin{bmatrix} X (s) \\ Y (s)\end{bmatrix} = \begin{bmatrix}\frac{1}{s-1}\\ 0\end{bmatrix}$$
Hence,
$$\begin{array}{rl} \begin{bmatrix} X (s) \\ Y (s)\end{bmatrix} &= \begin{bmatrix}s+5 & 1\\ -2 & s+3\end{bmatrix}^{-1} \begin{bmatrix}\frac{1}{s-1}\\ 0\end{bmatrix}\\ &= \dfrac{1}{(s+5)(s+3)+2} \begin{bmatrix}s+3 & -1\\ 2 & s+5\end{bmatrix} \begin{bmatrix}\frac{1}{s-1}\\ 0\end{bmatrix}\\ &= \dfrac{1}{s^2 + 8 s + 17} \begin{bmatrix}\frac{s+3}{s-1}\\ \frac{2}{s-1}\end{bmatrix}\\ &= \dfrac{1}{(s+4)^2 + 1} \begin{bmatrix}\frac{s+3}{s-1}\\ \frac{2}{s-1}\end{bmatrix}\end{array}$$
Now do partial fraction expansion and then take the inverse Laplace transform.
