Give the general solution of the $2\times 2$ inhomogeneous system of differential equations

Question

Give the general solution of the system: $$X'(t) = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix} X(t)+\begin{pmatrix} 2e^{2t} \\ 0 \end{pmatrix} $$

I manage to come to the general solution to the homogenous but when I get finding a particular solution I'm messing up.

Answer 1

The eigenvalues of the matrix are $2$ and $4$, one of which is in $e^{2t}$. So try $$X(t)=\left(\begin{array}{c}ate^{2t}\\bte^{2t}\end{array}\right)$$