I found this problem and I can't solve it. If anyone can help me with this , it'd be a lot appreciated.
Solve the following non-homogeneous system: $$\begin{pmatrix}x\\y \end{pmatrix}^{'}=\begin{pmatrix}4 &1 \\ 4&-2 \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix} e^{-2t}\\-2e^{t} \end{pmatrix}$$
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The characteristic polynomial of $A = \begin{pmatrix}4 &1 \\ 4&-2 \end{pmatrix}$ and got the eigenvalues , the struggle begin with the eigenvectors. Below is what I found for the eigenvectors is it correct? $$\begin{pmatrix}-\frac{-3-\sqrt{13}}{4}\\ 1\end{pmatrix},\begin{pmatrix}-\frac{\sqrt{13}-3}{4}\\1 \end{pmatrix}$$ I checked their products with $A$ and it worked, but I ended up with a terrifying integral.