We wish to solve the system
${\vec{x}\,}' = \left[\begin{array}{cc} 5 &1\cr -2 &2 \end{array}\right] \vec{x} + \begin{bmatrix} \sin(t) \\ e^t \end{bmatrix}$ ,
via eigenvector decomposition.

Let $\vec{v}_1$ be an eigenvector for the smaller eigenvalue of the coefficient matrix and $\vec{v}_2$ be an eigenvector for the larger eigenvalue. Let us pick the eigenvectors such that $\vec{v}_1 = \begin{bmatrix} -1 \\ ? \end{bmatrix}$ and $\vec{v}_2 = \begin{bmatrix} 1 \\ ? \end{bmatrix}$. What are these eigenvectors:
$\vec{v}_1 = {}$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
$\vec{v}_2 = {}$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

Then fill in the equation to write it in the eigenvector decomposed form.
$\vec{v}_1 \xi_1' + \vec{v}_2 \xi_2' = {}$ $\vec{v}_1 \xi_1 + {}$ $\vec{v}_2 \xi_2 + \vec{v}_1$ $+ \vec{v}_2$