Find the general solution for: $$y_1'=-y_1+3y_2 $$ $$y_2'=2y_1-2y_2 $$
Find the solution satisfying the initial condition $y(0)=\begin{pmatrix} 5 \\ 0 \\ \end{pmatrix}$
From the equation above, $\lambda_1=1$ and $\lambda_2=-4$,
The general soltution is: $$c_1e^x\begin{pmatrix} 3 \\ 2 \\ \end{pmatrix} + c_2e^{-4x}\begin{pmatrix} 1 \\ -1 \\ \end{pmatrix}$$
I'm not sure how to use the condition above to find the particular solution.
Hint: solve the linear system $$c_1e^0\begin{pmatrix} 3 \\ 2 \\ \end{pmatrix} + c_2e^0\begin{pmatrix} 1 \\ -1 \\ \end{pmatrix}=y(0)=\begin{pmatrix} 5 \\ 0 \end{pmatrix},$$ that is $$\begin{cases} 3c_1+c_2=5\\ 2c_1-c_2=0 \end{cases},$$ for the constants $c_1,c_2$.