# System of differential equations (particular solution)

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 $$c_1,c_2$$.