The system

$$\left\{\begin{array}{c}x'=x-5y\\\ y'=2x-y\end{array}\right.$$

can be reduced to a single equation. By the second equation, we have $x=\frac{1}{2}(y'+y)$.Plugging this into the first:

$$\frac{1}{2}(y''+y')=\frac{1}{2}(y'+y)-5y$$

...or:

$$y''+9y=0$$

The general solution for $y$ is given by: