How do I find the third solution of the system of differential equations, when I was able to find two vector solutions with the eigenroot 5?

Question

x' = 5x+2y

y' = 5y + e^5t

z' = 3y + 5z

I placed the system in a matrix to get solutions for the homogeneous system. I received two eigenvectors V1=(1,0,0) and V2=(0,0,1) corresponding to the solution x(t)=e^5t (C1V1 +C2V2) But I can't find the third solution (so that I have a linear dependence) Thanks for any direction or help