$\vec Y''=A\vec Y , A_{n\times n}$. I have to find a requirement such that $\vec Y(x)=\vec Ve^{ax}$ is a solution.
If $V e^{ax}$ is a solution then (plugging it into the equation) you must have $a^2V e^{ax} = AV e^{ax}$ so $a^2V = AV$ and $V$ is an eigenvector of $A$ with eigenvalue $a^2$.