Systems of Differential Equations · Hard problem

Two pairs of coupled springs are released from rest at $(1,1)$: pair A obeys $x''=-2x+y,\ y''=x-2y$, and pair B obeys $x''=-3x+y,\ y''=x-3y$. Each oscillates at a single frequency (since $(1,1)$ is a shared eigenvector of both coupling matrices). Find the ratio of the longer period to the shorter period.