Nonhomogeneous Equations & Resonance · Hard problem

Solve $y''+4y=6\sin2t\cos t$ with $y(0)=0$, $y'(0)=0$. Find the long-run time-average of $y(t)^2$, that is, $\displaystyle\lim_{T\to\infty}\dfrac1T\int_0^Ty(t)^2\,dt$.