Series Solutions · Hard problem

A forced oscillator satisfies $y'' + y = 2\cos x$, $y(0) = 0$, $y'(0) = 0$. Expanding $y$ as a power series using the recurrence from matching coefficients (where $2\cos x$ is also written as a series) shows the amplitude of the oscillation growing — resonance. Find the exact value of $y\!\left(\dfrac{\pi}{2}\right)$.