Boundary Values & Fourier Series · Challenge problem

A string $0 \le x \le \pi$ with fixed ends starts flat and at rest, and is driven at its fundamental frequency: $u_{tt} = u_{xx} + \sin x\cos t$, $u(x, 0) = u_t(x, 0) = 0$. Find $u\!\left(\dfrac\pi2, \dfrac{5\pi}{2}\right)$.