Boundary Values & Fourier Series · Superchallenge problem

A string $0 \le x \le \pi$ with both ends free, $u_x(0, t) = u_x(\pi, t) = 0$, obeys $u_{tt} = u_{xx}$. It starts flat, $u(x, 0) = 0$, with initial velocity $u_t(x, 0) = x$. Find $u\!\left(\dfrac\pi3, \dfrac{3\pi}{2}\right)$.