5. [10 pts] Solve the following inhomogeneous heat equation for a heat ring of length $2\pi$
$u_t = 3u_{xx} + e^{-t} \sin(2x)$
$u(-\pi, t) = u(\pi, t); \quad u_x(-\pi, t) = u_x(\pi, t); \quad t \ge 0$
$u(x, 0) = 4 + 3\cos(3x); \quad x \in [-\pi, \pi].$