Exercise 101

Find a series solution of $$\begin{array}{ll} u_t = 3 u_{xx} & \qquad \text{for } \; 0 < x < \pi \text{ and } t > 0, \\ u(0,t) = u(\pi,t) = 0 & \qquad \text{for } \; t > 0, \\ u(x,0) = 5\sin (x) + 2\sin (5x) & \qquad \text{for } \; 0 < x < \pi . \end{array}$$