Exercise 103

Solve $$\begin{array}{ll} y_{tt} = 2 y_{xx} & \qquad \text{for } \; 0 < x < \pi \text{ and } t > 0, \\ y(0,t) = y(\pi,t) = 0 & \qquad \text{for } \; t > 0 , \\ y(x,0) = x & \qquad \text{for } \; 0 < x < \pi , \\ y_t(x,0) = 0 & \qquad \text{for } \; 0 < x < \pi . \end{array}$$