Exercise 102

Solve $$\begin{array}{ll} y_{tt} = 25 y_{xx} & \qquad \text{for } \; 0 < x < 2 \text{ and } t > 0, \\ y(0,t) = y(2,t) = 0 & \qquad \text{for } \; t > 0, \\ y(x,0) = 0 & \qquad \text{for } \; 0 < x < 2 , \\ y_t(x,0) = \sin(\pi x) + 0.1 \sin(2\pi x) & \qquad \text{for } \; 0 < x < 2 . \end{array}$$