11.5.2.5 Fourier method

Problem: Solve the equation \( u_{x x}-\frac{1}{a^{2}} u_{t t}=0,(0 \leq x \leq l) \) under the given initial and boundary conditions. \begin{tabular}{|c|c|c|c|} \hline\( u(0, t) \) & \( u(l, t) \) & \( u(x, 0) \) & \( u_{t}^{\prime}(x, 0) \) \\ \hline 0 & 0 & \( 3 \sin \frac{5 \pi}{l} x \) & \( 7 x \) \\ \hline \end{tabular}