Exercise 1. Let $a > 0$ be a constant. Solve the system \begin{cases} u_{tt} = a^2 u_{xx}, & 0 \le x \le L, \quad -\infty < t < +\infty, \\ u(0, t) = 0, \quad u(L, t) = 0, & -\infty < t < +\infty, \\ u(x, 0) = 3\sin\left(\frac{\pi x}{L}\right) - \sin\left(\frac{4\pi x}{L}\right), & 0 \le x \le L, \\ u_t(x, 0) = \frac{1}{2}\sin\left(\frac{2\pi x}{L}\right), & 0 \le x \le L. \end{cases}