11.5.2.24 Fourier method

Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=25 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(8, t)=0, \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\left\{\begin{array}{ll} x, & 0 \leq x<4 \\ 8-x, & 4 \leq x \leq 8 \end{array}\right. \end{array} \]