11.5.2.23 Fourier method

Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables \( x \) (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=16 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=\frac{8}{9}\left(3 x-x^{2}\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0 \end{array} \]