11.5.2.16 Fourier method

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=25 u_{x x}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=5 \sin 3 \pi x, \quad u_{t}(x, 0)=20 \pi \sin 4 \pi x . \end{array} \]