11.5.2.18 Fourier method

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=9 u_{x x}, \quad u(x, 0)=5 \sin 5 \pi x, \\ u_{t}(x, 0)=0, \quad u(0, t)=0, \quad u_{x}(2.5, t)=0 . \end{array} \]