11.5.2.7 Fourier method

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=2, \quad u(x, 0)=2 \cos \frac{5}{2} x, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]