11.5.5.9 Mixed problems

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t t}=\frac{1}{16} u_{x x}+50 e^{-7 t} \cdot \sin 4 x, \\ u(x, 0)=u_{t}(x, 0)=0, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]