11.5.5.14 Mixed problems

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{16} u_{x x}+10 \sin 3 t \cdot \sin 4 x, \quad u(x, 0)=0, \\ u(0, t)=0, \quad u(\pi, t)=0 . \end{array} \]