11.5.5.6 Mixed problems

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{36} u_{x x}+17 \cos 4 t \cdot \sin 6 x \\ u(x, 0)=5 \sin 18 x, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]