11.5.5.5 Mixed problems

Problem: Solve the problem with stationary inhomogeneities using the Fourier method: \[ \begin{array}{ll} u_{t t}=u_{x x}+6 x, & u(x ; 0)=\sin ^{2} x-x^{3}, \\ u_{t}=(x ; 0)=0, & u(0 ; t)=0, \quad u(\pi ; t)=-\pi^{3} . \end{array} \]