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