Slove $$ \left\{ \begin{array}{rl} u_{tt}=u_{xx}+\sin 2x &\mbox{ $0\lt x\lt\pi,t\gt0$} \\ u(0,t)=u(\pi,t)=0 \\ u(x,0)=u_t(x,0)=0 \end{array} \right. $$
How to solve these kind equations? I already know the solution like $$ \left\{ \begin{array}{rl} u_{tt}=a^2u_{xx}\\ u(x,0)=\phi(x),u_t(x,0)=\psi(x) \end{array} \right. $$
Is there a general solution for the equation above?
On
For this kind of problem in particular, Duhamel's principle works very nicely. Incidentally, there happens to be an online source with a walkthrough of almost exactly the same problem. See if you can follow through it to get the general procedure for these kinds of problems.
Hint: make the substitution $v=u-\frac 14 \sin 2x$ to reduce the equation to the form $v_{tt}=v_{xx}$. Do you know what to do from here?