Wave equation $u_{tt}=u_{xx}+e^{-t}\cos{\frac{\pi x}{2l}}$, when $u(x,0)=u_t(x,0)=0$, $u_x(0,t)=u(l,t)=0.$

Question

How to do following:

In region $D=\{(x,t):0<x<l,\hspace{2mm} t>0\}$ solve $$u_{tt}=u_{xx}+e^{-t}\cos{\frac{\pi x}{2l}}$$ with initial conditions $u(x,0)=u_t(x,0)=0$ and boundary conditions $u_x(0,t)=u(l,t)=0.$

I have encontered similar problems (when $u(0,t)=u(l,t)=0$) and I know how to do this by separation of variables, but I'm stuck with this problem. Could anyone just write me a short steps in solving this problem, because I don't get solution that is written in book?

Thanks in advance.