I've just arrived home from an exam and cannot come to terms with the fact I couldn't solve the following question:
Find a solution to
$$ \left\{ \begin{array}{ll} u_{tt} - u_{xx}=0, & x>0, t>0 \\ u(x,0)=u_t(x,0)=0, & x>0 \\ u(0,t)=e^t \sin{t}, & t>0 \\ \end{array}. \right. $$
I tried separation of variables and failed miserably. I have no idea how to approach the problem, but maybe it's because I'm tired after looking at it for 3,5h hours straight. At some point I even started questioning if the problem was well-posed.
Any hint is appreciated.
It is true that twice differentiable wave solutions defined for all $x$ are of the form $f(x+t)+g(x-t)$, but I think something else applies here. The function $u(x,t) = e^{t-x}\sin(t-x)$ for $0\le x\le t$ and $u(x,t) = 0$ for $t\le x$, $\ t>0$ is probably what was intended. It is a weak solution not differentiable along the ray $x=t$.