I'm not very good at PDEs but this particular problem seems... Strange. It requires that the answer be "continuous (!!)" all in bold.
\begin{align} u_{tt}&=9u_{xx},\quad x>0,\, t>0, \\ u(x,t)&=e^{-x} \\ u_t(x,0)&=\cos(5x)\\ u_x(0,t)&=u(0,t)+t\\ \end{align}
That last one is giving me trouble, and I'm certain how to interact with it properly... I found a solution to the first 2 restrictions with separation of variables, but I can't see any way to manipulate it to find the third as well.
Could someone point me in the right direction, or barring that, spell out the answer for me?