So now the boundary condition amounts to an ODE with initial condition $\psi(0)=\phi(0)/2$? And then do we just leave the integral as it is after variation of parameter?

I am little diffident about the change of variable in the integral I made, so should I get

$$\psi(x-ct)=e^{\alpha(ct-x)}\int_0^{ct-x} e^{\alpha y}(\phi'(y)+\alpha \phi(y)) \,dy +e^{\alpha(ct-x)}\phi(0)/2 ?$$

It seems to me that I can be simplified further but I don't know how.

EDIT: This is solved in today's tutorial. So in fact it simplifies a little bit. But how about the integration constant, the initial condition for this ODE? On textbook it says (in our case) $\psi(0)=\phi(0)/2$, so then should I still have the term $e^{\alpha(ct-x)}\phi(0)/2$?