It's annoying for me too Yung. There is a way out of this quagmire of course: Solve it for a particular case and then back-substitute the solution into the PDE and see if satisfies it. You can do this in Mathematica very nicely.

This is what I did which is inadequate: Let

Then using variation of parameters, I obtain for the y-part:

which is similar to what you posted but I do not understand how to set up the limits of integration and again, in order to confirm the answer, I would figure out how to set up the limits, then solve if for a particular IBVP, back-substitute the solution into the PDE and see if my solution or the book's solution satisfies it.