I think you must be missing a lot of information. For one thing, you can solve this PDE using direct integration...
I agree with Prove It. The equation, as given, can be directly integrated while making the "indicated transformation" gives a much harder equation!
But if you must transform, use the fact that, for any function, F(x,y), and
So, letting , and, letting , .
The equation becomes .
As Prove It said, the original equation is much easier to solve than the transformed equation!