Let me put in some details that might help. In your problem you are actually transforming coordinates from . So by the chain rule

Choosing , and gives

(from your PDE).

Thus, we are required to solve

(these are PDEs but independent of so really ODEs).

What about the IC where We need some kind of IC to go with this in the plane. Here we choose simple and let and connect the two boundary curves via (as you've done).

Thus, solve

subject to when then

.

Remember when interating (*) we get functions of integration and not constants of integration. Use (**) to find these.