The solutions is
Should this be set equal to
You got two things going on that seems to be confusing you - (1) a rotation of coordinates and (2) a general change of coordinates. Let me see if I can help.
Your problem is
You have introduced a linear change of variables (this can be more general but it's not necessary here)
What you want is to transform your PDE to essentially an ODE by choosing the constant and with the requirement that (why?).
As you've said
and picked and (that's good!) so
You still have the flexibility to choose and anything. You further chose and so
This is also good b/c the boundary
Now you must switch everthing over in terms of your new variables so since
then your new problem is
with when .
Now follow what you did in post #2.