Could be way off here, but couldn't you just use this:
for some function u(x,y)
Rewrite the PDE and IC's in terms of by expressing
and in terms of and and solve.
My attempt: I believe the first equation is of the form
therefore but they are not in terms of and
Have I gone one step too far?
. Rewriting the pde becomes
The IC being for
Is this correct...Can this be solved using the method of characteristics? ill try..
It's because the integration you did was sort of "partial integration w.r.t. ", not a total integration like you'd have in an ODE setting. You remember doing exact ODE's? When you integrated w.r.t. , you'd get an arbitrary function of . In your case, period. It is a function of two variables. If you doubt the extra function , then just do a partial differentiation and see if you get back to the original DE. You do, right?
What is the IC, exactly?