Maybe I wasn't clear in my first post, f is a function of x and t and with df/dx I mean the partial derivative of f to x. The question is to rewrite the PDE into a PDE in g(y,s), y and s.
It wasn't clear to me. In general, for the PDE:
, we solve the characteristic DE:
Assume the characteristic solution can be put into the form . Then the change of variable will convert the PDE to an ODE in for fixed . Probably not what you want. That's why I didn't say anything. Also while I'm sayin', this is right out of "Basic Partial Differential Equations" by Bleecker and Csordas. It's a nice PDE book that's easy to read. They go over first order PDEs like this pretty good.
This is how I'd do it; maybe someone can suggest a better approach to both of us. I first write it in standard form:
The characteristic equation becomes then:
So I make the change of variables:
Then define:
and under these transformations it can be shown:
Thus I have: