I have the PDE a(t)*(df/dx) + df/dt + b(t)f=0.
I have to make a change of variables now where g=fc(t), y=xd(t) and s=e(t). I have no idea how this works with coefficients depending on t.
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:
and under these transformations it can be shown:
Thus I have: