First-Order integrating factor of the form f(xy)

[PoleVault12]

(Moderator's note: thread moved from "Differential Equations")

M(x,y) + N(x,y)(dy/dx) = 0

f'(xy) = G(xy)f(xy) where G(xy) = (N_x - M_y)/(xM - yN)

Replace xy with a single variable to obtain a simple 1st order differential equation and find f(xy).

I got to:

ln|f| = Integral(G(xy)) by seperating the variables

But I am unsure how to integrate G(xy) with respect to a single variable.

Any Suggestions?

 

[matematikawan]

What does f'(xy) denote? Is it $$\frac{d}{dx}f(xy)$$ or $$\frac{d}{d(xy)}f(xy)$$