Nonhomogeneous PDE with non-constant coefficients

[kingwinner]

This is a question from a book in which I can't figure out, but it has no solutions at the back.

Find the general solution to the PDE:
xy u_x + y² (u_y) - y u = y - x

I've learned methods such as change of variables and characteristic curves, but I'm not sure how I can apply them in this situation. The PDE is nonhomogeneous, with non-constant coefficients, and there is an "u" term.

Can somebody help?

Thank you!

 

[kosovtsov]

If change variables as {x = X, y = YX} we get ODE

XYu_X-Yu-Y+1=0.

Its solution is

u(X,Y) = -1+1/Y+XF(Y),

where F(Y) is an arbitrary function.
So the general solution to initial PDE is as follows

u(x,y) = -1+x/y+xF(y/x)

 

[kingwinner]

Thanks, but what motivates that particular change of variables? (i.e. how did you derive them?) How did you get x = X and y = YX? (Is there a systematic way to derive these?)