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Nonhomogeneous PDE with non-constant coefficients

  1. Sep 18, 2009 #1
    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 ux + y2 (uy) - y u = y - x

    I've learnt 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!
     
  2. jcsd
  3. Sep 18, 2009 #2
    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)
     
  4. Sep 19, 2009 #3
    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?)
     
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