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Does this P.D.E have analytic solutions?

  1. Jun 15, 2013 #1
    I would like to know whether this P.D.E is solvable and if yes, for which values of a.

    2 (y Cos[a] + x Sin[a]) - 4 (y Sin[a] - x Cos[a]) f[x, y]
    - 2 (y Cos[a] + x Sin[a]) (f[x, y])^2
    + ((x^2) Cos[a] - 2 x y Sin[a] - (y^2) Cos[a]) D[f[x, y], x]
    - ((x^2) Sin[a] + 2 x y Cos[a] - (y^2) Sin[a]) D[f[x, y], y] == 0

    I have tried to solve it for the case where a=0
    2 y + 4 x f[x,y] - 2 y f[x,y]^2 - 2 x y (f^(0,1))[x,y] + (x^2-y^2) (f^(1,0))[x,y]==0,
    on MATHEMATICA but after 3 hours it gave me no result.

    Then I utilised the method of characteristics and expressed the a=0 case as a system of three ODES
    dx/dt = (x^2 - y^2)
    dy/dt = (-2xy)
    dw/dt = 2yf^2 - 2y - 4xf
    but MATHEMATICA still gave me no results. So I suspect that this PDE might not have an analytic solution.
    Is this PDE solvable, and if not could you please indicate a reference which indicates which cases of PDES are non solvable? I am only interested in Analytic solutions.

    Any kind of help will be most appreciated.
    Thank you in advance.
     
  2. jcsd
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