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Graphic and solution of PDE

  1. Jul 29, 2015 #1
    y = a x² + b x + c is a parabola. But, a parabola is just a kind of conic.

    All conics are given by a x² + b x y + c y² + d x + e y + f = 0

    The same way, the graphic y = f(x), with f(x) satisfying a d²f/dx² + b df/dx + c f = 0, is just a particular graphic of F(x,y) = 0 with F(x,y) satisfying

    a d²F/dx² + b d²F/dxdy + c d²F/dy² + d dF/dx + e dF/dy + f F = 0

    OBS: a, b, c... are constants.

    So, which is the general solution for the PDE above? And, where I can visualize the graphic?
     
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  3. Jul 29, 2015 #2

    HallsofIvy

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    I am not sure I understand your question. It should be clear that the "general solution" to "[itex]a F_{xx}+ b F_{xy}+ c F_{yy}+ d F_x+ e F_y+ fF= 0[/itex]" is simply "[itex]ax^2+ bxy+ cy^2+ dx+ ey+ f= 0[/itex]".
     
  4. Jul 29, 2015 #3
    I simply want to know which is the funcion F(x,y) that satisfies a PDE of second order with constant coeficients.

    If this function F don't exist, or wasn't discoverd yet, so, exist some program that outline the graphic(approximately) of the solution of the PDE?
     
  5. Aug 7, 2015 #4

    HallsofIvy

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    That is basically the opposite of the question you originally asked! Second order partial differential equations can have very different solutions and are basically divided into "elliptic", "hyperbolic", and "parabolic" equations. Any introductory PDE text book will have a discussion of what those type of equations and how their solutions differ.
     
  6. Aug 8, 2015 #5
    Still today, none made some effort for try to unify these 3 kinds of solutions. Don't exist a general solution?
     
  7. Aug 8, 2015 #6

    micromass

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    No, it doesn't exist.
     
  8. Aug 8, 2015 #7
    Why not? Is it impossible?
     
  9. Aug 8, 2015 #8

    micromass

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    Yes, I think it is impossible to give a nice unified solution. But I have not seen this in a theorem/proof form yet. You can of course solve any PDE as in the original post if you are given specific values and specific boundary values. Just as conic section can have very different behavior, so do PDE's.
     
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