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First order linear non-homogeneous PDE

  1. Sep 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the general solution to the PDE and solve the initial value problem:
    y2 (ux) + x2 (uy) = 2 y2, initial condition u(x, y) = -2y on y3 = x3 - 2


    2. Relevant equations/concepts
    First order linear non-homogeneous PDEs


    3. The attempt at a solution
    I know that the general solution to the non-homogeneous PDE = a particular soltuion to it + the general solution to the assoicated homogenous PDE, so I first consider to assocatied homogeneous equation:
    y2 (ux) + x2 (uy) = 0

    The characteristic equation is dy/dx = x2/y2
    And the solution to this ODE is y3 - x3 = C where C is an arbitrary constant.
    So the general solution to the homogeneous equation is u = f(y3 - x3) where f is arbitrary differentiable function of one variable.
    But then I am stuck. How can I find a particular solution to the original non-homogeneous PDE? Is there a systematic way to find one such solution?

    Any help is appreciated! :)
     
  2. jcsd
  3. Sep 20, 2009 #2
    Can someone please help? I am still stuck on this problem... :(
     
  4. Sep 20, 2009 #3

    LCKurtz

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    I don't know about any systematic ways, but just looking at it, if you could find a u such that ux = 2 and uy = 0 that would do it. And that's easy.
     
  5. Sep 20, 2009 #4
    Then, I think u=2x will work as a particular solution.

    But really, can anyone suggest a more general/systematic way to solve it that would work also for more complicated first order linear PDEs?

    Thanks!!!
     
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