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Characteristics and PDEs

  1. Dec 21, 2011 #1
    1. The problem statement, all variables and given/known data

    For the system [itex]\psi_{xx}+y\psi_{yy}+{1\over 2}\psi_y=0[/itex] defined on [itex]y<0[/itex].
    Show that [itex]\psi(x,y)=f(x+2\sqrt{-y})+g(x-2\sqrt{-y})[/itex] for any functions [itex]f,g[/itex].

    Please help

    2. Relevant equations

    See above.

    3. The attempt at a solution

    I think that the characteristics for the system are [itex]\xi={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\eta={2\over 3}(-y)^{3\over 2}+x[/itex] .

    So we can reduce the system to [itex]{\partial^2 \psi\over\partial \xi\partial\eta}=0[/itex]

    It is clear to me that [itex]\psi(x,y)=f(\xi)+g(\eta)[/itex] for any functions [itex]f,g[/itex], but I cannot see how I might get the form required.

    Perhaps I have done something wrong?
    Last edited: Dec 21, 2011
  2. jcsd
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