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Characteristics and PDEs

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Homework Statement



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


Homework Equations



See above.

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:

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