# Characteristics and PDEs

## Homework Statement

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

Please help

See above.

## The Attempt at a Solution

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

So we can reduce the system to ${\partial^2 \psi\over\partial \xi\partial\eta}=0$

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

Perhaps I have done something wrong?

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## Answers and Replies

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