# Characteristics and PDEs

1. Dec 21, 2011

### c299792458

1. The problem statement, all variables and given/known data

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 $f,g$.

2. Relevant equations

See above.

3. 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?

Last edited: Dec 21, 2011