# Why is this change of variables true?

I have an equation $F_{xx}+yF_{yy}+{1\over 2}F_y=0$ defined on $y<0$.

I found that the characteristics are $\alpha={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\beta={2\over 3}(-y)^{3\over 2}+x$ and that ${\partial^2 F\over\partial \alpha\partial\beta}=0$

I wish to show that $F(x,y)=f_1(x+2\sqrt{-y})+f_2(x-2\sqrt{-y})$ for any functions $f_1,f_2$.

I get how $F(x,y)=f_1(\alpha)+f_2(\beta)$ for any functions $f_1,f_2$, but how does ${\partial^2 F\over\partial \alpha\partial\beta}=0$ imply that $F(x,y)=f_1(x+2\sqrt{-y})+f_2(x-2\sqrt{-y})$ for any functions $f_1,f_2$?

Thanks.

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If a function of two variable $(a,b)$ $g$ is such that $g_{ab}=0$ then $g_a(a,b)=g_1(a)$ since $g_b$ is constant in $b$ (but we have a constant for each $a$) and $g(a,b):=h_1(a)+h_2(b)$. –  Davide Giraudo Dec 18 '11 at 19:28