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.