Consider second order PDE: $x\phi_{xx}+2x^2\phi_{xy}-\phi_x=x^3$
My characteristic variables are: $\xi=y-x^2$ and $\eta=y$
$\phi_x = \Phi_\xi.\xi_x+\Phi_\eta.\eta_x=-2x\Phi_\xi$
$\phi_{xx} = -2\Phi_\xi+4x^2\Phi_{\xi\xi}$
$\phi_y = \Phi_\xi+\Phi_\eta$
$\phi_{xy}$ = $-2x(\Phi_{\xi\xi}+\Phi_{\xi\eta})$
I am having trouble understanding how in $\phi_{xx}$, where the second term came from and how they deduced the chain rule for $\phi_{xy}$.