Question: Consider $\Phi (x,h)= (1-2xh+h^2)^{-1/2}$. Show that $$(1-x^2){\partial^2 \Phi\over\partial x^2} -2x{\partial\Phi\over\partial x} +h{\partial^2\over\partial h^2} (h\Phi)=0$$
This is what I’m able to come up with (again, now third time): \begin{align} \Phi_x&=h\Phi^3\\ \Phi_{xx}&=3h^2\Phi^5\\ \ (h \Phi)_{hh}&=\Phi^3(2x-3h)+3h\Phi^5(x-h)^2 \end{align}
But this makes the LHS of the equation in the question to evaluate to $3h^2\Phi^3(\Phi^4-1)$ which is clearly not zero.
What went wrong?
Got that: I was writing wrongly that $\Phi^2(1-2xh+h^2)$ was equal to $\Phi^4-1$. It is $0$, in fact.