I have an ordinary differential equation in which I substitute $\xi = \xi(x), \eta = \eta(x,y)$ and I wish to express $y''_{xx}$ in terms of $\eta''_{\xi\xi}$.
So I get an equation $$d^2\eta=\frac{\partial^2 \eta}{\partial x^2}dx^2+2\frac{\partial \eta}{\partial x\partial y}dxdy+\frac{\partial^2 \eta}{\partial y^2}dy^2=\frac{d\eta^3}{d\xi^2}(\xi')^2dx^2$$
If it is correct that $dxdy=d(ydx)=d^2y$, then I could divide by $dx^2$ and get the expression. Although it seems to be far from correct and I'm asking for your advice.