Given that $y=x\varphi(z)+\psi(z)$ where $z$ is an implicit function of $x,y$, and $x\cdot\varphi'(z)+\psi'(z)\neq0$. Try to prove that $\frac{\partial^2z}{\partial x^2}\cdot\left(\frac{\partial z}{\partial y}\right)^2-2\cdot\frac{\partial z}{\partial x}\cdot\frac{\partial z}{\partial y}\cdot\frac{\partial^2 z}{\partial x\partial y}+\frac{\partial^2 z}{\partial y^2}\cdot\left(\frac{\partial z}{\partial x}\right)^2=0$
The outline of the proof from the book Григорий Михайлович Фихтенгольц:
Differentiating $y=x\varphi(z)+\psi(z)$ with $\dfrac{\partial^2z}{\partial x^2}$, $\dfrac{\partial^2z}{\partial x\partial y}$, $\dfrac{\partial^2z}{\partial y^2}$, and then multiplying special coefficients, we can get the answer.
I wonder whether there's some sensible ways to check these equations, or even more, some sysmatical ways to produce such equations.
Any help? Thanks a lot!