2.2.11 Derivatives and differentials

Problem: Function \( z=f(x, y) \) is given. Show that \[ \begin{array}{l} F\left(x ; y ; z ; \frac{\partial z}{\partial x} ; \frac{\partial z}{\partial y} ; \frac{\partial^{2} z}{\partial x^{2}} ; \frac{\partial^{2} z}{\partial y^{2}} ; \frac{\partial^{2} z}{\partial x \partial y}\right)=0 . \\ z=\frac{y^{2}}{3 x}+\sin ^{-1}(x y), \quad F=x^{2} \frac{\partial z}{\partial x}-x y \frac{\partial z}{\partial y}+y^{2} . \end{array} \]