问题

设 $z=z(x,y)$ 可微, 且满足 $x^2\frac{\partial z}{\partial x}+y^2\frac{\partial z}{\partial y}=z^2$. 作变换

$$\{\begin{array}{ll}u& =x,\\ v& =\frac{1}{y}-\frac{1}{x},\end{array}\text{及}w=\frac{1}{z}-\frac{1}{x},$$

证明: $\frac{\partial w}{\partial u}=0$.

注: 题目来自于 https://www.bilibili.com/video/BV1kv4y1E7kK/