设 $z=z(x,y)$ 可微, 且满足 $x^2\frac{\partial z}{\partial x}+y^2\frac{\partial z}{\partial y}=z^2$.
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
\[
\begin{cases}
u&=x,\\
v&=\frac{1}{y}-\frac{1}{x},
\end{cases}\quad\text{及}\quad w=\frac{1}{z}-\frac{1}{x},
\]
证明: $\frac{\partial w}{\partial u}=0$.
注: 题目来自于 https://www.bilibili.com/video/BV1kv4y1E7kK/