📝 题目
5.3.5 设 $z = z\left( {x,y}\right)$ 由方程 ${x}^{2} + {y}^{2} + {z}^{2} = {yf}\left( \frac{z}{y}\right)$ 所确定,证明:
$$ \left( {{x}^{2} - {y}^{2} - {z}^{2}}\right) \frac{\partial z}{\partial x} + {2xy}\frac{\partial z}{\partial y} = {2xz}. $$
5.3.5 设 $z = z\left( {x,y}\right)$ 由方程 ${x}^{2} + {y}^{2} + {z}^{2} = {yf}\left( \frac{z}{y}\right)$ 所确定,证明:
$$ \left( {{x}^{2} - {y}^{2} - {z}^{2}}\right) \frac{\partial z}{\partial x} + {2xy}\frac{\partial z}{\partial y} = {2xz}. $$