📝 题目
5.3.10 证明: 由方程组
$$ \left\{ \begin{array}{l} z = {ax} + {y\varphi }\left( a\right) + \psi \left( a\right) , \\ 0 = x + {y\varphi }\left( a\right) + {\psi }^{\prime }\left( a\right) \end{array}\right. $$
所确定的函数 $z = z\left( {x,y}\right)$ 满足方程
$$ \frac{{\partial }^{2}z}{\partial {x}^{2}} \cdot \frac{{\partial }^{2}z}{\partial {y}^{2}} - {\left( \frac{{\partial }^{2}z}{\partial x\partial y}\right) }^{2} =