Is there partial laplace derivative equations? I am so confused.
Show that the function provides the equation.
\begin{equation} \label{simple_equation0} u = {\varphi }(xy)+\sqrt{xy}{\psi}(\frac{y}{x}) \end{equation}
\begin{equation} \label{simple_equation1} x^{2}\frac{\partial^2 u}{\partial x^{2}}-y^{2}\frac{\partial^2 u}{\partial y^{2}}= 0 \end{equation}
Hint: \begin{align*} \frac{\partial^2 u}{\partial x^2} &= -\frac{y^2 \psi '\left(\frac{y}{x}\right)}{x^2 \sqrt{x y}}+\sqrt{x y} \left(\frac{y^2 \psi ''\left(\frac{y}{x}\right)}{x^4}+\frac{2 y \psi '\left(\frac{y}{x}\right)}{x^3}\right)-\frac{y^2 \psi \left(\frac{y}{x}\right)}{4 (x y)^{3/2}}+y^2 \varphi ''(x y) \\ &= \frac{\sqrt{x y} \left(4 y \left(x^3 \sqrt{x y} \varphi ''(x y)+y \psi ''\left(\frac{y}{x}\right)+x \psi '\left(\frac{y}{x}\right)\right)-x^2 \psi \left(\frac{y}{x}\right)\right)}{4 x^4} \text{.} \end{align*}