2.2.21 Derivatives and differentials

Problem: Let \( u=\frac{1}{x}(\varphi(x-y)+\psi(x+y)) \), where \( \varphi \) and \( \psi \) are differentiable functions. Show that \[ \frac{\partial}{\partial x}\left(x^{2} \frac{\partial u}{\partial x}\right)=x^{2} \frac{\partial^{2} u}{\partial y^{2}} . \]