Exercise 47

Assume that all the given functions are differentiable. If \(z = \frac{1}{x}[f(x - y) + g(x + y)]\), show that \[ \frac{\partial}{\partial x}\left(x^2 \frac{\partial z}{\partial x}\right) = x^2 \frac{\partial^2 z}{\partial y^2} \]