I was wondering if there's any other way of doing this exercise, other than calculating the partial derivatives and then replacing them in the given equation. The calculus can get pretty ugly, and I didn't exactly see the point of this, so that's why it got me wondering if maybe I should look for a different simpler approach.
Show that the function $u=\varphi(xy) + \sqrt{xy} * \psi(\frac{y}{x}), \varphi,\psi \in C^2$
satisfies the equation: $ x^2\frac{\partial^2u}{\partial x^2} -y^2\frac{\partial^2u}{\partial y^2}=0$
Looking forward for your answers, and thank you!