Is it true that $x^2u_x+x^2u_y=G(x,y)u(x,y)$ if $u(x,y)=xyf\left(\frac{x+y}{x-y}\right)$ for some differentiable $f(x,y)$ defined from $\mathbb{R}\to\mathbb{R}$ and a two variable function $G(x,y)$?
I dont think so, because when we differentiate partially, we get the term $f'\left(\frac{x+y}{x-y}\right)$ which cannot be expressed in terms of product of $u(x,y)$