BG-MO 2013 Problem 2

Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying: $x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$ for all $x,y \in \mathbb{R}$ Proposed by Nikolay Nikolov