Multiplicative monomial functions of third degree (Q1345862)

The author proves that the general multiplicative solution of \[ f(x + 3y) + f(x - 3y) = 3f(x + y) + 3f(x - y) + 6f(2y)\qquad (x,y \in \mathbb{R}) \] in given by \(f(x) = xg(x)\) where \(g\) is an arbitrary multiplicative solution of \[ g(x + y) + g(x - y) = 2g(x) + 2g(y)\qquad (x,y \in \mathbb{R}). \]