Question 127
Let $\mathbb{R}^{\ge 0}$ be the set of non-negative real numbers. Find all functions $f : \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that for all $x, y \in \mathbb{R}^{\ge 0}$,
$$
f(x^3 + x f(xy)) = f(xy) + x^2 f(x + y).
$$
$$
f(x^3 + x f(xy)) = f(xy) + x^2 f(x + y).
$$