Question 107
Let $\mathbb{R}^+$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying the equation
$$
xf(x^2)f(f(y)) + f(yf(x)) = f(xy)(f(f(x^2)) + f(f(y^2))).
$$
for all $x, y \in \mathbb{R}^+$.
$$
xf(x^2)f(f(y)) + f(yf(x)) = f(xy)(f(f(x^2)) + f(f(y^2))).
$$
for all $x, y \in \mathbb{R}^+$.