KöMaL 2018 A725

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f\colon \mathbb{R}^+\to \mathbb{R}^+$ satisfying the following equation for all $x,y\in\mathbb{R}^+$: $f\big(xy+f(y)^2\big) = f(x)f(y)+yf(y).$