Question 138
Let us denote the set of positive real numbers by $\mathbb{R}_{\gt 0} = \{x \in \mathbb{R} \mid x \gt 0\}$. Find all functions $f: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ such that for all $x, y \in \mathbb{R}_{\gt 0}$, the following holds:
$$f(x f(x) + y) = x^2 + f(y).$$
(Bilegdemberel Bat-Amgalan)
$$f(x f(x) + y) = x^2 + f(y).$$
(Bilegdemberel Bat-Amgalan)