Problem

Source: India TST 2023 Practice Test 2 P2

Tags: algebra, functional equation



Let $\mathbb R^+$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying \[f(x+y^2f(x^2))=f(xy)^2+f(x)\]for all $x,y \in \mathbb{R}^+$. Proposed by Shantanu Nene