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Functional equation from best Ukrainian author

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.4

March 19, 2024
functionalgebrafunctional equation

Problem Statement

Find all functions f:R→Rf:\mathbb{R} \to \mathbb{R}, such that for any x,y∈Rx, y \in \mathbb{R} holds the following:
f(x)f(yf(x))+yf(xy)=xf(xy)+y2f(x)f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)
Proposed by Mykhailo Shtandenko
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