Problem

Source: Vietnam TST 2022 P1

Tags: function, algebra, phi function, functional equation



Given a real number $\alpha$ and consider function $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$. Find all function $f:\mathbb R\to\mathbb R$ that satisfy: $$f(\varphi(x)+f(y))=y+\varphi(f(x))$$forall $x,y\in\mathbb R$