Question 184
Given a real number $\alpha$, consider the function $\varphi(x) = x^2 e^{\alpha x}$ for all real numbers $x$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy
$$
f(\varphi(x) + f(y)) = y + \varphi(f(x))
$$
for all real numbers $x, y$.
$$
f(\varphi(x) + f(y)) = y + \varphi(f(x))
$$
for all real numbers $x, y$.