Question 20
Let a real number $\alpha$ be given.
Determine, depending on $\alpha$, all functions $f: \mathbb{R} \to \mathbb{R}$ with
$$
f(f(x+y)f(x-y)) = x^2 + \alpha y f(y)
$$
for all $x, y \in \mathbb{R}$.
Determine, depending on $\alpha$, all functions $f: \mathbb{R} \to \mathbb{R}$ with
$$
f(f(x+y)f(x-y)) = x^2 + \alpha y f(y)
$$
for all $x, y \in \mathbb{R}$.