Question 42
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x+y)f(x-y) = (f(x)+f(y))^2 - 4x^2f(y), \quad (1)
$$
for all $x, y \in \mathbb{R}$, where $\mathbb{R}$ denotes the set of all real numbers.
$$
f(x+y)f(x-y) = (f(x)+f(y))^2 - 4x^2f(y), \quad (1)
$$
for all $x, y \in \mathbb{R}$, where $\mathbb{R}$ denotes the set of all real numbers.