Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$:

$$
(x-y) f(x+y)-(x+y) f(x-y)=4 x y\left(x^{2}-y^{2}\right)
$$
The expression seems complicated at first glance. However, we can notice the strong presence of $x+y$ and $x-y$. We then set $a=x+y$ and $b=x-y$, so we have $x=\frac{a+b}{2}$ and $y=\frac{a-b}{2}$. The initial equation is then equivalent to:

$$
b f(a)-a f(b)=a b\left(a^{2}-b^{2}\right)
$$

Taking $b=1$, we have $f(a)=a^{3}+a(f(1)-1)$. Thus, there exists a real number $c$ such that $f(x)=x^{3}+x c$ for all real numbers $x$, and we verify that these functions satisfy the initial equation.