Example 2 Let $f(x)$ be defined on $[0,+\infty)$, and satisfy the functional inequality:
$$
(x-y) f(x+y)-(x+y) f(x-y) \geqslant 4 x y\left(x^{2}-y^{2}\right) .
$$

Find all such functions $f(x)$.
Let's assume $x>y$, then the given inequality can be equivalently transformed into
$$
\frac{f(x+y)}{x+y}-\frac{f(x-y)}{x-y} \geqslant 4 x y,
$$

Let $g(x)=\frac{f(x)}{x}$, inequality (1) is equivalently transformed into $g(x+y)-g(x-y) \geqslant 4 x y$.
Let $u=x+y \cdot v=x-y$, then $u \geqslant v$, the above inequality is equivalently transformed into $g(u)-g(v) \geqslant u^{2}-v^{2}$. That is,
$$
g(u)-u^{2} \geqslant g(v)-v^{2},
$$

Let $h(u)=g(u)-u^{2}$, then $h(u) \geqslant h(v)$. By induction, we can obtain the general solution of the functional inequality
$$
f(x)=x^{3}+x h(x), x \in[0,+\infty) .
$$

where $h(x)$ is a monotonically increasing function on $[0,+\infty)$.