Trigonometric functional equation $g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$

Question

Find all functions $ f,g : \mathbb R \to \mathbb R $ that satisfy the functional equation $$g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$$ for all $x,y \in \mathbb R$.

I need hints for this problem.

Answer 1

Some hints/remarks about necessity: Play around with the substitutions $y = x$, $y = -x$ to show that necessarily either

Case 1: $f$ must be an odd function, or

Case 2: $g \equiv 0$.

If in Case 1, go on to show either

Case 1(a): $g \equiv f$, or

Case 1(b): $f \equiv 0$. In this case $g$ can be any arbitrary function.

If in Case 2, show that necessarily $f = c$ for some constant $c$.

The remaining bit which I haven't any suggestions is Case 1(a). In this case, it is sufficient if $f$ is a linear function: $f(ax + y) = af(x) + f(y)$ for $a,x,y \in \mathbb{R}$, but I don't know if it is necessary.