Find all function such that $f(x)-f(y) = (x -y)g(\sqrt{xy})$

Question

Find all functions $f, g$ that satisfy the functional equation $$ f(x)-f(y)= (x -y)g(\sqrt{xy}) \quad \forall\ x,y>0. $$

Answer 1

Hint: First fix $y=1$, this shows what $f$ has to be, in terms of $g$. Then fix another $y$ as well.