What approaches can we take to solve the functional equation,
$g:\mathbb{R}\to \mathbb{R}$ is a differentiable function, such that,
$$g'(x)=\dfrac{g(bx)-g(x)}{(b-1)x}$$
Where, $b \in \mathbb{R}$ is a constant.
My question arises from trying to solve this question.
If $g(x)$ is an $n$th-order polynomial with $g(0) = 0$, then $$\frac{g(bx)-g(x)}{x}$$ will be a polynomial of order $n-1$. Conveniently, differentiation of an $n$th-order polynomial also results in a polynomial of order $n-1$.
With this in mind, we try $g(x) = ax^n$. As it turns out, $ax^n$ is a solution for any $a\in\mathbb{R}$, where $n$ is given implicitly by $$n = \frac{b^n-1}{b-1}.$$ As an example, when $b = -1$, we can have both $n = 0$ and $n = 1$.
