A simple difference equation or functional equation that has a solution $(ax+b)^c$

Question

I am looking for a (system of) functional equation or difference equation that has a solution: $f(x)=(ax+b)^c$, where $a,b,c$ are constants. Constant $a,b,c$ cannot appear in the equation.

For example, the closest possible equation I found is a differential equation: $\frac{f'''f'}{(f'')^2}=C$.

For another example, consider a special case. The simple function $g(x)=x^c$ is the unique solution to a very simple functional equation: $g(xy)=g(x)g(y)$. However, a differential equation is considerably more complicated. The motivation of this question is to find an elegant functional equation that has the same solution as a more complicated differential equation.

Answer 1

$$f’(x)=c(ax+b)^{c-1}a=ac\dfrac{f(x)}{ax+b}$$

Answer 2

Why, you just have to differentiate away all three constants to end up with a third degree equation: $$\left({f\cdot f''\over(f')^2}\right)'=0$$