Functional equations of the form $f(g(x))=(f(x))^p$, where $g(x)$ is known, is called Bottcher equation. Generally, we have only asymptotic formula for the solution $f(x)$ under certain conditions. In one of my study, I come across the following problem,

How to find a nontrivial (i.e., nonconstant) function $f(x)\in C^\infty(0 ,+\infty)$ satisfying the following functional equation $$f(\frac{2x^3+a}{3x^2})=f(x)^2,$$ where $a>0$ is a constant.

Or can you tell me finding a explicit form for $f(x)$ is impossible?