Question about solutions of nonlinear functional equation $f(x)^2=xf(2x)$

Question

One of the basic nonlinear functional equations is the following one:
$$f(x)^2=xf(2x),\quad x>0\text.$$ I found out that functions $f(x)=2^{1-x}x\exp(cx)$ form the family of solutions of this equation. But do this family cover all possible solutions to this equation? Truly speaking, I have no idea how to answer to this question. Thank you.

Answer 1

Indeed, this family does not cover all solutions, nor even all continuous ones. The general solution is: take any function defined for $x\in[1,2)$ (which means an awful lot of possibilities, mind you!) and continue it both ways, up and down, using the expressions for $f(2x)$ via $f(x)$ and vice versa.