Question about solutions of nonlinear functional equation

One of the basic nonlinear functional equations is the following one:
$$(f(x))^{2}=xf(2x),\ \ \ x>0.$$
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

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.