I'm struggling to solve this problem: Let $f$ be a continuous function defined on $[0,+\infty[$, derivable in every point such that $$ f'(x)=2f(2x)-f(x), \ \ \ \ \ \ \forall x>0, $$ and $$ M_n=\int_{0}^{\infty}x^nf(x)dx<+\infty, \ \ \ \ \ \ \forall n\in\Bbb{N}. $$ Prova that exists a non zero function that satisfies this conditions. For which sequences of real numbers $\{a_n\}$ it's true that $a_n=M_n, \ \ \forall n\in\Bbb{N}$.
All I managed to get is that $$ M_n=n!\prod_{i=1}^{n}\frac{2^i}{2^i-1}M_0=n!\prod_{i=1}^{n}\frac{2^i}{2^i-1}\int_{0}^{\infty}f(x)dx. $$
