Let $f$ be a function on $[0,\infty)$, differentiable and satisfying  $$f'(x)=-3f(x)+6f(2x)$$  for $x>0.$ Assume that $|f(x)|\leqslant e^{-\sqrt{x}}$ for $x\geqslant0$ (so that $f(x)$ tends rapidly to 0 as  $x$ increases). For $n$ a non-negative integer, define  $$\mu_n=\int_0^\infty x^nf(\:x\:)\:dx$$  (sometimes called the nth moment of $f).$  a. Express $\mu_n$ in terms of $\mu_0.$ Given that $n = 1$, please provide the value of m + n where the expression is in the format \(\frac{m}{n}\).

Let's think step by step and output the final answer within \boxed{}.
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