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f'(x)=-3f(x)+6f(2x), find integral of x^nf(x)

Source: Putnam 1989 B3

August 25, 2021
calculusintegrationdedifferential equation

Problem Statement

Let f:[0,∞)→Rf:[0,\infty)\to\mathbb R be differentiable and satisfy f′(x)=−3f(x)+6f(2x)f'(x)=-3f(x)+6f(2x)for x>0x>0. Assume that ∣f(x)∣≤e−x|f(x)|\le e^{-\sqrt x} for x≥0x\ge0. For n∈Nn\in\mathbb N, define μn=∫0∞xnf(x)dx.\mu_n=\int^\infty_0x^nf(x)dx. a.a. Express μn\mu_n in terms of μ0\mu_0. b.b. Prove that the sequence 3nμnn!\frac{3^n\mu_n}{n!} always converges, and the the limit is 00 only if μ0\mu_0.
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