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Found problems: 2

1982 Putnam, A4
Tags: system of differential equatio , de , differential equation , function
Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, $$f(x+L)=f(x),\enspace g(x+L)=g(x).$$

1989 Putnam, B3
Tags: integration , calculus , de , differential equation
Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy $$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define $$\mu_n=\int^\infty_0x^nf(x)dx.$$ $a.$ Express $\mu_n$ in terms of $\mu_0$. $b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.