Olympiad problems

1989 Putnam, B3

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 $\mu_0$.

1987 Putnam, A3

Tags: differential equation

For all real $x$, the real-valued function $y=f(x)$ satisfies \[ y''-2y'+y=2e^x. \] (a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain. (b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.

1942 Putnam, B3

Tags: partial differential equations , differential equation

Given $x=\phi(u,v)$ and $y=\psi(u,v)$, where $ \phi$ and $\psi$ are solutions of the partial differential equation $$(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$ By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to $$(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$ Integrate $(2)$ and show how this effects in general the solution of $(1)$. What other solutions does $(1)$ possess?

2020 IMC, 5

Tags: function , calculus , differential equation , inequality

Find all twice continuously differentiable functions $f: \mathbb{R} \to (0, \infty)$ satisfying $f''(x)f(x) \ge 2f'(x)^2.$

1958 November Putnam, A3

Tags: differential equation

Under the assumption that the following set of relations has a unique solution for $u(t),$ determine it. $$ \frac{d u(t) }{dt} = u(t) + \int_{0}^{t} u(s)\, ds, \;\;\; u(0)=1.$$