This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 30

1941 Putnam, B3
Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$.
differential equation algebra

1983 Putnam, B3
Assume that the differential equation $$y'''+p(x)y''+q(x)y'+r(x)y=0$$has solutions $y_1(x)$, $y_2(x)$, $y_3(x)$ on the real line such that $$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$for all real $x$. Let $$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y'+Ap(x)y=Br(x).$$
calculus differential equation algebra

1954 Putnam, A3
Prove that if the family of integral curves of the differential equation $$ \frac{dy}{dx} +p(x) y= q(x),$$ where $p(x) q(x) \ne 0$, is cut by the line $x=k$ the tangents at the points of intersection are concurrent.
differential equation algebra calculus

1957 Putnam, B6
The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$-axis between $$x= \frac{3}{2} \pi \;\;\; \text{and} \;\;\; x= \sqrt{\frac{63}{53}} \pi.$$
graphing lines differential equation analytic geometry algebra

1958 November Putnam, A3
Under the assumption that the following set of relations has a unique solution for $u(t),$ determine it. $$ \frac{d u(t) }{dt} = u(t) + \int_{0}^{t} u(s)\, ds, \;\;\; u(0)=1.$$
differential equation algebra