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Putnam 1941 B1

Source: Putnam 1941

February 23, 2022
Putnamconicshyperbola

Problem Statement

A particle (x,y)(x,y) moves so that its angular velocities about (1,0)(1,0) and (−1,0)(-1,0) are equal in magnitude but opposite in sign. Prove that y(x2+y2+1)  dx=x(x2+y2−1)  dy,y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy, and verify that this is the differential equation of the family of rectangular hyperbolas passing through (1,0)(1,0) and (−1,0)(-1,0) and having the origin as center.
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