Putnam 1941 B1

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