Free solution of Hill equations

The planar Hill equations describe small displacements about a circular Kepler orbit in a frame rotating at angular frequency $n$. With $x$ radial and $y$ along the orbit, their unforced solution isDirect differentiation gives $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$.

The $A,B$ terms describe epicyclic motion with tangential amplitude twice the radial amplitude. The constant radial offset $C$ causes an accumulating along-track displacement through Keplerian shear, while $D$ is an initial along-track offset. This separation makes the solution useful for comparing nearly neighbouring orbits and for identifying which motion requires additional encounter forcing.

The oscillatory integration constants determine the orbital eccentricity vector, the constant radial offset determines the semi-major axis, and the constant along-track offset fixes the mean longitude. The radial and tangential epicycle amplitudes have ratio $1:2$.