 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.