Completely general closed-form solution for coordinates and partial derivative of the two-body problem

Completely general closed-form solutions are given for both the rectangular coordinates of the two-body problem and their partial derivatives with respect to their initial values. The solutions are valid for the circular, elliptic, parabolic, hyperbolic, and rectilinear orbits of the attractive force, and also for the hyper- bolic and rectilinear orbits of the repulsive force. The complete generality and relative simplicity of the solutions make them valuable for many practical applications of the two-body problem.