This paper introduces a different approach to obtain the exact solution of the relative equations of motion of a deputy (follower) satellite with respect to a chief (leader) satellite that both rotate about central body (Earth) in elliptic orbits by using Laplace transformation. Moreover, the paper will take the perturbation due to the oblateness of the Earth into consideration and simulate this problem with numerical example showing the effect of the perturbation on the Keplerian motion. The solution of such equations in this work is represented in terms of the eccentricity of the chief orbit and its true anomaly as the independent variable.