Abstract: Most of the motion equations of particles in relative motion are nonlinear, and their exact explicit solutions are difficult to obtain. Based on the method of homotopy perturbation analysis, the approximate explicit solutions of a class of nonlinear differential equations of particle motion in relative motion are studied. First, the homotopy equation of the system is constructed, then the natural frequency of the free vibration of the system is derived by combining the Lindstedt–Poincare method and the initial conditions of the system, and the approximate displacement response of the system is solved. The correctness of the analytical solution is verified by numerical simulation, which provides a new solution method for solving the motion equations of relative moving particles.