In this paper, we relate with the stability, uniform stability, asymptotic stability, integrability and boundedness of solutions of a system of nonlinear differential equations of second order by the second method of Lyapunov. We obtain new sufficient conditions under which these properties of solutions of the equations considered are verified. Two examples are given to show the applicability of the results obtained and for illustrations. In the particular cases, the behaviors of the paths of that equations are exhibited by MATLAB-Simulink. The results of this paper extend and improve some results that can be found in the literature.