Abstract
In this paper an extension, of the method of Krylov-Bogolioubov for a set of purely non-linear autonomous first-order differential equations is presented. As a special case the results of Krylov-Bogolioubov can be obtained in an easy way. The proposed method enables the error of the approximate solution to be determined and can also be used for the determination of the existence and the corresponding stability of a limit-cycle of the system.