Here we consider some non-autonomous ordinary differential equations of order n and present some results and theorems on the existence of periodic solutions for them, which are sufficient conditions, section 1. Also we include generalizations of these results to vector differential equations and examinations of some practical examples by numerical simulation, section 2. For some special cases that extendibility of the solutions can be verified and under other suitable conditions, we show that the frequency of the periodic solutions can be arbitrary small†.