The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions
of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).
We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous
ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations.
In the present paper the method of the equivalent differential-operator equation has been applied in the study of the existence
and asymptotic representation of periodic solutions of autonomous systems of the form
wheree1 ande2 are real constants ande1 ande2 are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case
. They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case
either. Our method of proof consists essentially of constructing a Dulac function (see  and ) and using the conception of Duff's rotated vector field (see , , , , and ).