The Integrability and Existence of Periodic Solutions on a First‐Order Nonlinear Differential Equation with a Polynomial Nonlinear Term

Ni Hua,Tian Li-Xin

doi:10.1155/2012/703474

Abstract

This paper deals with a first‐order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

Highlights

IntroductionConsider the following first-order nonlinear differential equation: dx dt n ak t xn−k k0 n ∈ N, n ≥ 2 , 1.1 when n 2, 1.1 becomes Ricatti’s equation, when n 3, 1.1 becomes the following nonlinear Abel type differential equation: dx a t x3 b t x2 ctxdt . dt
Consider the following first-order nonlinear differential equation: dx dt n ak t xn−k k0 n ∈ N, n ≥ 2, 1.1 when n 2, 1.1 becomes Ricatti’s equation, when n 3, 1.1 becomes the following nonlinear Abel type differential equation: dx a t x3 b t x2 ctxdt . dtThe nonlinear Abel type differential equation plays an important role in many physical and technical applications 1–9
Matsuno 10 analyzed a twodimensional dynamical system associated with Abel nonlinear equation

Summary

Introduction

Consider the following first-order nonlinear differential equation: dx dt n ak t xn−k k0 n ∈ N, n ≥ 2 , 1.1 when n 2, 1.1 becomes Ricatti’s equation, when n 3, 1.1 becomes the following nonlinear Abel type differential equation: dx a t x3 b t x2 ctxdt . dt. The nonlinear Abel type differential equation plays an important role in many physical and technical applications 1–9. Reid , Reid and Strobel have obtained superposition rules prescriptions for combining a finite number of known particular solutions in such a way to obtain the general solution to a system of differential equation s without operation of integration for the Abel type equation, involving four or two particular solutions. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. To the best of authors’ knowledge, this is the first paper considering the three periodic solutions of 1.1 , some new results are obtained.

Preliminary Lemmas

Concluding Remarks