Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two‐Dimensional Autonomous Systems

M P Markakis

doi:10.1155/2010/436860

Abstract

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab‐1) equations). Based on the results of a previous work, concerning a closed‐form solution of a general (Ab‐1) equation, and introducing an arbitrary function, exact one‐parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two‐dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

Highlights

Autonomous equations, as it is well known, often arise in mechanics, physics, and chemical engineering since a considerable number of problems are governed by weakly or strongly nonlinear equations of this kind
See for instance the work of Kooij and Christopher for the integrability of planar polynomial systems by means of algebraic invariant curves, as well as the works of Denman and Van Horssen, where approximate invariants are obtained via perturbation techniques
By considering autonomous equations of a polynomial structure for dy/dx up to the second degree, with coefficients of a not necessarily polynomial form for y see, e.g., the Langmuir equation 18, in the present work we investigate analytically this generalized polynomial form, aiming at the construction of proper techniques, capable of removing the difficulties arising in the derivation of exact solutions

Summary

Introduction

Autonomous equations, as it is well known, often arise in mechanics, physics, and chemical engineering since a considerable number of problems are governed by weakly or strongly nonlinear equations of this kind. The classic transformation yx q y , usually applied to autonomous nonlinear ordinary differential equations of the second order results in Abel equations of the second kind see, e.g., 20, Section 2.2.3 , which in general cannot be solved analytically, except in special cases, most of which accept only parametric solutions see 20, Sections 1.3.1–1.3.4 a parametric solution for y, yx is derived as regards the considered autonomous equation.

Reduction of a General Autonomous Equation

Construction of Exact Solutions

Discussion and Conclusion