scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations

2006 ◽  
Vol 462 (2070) ◽  
pp. 1831-1852 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Powerful Method ◽  
Integrals Of Motion ◽  
Third Order ◽  
Non Local ◽  
The Given

We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle–Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation, so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation. In addition to the above, we provide an algorithm to invert the non-local linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.

scholarly journals On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

2005 ◽  
Vol 461 (2060) ◽  
pp. 2451-2477 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Free Particle ◽  
First Integrals ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Integrating Factors

A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 609-629 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Nonlinear Odes ◽  
Second Order Equations ◽  
Integrating Factors

Coupled second-order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the method of deriving a general solution for two coupled second-order nonlinear ODEs through the extended Prelle–Singer procedure. We describe a procedure to obtain integrating factors and the required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above-mentioned method, we introduce a new method of transforming coupled second-order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations

2009 ◽  
Vol 465 (2108) ◽  
pp. 2369-2389 ◽  
Author(s):  
V. K. Chandrasekar ◽  
M. Senthilvelan ◽  
M. Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Straightforward Method ◽  
The Given ◽  
Second Order Equations

Linearization of coupled second-order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper, we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled SNODEs. Our procedure gives several new types of linearizing transformations of both invertible and non-invertible kinds. In both cases, we provide algorithms to derive the general solution of the given SNODE. We illustrate the theory with potentially important examples.

scholarly journals Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

2017 ◽  
Vol 20 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Jianfang Gao
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Differential Algebraic Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Third Order ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

scholarly journals Elastic transformation method for solving ordinary differential equations with variable coefficients

2021 ◽  
Vol 7 (1) ◽  
pp. 1307-1320
Author(s):  
Pengshe Zheng ◽  
◽  
Jing Luo ◽  
Shunchu Li ◽  
Xiaoxu Dong
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transformation Method ◽  
Variable Coefficients ◽  
Nonlinear Ordinary Differential Equations ◽  
Third Order ◽  
First Order ◽  
First Time

<abstract><p>Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.</p></abstract>

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

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