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

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2008.0240 ◽

2008 ◽

Vol 465 (2102) ◽

pp. 609-629 ◽

Cited By ~ 7

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2009.0041 ◽

2009 ◽

Vol 465 (2108) ◽

pp. 2369-2389 ◽

Cited By ~ 7

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1648 ◽

2006 ◽

Vol 462 (2070) ◽

pp. 1831-1852 ◽

Cited By ~ 19

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.

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FIRST INTEGRALS OF FIN EQUATIONS FOR STRAIGHT FINS

Modern Physics Letters B ◽

10.1142/s0217984909021697 ◽

2009 ◽

Vol 23 (30) ◽

pp. 3659-3666 ◽

Cited By ~ 3

Author(s):

E. MOMONIAT ◽

C. HARLEY ◽

T. HAYAT

Keyword(s):

Temperature Distribution ◽

Temperature Gradient ◽

Differential Equations ◽

Boundary Conditions ◽

Ordinary Differential Equations ◽

First Integrals ◽

Second Order ◽

Nonlinear Ordinary Differential Equations

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.

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