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Research Papers

# New Conservation Laws, Lagrangian Forms, and Exact Solutions of Modified Emden Equation

[+] Author and Article Information
Gülden Gün Polat

Faculty of Science and Letters,
Department of Mathematics,
İstanbul Technical University,
Maslak, İstanbul 34469, Turkey

Teoman Özer

Faculty of Civil Engineering,
Division of Mechanics,
İstanbul Technical University,
Maslak, İstanbul 34469, Turkey
e-mail: tozer@itu.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 1, 2016; final manuscript received November 15, 2016; published online January 19, 2017. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 12(4), 041001 (Jan 19, 2017) Paper No: CND-16-1047; doi: 10.1115/1.4035408 History: Received February 01, 2016; Revised November 15, 2016

## Abstract

This study deals with the determination of Lagrangians, first integrals, and integrating factors of the modified Emden equation by using Jacobi and Prelle–Singer methods based on the Lie symmetries and λ-symmetries. It is shown that the Jacobi method enables us to obtain Jacobi last multipliers by means of the Lie symmetries of the equation. Additionally, via the Lie symmetries of modified Emden equation, we analyze some mathematical connections between λ-symmetries and Prelle–Singer method. New and nontrivial Lagrangian forms, conservation laws, and exact solutions of the equation are presented and discussed.

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## Figures

Fig. 1

Solution plot of Eq. (5.6) for three different integration constants

Fig. 2

Phase portrait of Eq. (5.6) for three different integration constants

Fig. 3

The state-simulation representing position x(t), velocity x˙, and acceleration x¨ of Eq. (5.6)

Fig. 4

Plot of two solutions corresponding to Eq. (5.6) as x2(t) and corresponding for f(x)=g(x)=constant in Eq. (1.2) as x1(t)

Fig. 5

Solution plot of Eq. (5.17) for three different integration constants

Fig. 6

Phase portrait of Eq. (5.17) for three different integration constants

Fig. 7

The state-simulation representing position x(t), velocity x˙, and acceleration x¨ of Eq. (5.17)

Fig. 8

Plot of two solutions corresponding to Eq. (5.17) as x2(t) and corresponding for f(x)=g(x)=constant in Eq. (1.2) as x1(t)

Fig. 9

Solution plot of Eq. (6.5) for three different integration constants

Fig. 10

Phase portrait of Eq. (6.5) for three different integration constants

Fig. 11

The state-simulation representing position x(t), velocity x˙, and acceleration x¨ of Eq. (6.5)

Fig. 12

Plot of two solutions corresponding to Eq. (6.5) as x2(t) and corresponding for f(x)=g(x)=constant in Eq. (1.2) as x1(t)

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