Cite the paper
Aparna Saha & Benoy Talukdar (2016). Lagrangian Representations and Solutions of Modified Emden-Type Equations. Mechanics, Materials Science & Engineering, Vol 5. doi:10.13140/RG.2.1.2683.0323
Authors: Aparna Saha, Benoy Talukdar
ABSTRACT. We derive two novel methods to construct solutions of the physically important modified Emden-type equations (MEEs). Following the basic philosophy implied in the Lagrange’s method of variation of parameters, we make use of a trivial particular solution of the problem to construct expressions for the nontrivial general solutions. Secondly, we judiciously adapt the factorization method of differential operators to present solutions of certain oscillator equations obtained by adding a linear term to the MEEs. We provide Lagrangian and Hamiltonian formulations of these equations in order to look for another useful theoretical model for solving MEEs.
Keywords: variation of parameters, factorization method, modified Emden-type equations, solutions, Lagrangian representation
 E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
 S. Chandrasekher, An Introduction to the study of Stellar Structure (Dover Pub., New York, 1957).
 P. G. Leach, J. Math. Phys. Vol. 26, 2310 (1985).
 V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the general solution for the modified Emden-type equation, J. Phys. A : Math. Theor. 40, 4717 (2007).
 M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279, 215 (1983).
 R. Iacono 2008, Comment on ‘On the general solution for the modified Emden-type equation, J. Phys. A : Math. Theor. 41, 068001(2008).
 V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Reply to ‘Comment on “On the general solution for the modified Emden type equation, J. Phys. A : Math. Theor. 41, 0680002(2008).
 V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Liénard-type nonlinear oscillator, Phys. Rev. E 72, 066203(2005).
 H. Goldstein Classical Mechanics, 7th reprint (Narosa Pub. House, New Delhi, India, 1998).
 H. C. Rosu and O. Cornejo-Pérez, Supersymmetric pairing of kinks for polynomial nonlinearities, Phys. Rev. E 71, 046607(2005).
 O. Cornejo-Pérez and H. C. Rosu, Nonlinear second order ODE’s: Factorizations and particular solutions, Prog. Theor. Phys. 114, 533 (2005).
 J. F. Cariñena, M. F. Rañada and F. Santander, Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability, J. Math. Phys. 46, 062703 (2005).
 P. Caldirola Nuovo. Cim. 18. 393 (1941); E. Kanai Prog. Theor. Phys. 20,440(1948).
 A. Sommerfeld, 1932 Z. Phys. 78,19 (1932).
 S. Esposito, Majorana solution of the Thomas-Fermi equation, arXiv : physics/0111167v1(2001) [physics. atom-ph]
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