Exact analytic solutions of the Abel, Emden-Fowler and generalized Emden-Fowler nonlinear ODEs
type: Item
instance of: scholarly article
title: Exact analytic solutions of the Abel, Emden-Fowler and generalized Emden-Fowler nonlinear ODEs
zbMATH Open document ID: 1114.34002
author: D. E. Panayotounakos
author: Dimitrios Kravvaritis
published in: Nonlinear Analysis. Real World Applications
publication date: 2007-01-16T00:00:00Z
review text: Eight particular so far unsolvable classical second order nonlinear ordinary differential equations (ODEs) in mathematical physics and nonlinear mechanics (the Rayleigh, Emden's, simplified Blasius, Duffing, Langmuir's, van der Pol, Thomas-Fermi and Kidder's equations, respectively) are reduced by several admissible functional transformations to an equivalent Abel's equation of the second kind of the normal form \(yy' - y = f(x)\). A mathematical technique leading to the construction of the exact analytic solutions concerning the above type of Abel equation is provided. The presented method can also be used for the Emden-Fowler equations \(y''_{xx} = A x^n y ^m\) and the generalized Emden-Fowler equations \(y''_{xx} = A x^n y^m (y'_x)^{\ell}\) because there exist admissible functional transformations [\textit{A. D. Polyanin, V. F. Zaitsev}, Handbook of Exact Solutions for Ordinary Differential Equations, (Boca Raton), FL: CRC Press, ( 1995; Zbl 0855.34001)]) that can reduce these equations to Abel's equation of the second kind of the normal form.
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 5116300
zbMATH Keywords: Abel's equations
zbMATH Keywords: Emden-Fowler's equations
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2005.03.025
OpenAlex ID: W2007422115
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DOI: 10.1016/J.NONRWA.2005.03.025
description: scientific article; zbMATH DE number 5116300