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International Journal of Differential Equations
Volume 2013 (2013), Article ID 929286, 9 pages
http://dx.doi.org/10.1155/2013/929286
Research Article

On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations

1Department of Mathematics, University of Aegean, 83200 Samos, Greece
2Department of Computer Engineering and Informatics, University of Patras Rio, 26504 Patras, Greece
3Computer Technology Institute and Press “Diophantus” Rio, 26504 Patras, Greece
4Department of Business Administration, University of Patras Rio, 26504 Patras, Greece
5Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

Received 9 March 2013; Accepted 30 May 2013

Academic Editor: Elena Braverman

Copyright © 2013 Panayotis E. Nastou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the properties of a general class of differential equations described by with a positive integer and , with , real functions of . For , these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values of connects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values of , we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology.