On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations
We investigate the properties of a general class of differential equations described by dy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t), with k>1 a positive integer and fi(t), 0≤i≤k+1, with fi(t), real functions of t. For k=2, these equations reduce to the class of Abel different...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2013/929286 |
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| author | Panayotis E. Nastou Paul Spirakis Yannis C. Stamatiou Apostolos Tsiakalos |
| author_facet | Panayotis E. Nastou Paul Spirakis Yannis C. Stamatiou Apostolos Tsiakalos |
| author_sort | Panayotis E. Nastou |
| collection | DOAJ |
| description | We investigate the properties of a general class of differential equations described by dy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t), with k>1 a positive integer and fi(t), 0≤i≤k+1, with fi(t), real functions of t. For k=2, these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for k>2 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 k 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 k, 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. |
| format | Article |
| id | doaj-art-b324601e12ee48c9acc7cd32d63646a0 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-b324601e12ee48c9acc7cd32d63646a02025-02-03T01:31:11ZengWileyInternational Journal of Differential Equations1687-96431687-96512013-01-01201310.1155/2013/929286929286On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential EquationsPanayotis E. Nastou0Paul Spirakis1Yannis C. Stamatiou2Apostolos Tsiakalos3Department of Mathematics, University of Aegean, 83200 Samos, GreeceDepartment of Computer Engineering and Informatics, University of Patras Rio, 26504 Patras, GreeceComputer Technology Institute and Press “Diophantus” Rio, 26504 Patras, GreeceDepartment of Mathematics, University of Ioannina, 45110 Ioannina, GreeceWe investigate the properties of a general class of differential equations described by dy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t), with k>1 a positive integer and fi(t), 0≤i≤k+1, with fi(t), real functions of t. For k=2, these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for k>2 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 k 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 k, 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.http://dx.doi.org/10.1155/2013/929286 |
| spellingShingle | Panayotis E. Nastou Paul Spirakis Yannis C. Stamatiou Apostolos Tsiakalos On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations International Journal of Differential Equations |
| title | On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations |
| title_full | On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations |
| title_fullStr | On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations |
| title_full_unstemmed | On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations |
| title_short | On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations |
| title_sort | on the derivation of a closed form expression for the solutions of a subclass of generalized abel differential equations |
| url | http://dx.doi.org/10.1155/2013/929286 |
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