Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy′=(αxm+k−1+βxm−k−1)y+γx2m−2k−1,y′=dydx where a, b, c ∈ ℂ, m, k ∈ ℤ and α=a(2m+k)β=b(2m−k),γ=−(a2mx4k+cx2k+b2m). This family is very important because...

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Autores:: Rodríguez-Contreras, Jorge
Acosta-Humánez, Primitivo B.
Reyes-Linero, Alberto

Tipo de recurso:

Fecha de publicación:: 2019

Institución:: Universidad Simón Bolívar

Repositorio:: Repositorio Digital USB

Idioma:: eng

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dc.title.eng.fl_str_mv	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
title	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
spellingShingle	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1 Critical points Integrability Gegenbauer equation Legendre equation Liénard equation
title_short	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
title_full	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
title_fullStr	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
title_full_unstemmed	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
title_sort	Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
dc.creator.fl_str_mv	Rodríguez-Contreras, Jorge Acosta-Humánez, Primitivo B. Reyes-Linero, Alberto
dc.contributor.author.none.fl_str_mv	Rodríguez-Contreras, Jorge Acosta-Humánez, Primitivo B. Reyes-Linero, Alberto
dc.subject.eng.fl_str_mv	Critical points Integrability Gegenbauer equation Legendre equation Liénard equation
topic	Critical points Integrability Gegenbauer equation Legendre equation Liénard equation
description	The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy′=(αxm+k−1+βxm−k−1)y+γx2m−2k−1,y′=dydx where a, b, c ∈ ℂ, m, k ∈ ℤ and α=a(2m+k)β=b(2m−k),γ=−(a2mx4k+cx2k+b2m). This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations.
publishDate	2019
dc.date.accessioned.none.fl_str_mv	2019-11-12T21:04:12Z
dc.date.available.none.fl_str_mv	2019-11-12T21:04:12Z
dc.date.issued.none.fl_str_mv	2019
dc.type.eng.fl_str_mv	article
dc.type.coar.fl_str_mv	http://purl.org/coar/resource_type/c_6501
dc.identifier.issn.none.fl_str_mv	23915455
dc.identifier.uri.none.fl_str_mv	https://hdl.handle.net/20.500.12442/4331
identifier_str_mv	23915455
url	https://hdl.handle.net/20.500.12442/4331
dc.language.iso.eng.fl_str_mv	eng
language	eng
dc.rights.*.fl_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 Internacional
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dc.rights.uri.*.fl_str_mv	http://creativecommons.org/licenses/by-nc-nd/4.0/
rights_invalid_str_mv	Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ http://purl.org/coar/access_right/c_abf2
dc.publisher.eng.fl_str_mv	De Gruyter
dc.source.eng.fl_str_mv	Open Mathematics Vol. 17, Issue 1 (2019)
institution	Universidad Simón Bolívar
dc.source.uri.eng.fl_str_mv	DOI: https://doi.org/10.1515/math-2019-0100
dc.source.bibliographicCitation.eng.fl_str_mv	Guckenheimer J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag New York, 1983. Guckenheimer J., Hoffman K., Weckesser W., The forced Van der Pol equation I: The slow flow and its bifurcations, SIAM J. Applied Dynamical Systems, 2003, 2, 1–35. Kapitaniak T., Chaos for Engineers: Theory, Applications and Control, Springer, Berlin, Germany, 1998. Nagumo J., Arimoto S., Yoshizawa S., An active pulse transmission line simulating nerve axon, Proc. IRE, 1962, 50, 2061– 2070. Nemytskii V.V., Stepanov V.V., Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960. Perko L., Differential equations and Dynamical systems, Third Edition, Springer-Verlag New York, Inc, 2001. Van der Pol B., Van der Mark J., Frequency demultiplication, Nature, 1927, 120, 363–364. Acosta-Humánez P.B., La Teoría de Morales-Ramis y el Algoritmode Kovacic, LecturasMatemáticas, Volumen Especial, 2006, 21–56. Acosta-Humánez P.B., Pantazi Ch., Darboux Integrals for Schrodinger Planar Vector Fields via Darboux Transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 2012, 8, 043, arXiv:1111.0120. Morales-Ruiz J., Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser, Basel, 1999. Van der Put M., Singer M., Galois Theory in Linear Differential Equations, Springer-Verlag New York, 2003. Weil J.A., Constant et polynómes de Darboux en algèbre différentielle: applications aux systèmes différentiels linéaires, Doctoral thesis, 1995. Acosta-Humánez P.B., Galoisian Approach to Supersymmetric Quantum Mechanics, PhD Thesis, Barcelona, 2009, arXiv:0906.3532. Acosta-Humánez P.B., Blázquez-Sanz D., Non-integrability of some Hamiltonians with rational potentials, Discrete Contin. Dyn. Syst. Ser. B, 2008, 10, 265–293. Acosta-Humánez P., Morales-Ruiz J., Weil J.-A., Galoisian approach to integrability of Schrödinger Equation, Rep. Math. Phys., 2011, 67(3), 305–374. Polyanin A.D., Zaitsev V.F., Handbook of exact solutions for ordinary differential equations, Second Edition, Chapman and Hall, Boca Raton, 2003. Acosta-Humánez P.B., Lázaro J.T., Morales-Ruiz J.J., Pantazi Ch., On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Discrete Contin. Dyn. Syst., 2015, 35(5), 1767–1800. Drumortier F., Llibre J., Artés J.C., Qualitative theory of planar differential systems, Springer-Verlag Berlin Heidelberg, 2006.
dc.source.bibliographicCitation.spa.fl_str_mv	Acosta-Humánez P.B., Perez J., Teoría de Galois diferencial: una aproximación, Matemáticas: Enseãnza Universitaria, 2007, 91–102. Acosta-Humánez P.B., Perez J., Una introducción teoría de Galois diferencial, Boletín deMatemáticas Nueva Serie, 2004, 11, 138–149.
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spelling	Rodríguez-Contreras, Jorge73696779-f908-4efd-bd6a-8a8f93f60f4cAcosta-Humánez, Primitivo B.99e6835c-988c-4178-9496-9266aab5f6c1Reyes-Linero, Albertof9429574-f2c4-4944-8987-8f6cd54649502019-11-12T21:04:12Z2019-11-12T21:04:12Z201923915455https://hdl.handle.net/20.500.12442/4331The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy′=(αxm+k−1+βxm−k−1)y+γx2m−2k−1,y′=dydx where a, b, c ∈ ℂ, m, k ∈ ℤ and α=a(2m+k)β=b(2m−k),γ=−(a2mx4k+cx2k+b2m). This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations.engDe GruyterAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/http://purl.org/coar/access_right/c_abf2Open MathematicsVol. 17, Issue 1 (2019)DOI: https://doi.org/10.1515/math-2019-0100Guckenheimer J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag New York, 1983.Guckenheimer J., Hoffman K., Weckesser W., The forced Van der Pol equation I: The slow flow and its bifurcations, SIAM J. Applied Dynamical Systems, 2003, 2, 1–35.Kapitaniak T., Chaos for Engineers: Theory, Applications and Control, Springer, Berlin, Germany, 1998.Nagumo J., Arimoto S., Yoshizawa S., An active pulse transmission line simulating nerve axon, Proc. IRE, 1962, 50, 2061– 2070.Nemytskii V.V., Stepanov V.V., Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960.Perko L., Differential equations and Dynamical systems, Third Edition, Springer-Verlag New York, Inc, 2001.Van der Pol B., Van der Mark J., Frequency demultiplication, Nature, 1927, 120, 363–364.Acosta-Humánez P.B., La Teoría de Morales-Ramis y el Algoritmode Kovacic, LecturasMatemáticas, Volumen Especial, 2006, 21–56.Acosta-Humánez P.B., Pantazi Ch., Darboux Integrals for Schrodinger Planar Vector Fields via Darboux Transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 2012, 8, 043, arXiv:1111.0120.Morales-Ruiz J., Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser, Basel, 1999.Van der Put M., Singer M., Galois Theory in Linear Differential Equations, Springer-Verlag New York, 2003.Weil J.A., Constant et polynómes de Darboux en algèbre différentielle: applications aux systèmes différentiels linéaires, Doctoral thesis, 1995.Acosta-Humánez P.B., Galoisian Approach to Supersymmetric Quantum Mechanics, PhD Thesis, Barcelona, 2009, arXiv:0906.3532.Acosta-Humánez P.B., Blázquez-Sanz D., Non-integrability of some Hamiltonians with rational potentials, Discrete Contin. Dyn. Syst. Ser. B, 2008, 10, 265–293.Acosta-Humánez P., Morales-Ruiz J., Weil J.-A., Galoisian approach to integrability of Schrödinger Equation, Rep. Math. Phys., 2011, 67(3), 305–374.Polyanin A.D., Zaitsev V.F., Handbook of exact solutions for ordinary differential equations, Second Edition, Chapman and Hall, Boca Raton, 2003.Acosta-Humánez P.B., Lázaro J.T., Morales-Ruiz J.J., Pantazi Ch., On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Discrete Contin. Dyn. Syst., 2015, 35(5), 1767–1800.Drumortier F., Llibre J., Artés J.C., Qualitative theory of planar differential systems, Springer-Verlag Berlin Heidelberg, 2006.Acosta-Humánez P.B., Perez J., Teoría de Galois diferencial: una aproximación, Matemáticas: Enseãnza Universitaria, 2007, 91–102.Acosta-Humánez P.B., Perez J., Una introducción teoría de Galois diferencial, Boletín deMatemáticas Nueva Serie, 2004, 11, 138–149.Critical pointsIntegrabilityGegenbauer equationLegendre equationLiénard equationAlgebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1articlehttp://purl.org/coar/resource_type/c_6501ORIGINALAlgebraic_and_qualitative_remarks_about_the_family.pdfAlgebraic_and_qualitative_remarks_about_the_family.pdfPDFapplication/pdf629638https://bonga.unisimon.edu.co/bitstreams/904f782b-e830-43c1-bd4b-b04c6eb72a5a/download9bb5cf6ee8358fb07eea709bdca8ae34MD54CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://bonga.unisimon.edu.co/bitstreams/b8b623de-5031-4cc5-8814-2530651838ed/download4460e5956bc1d1639be9ae6146a50347MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-8381https://bonga.unisimon.edu.co/bitstreams/ffc38321-3224-4932-83fb-6438a1d5c031/download733bec43a0bf5ade4d97db708e29b185MD53TEXTAlgebraic_and_qualitative_remarks_about_the_family.pdf.txtAlgebraic_and_qualitative_remarks_about_the_family.pdf.txtExtracted texttext/plain42427https://bonga.unisimon.edu.co/bitstreams/fe06eec4-8f47-4488-8832-7eb5eae0fadb/downloadc4220b82a14a829815bc05d3e423df19MD55THUMBNAILAlgebraic_and_qualitative_remarks_about_the_family.pdf.jpgAlgebraic_and_qualitative_remarks_about_the_family.pdf.jpgGenerated Thumbnailimage/jpeg1771https://bonga.unisimon.edu.co/bitstreams/30d7ca95-5ee7-4287-968a-6786511c4626/downloadeba38ff5376c48befe7212b285336dc3MD5620.500.12442/4331oai:bonga.unisimon.edu.co:20.500.12442/43312024-08-14 21:51:44.806http://creativecommons.org/licenses/by-nc-nd/4.0/Attribution-NonCommercial-NoDerivatives 4.0 Internacionalopen.accesshttps://bonga.unisimon.edu.coRepositorio Digital Universidad Simón Bolívarrepositorio.digital@unisimon.edu.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

Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

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