Abstract
We characterize the complex differential equations of the form
where aj(x) are meromorphic functions in the variable x for j = 0,…,n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Casas-Alvero, E.: Singularities of Plane Curves, London Math. Soc. Lecture Note Ser., Vol. 276, Cambridge University Press, Cambridge, 2000
Cheb-Terrab, E. S., Roche, A. D.: Abel ODE’s: equivalence and integrable classes. Comput. Phys. Comm. 130 (2000), 204–231.
Cheb-Terrab, E. S., Roche, A. D.: An Abel ordinary differential equation class generalizing known integrable classes. Eur. J. Appl. Math., 14, 217–229 (2003)
Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la Conjecture de Dulac, Hermann, Paris, 1992
Giné, J., Grau, M.: Weierstrass integrability of differential equations. Appl. Math. Lett., 23, 523–526 (2010)
Giné, J., Llibre, J.: On the integrable rational Abel differential equations. Z. Angew. Math. Phys., 61, 33–39 (2010)
Kamke, E.: Differentialgleichungen, Lösungsmethoden und Lösungen, Mathematik und ihre Anwendungen, Vol. 18, Akademische Verlagsgesellschaft Becker und Erler Kom-Ges., Leipzig, 1943
Llibre, J., Valls, C.: Generalized Weierstrass integrability of the Abel differential equations. Mediterr. J. Math., 10, 1747–1758 (2013)
Llibre, J., Valls, C.: Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse. Acta Mathematica Sinica, English Series, 30, 453–466 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013
Rights and permissions
About this article
Cite this article
Llibre, J., Valls, C. Weierstrass Integrability of Complex Differential Equations. Acta. Math. Sin.-English Ser. 37, 1497–1506 (2021). https://doi.org/10.1007/s10114-021-8374-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-021-8374-4