Abstract
We find the exact forms of meromorphic solutions of the nonlinear differential equations
where q, Q are nonzero polynomials, Q ≡ Const., and p1, p2, α1, α2 are nonzero constants with α1 ≠ α2. Compared with previous results on the equation p(z)f3 + q(z)f″ = − sin α(z) with polynomial coefficients, our results show that the coefficient of the term f(k) perturbed by multiplying an exponential function will affect the structure of its solutions.
Similar content being viewed by others
References
Cartan H. Sur les zé ros des combinaisons liné aires de p fonctions holomorphes donné es. Mathematica Cluj, 1933, 7: 5–31
Gundersen G G, Hayman W K. The strength of Cartan’s version of Nevanlinna theory. Bull London Math Soc, 2004, 36: 433–454
Gundersen G G, Lü W R, Ng T W, et al. Entire solutions of differential equations that are related to trigonometric identities. J Math Anal Appl, 2022, 507: Art 125788
Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin: Walter de Gruyter, 1993
Li P. Entire solutions of certain type of differential equations. J Math Anal Appl, 2008, 344: 253–259
Li P. Entire solutions of certain type of differential equations II. J Math Anal Appl, 2011, 375: 310–319
Liao L W, Yang C C, Zhang J J. On meromorphic solutions of certain type of non-linear differential equations. Ann Acad Sci Fenn Math, 2013, 38: 581–593
Lu X Q, Liao L W, Wang J. On meromorphic solutions of a certain type of nonlinear differential equations. Acta Math Sin, 2017, 33: 1597–1608
Liu H F, Mao Z Q. Meromorphic solutions of certain nonlinear difference equations. Results Math, 2021, 76(102): 1–14
Yang C C, Li P. On the transcendental solutions of a certain type of nonlinear differential equations. Arch Math, 2004, 82: 442–448
Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. New York: Kluwer Academic Publishers, 2003
Zhang J, Liao L W. Meromorphic solutions of one certain type of nonlinear complex differential equation. Complex Var Elliptic Equ, 2022, 67: 2553–2568
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
This work was supported by the NSFC (12261044) and the STP of Education Department of Jiangxi Province of China (GJJ210302).
Rights and permissions
About this article
Cite this article
Liu, H., Mao, Z. The exact meromorphic solutions of some nonlinear differential equations. Acta Math Sci 44, 103–114 (2024). https://doi.org/10.1007/s10473-024-0104-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-024-0104-4