General Solutions of Some Complex Third-order Differential Equations

Rong Liao; Zhibo Huang

doi:doi:10.11648/j.ajam.20200806.14

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General Solutions of Some Complex Third-order Differential Equations

Rong Liao, Zhibo Huang

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)

Received: 10 November 2020 Accepted: 26 November 2020 Published: 8 December 2020

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Abstract

According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs with entire coefficient in the neighborhood of z₀.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.ajam.20200806.14
Page(s)	319-326
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Ordinary Differential Equation, Local Series Method, Linearly Independent, Meromorphic General Solution

References

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[2]	Bank, S. B. Some results on analytic and meromorphic solutions of algebraic differential equations, Advances in Math., 15 (1975): 41–62.
[3]	Chiang, Y. M. and Halburd R. G., On the meromorphic solutions of an equation of Hayman, J.Math.Anal.Appl., 281 (2) (2003), 663–677.
[4]	Chen, W., Wang, Q. and Yuan, W., Meromorphic solutions of two certain types of nonlinear differential equations, Rocky Mountain J. Math., 50 (2) (2020), 479– 497.
[5]	Dilip Chandra, P., Jayuanta, R. and Kapil, R., On the growth of solutions of some non-linear complex differential equations, Korean J. Math., 28 (2) (2020), 295–309.
[6]	Gol’dberg, A. A. On single valued solution of first order differential equations (in Russian), Ukrain. Mat. Zh., 8, (1956), 254–261.
[7]	Halburd, R. and Korhonen, R., Growth of meromorphic solutions of delay differential equations, Proc. Amer. Math. Soc., 145 (6) (2017), 2513–2526.
[8]	Halburd, R. and Wang, J., All admissible meromorphic solutions of Hayman’s equation, Int. Math. Res. Not., 2015 (18) (2015), 8890–8902.
[9]	Hayman, W. K., The growth of solutions of algebraic differential equations, Mat. Appl., 7 (2) (1996), 67–73.
[10]	Hayman, W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.
[11]	Heittokangas, J., Ishizaki, K., Laine, I. andTohge, K., Complex oscillation and nonnscillation results, Trans. Amer. Math. Soc., 372 (9) (2019), 6161–6182.
[12]	Herold, H. Differentialglechungen im Komplexen, Vandenhoeck Ruprecht, Göttngen, 1975.
[13]	Laine, I., Nevanlinna theory and complex differental equations, de Gruyter, Berlin, 1993.
[14]	Long, J., Heittokangas, J., Ye, Z., On the relationship between the lower order of coefficients and the growth of solutions of differential equations, J. Math. Anal. Appl., 444 (1) (2016), 153–166.
[15]	Lü, F., Lü, W., Li, C. and Xu, J., Growth and uniqueness related to complex differential and difference equations, Results Math., 74 (1) (2019), 1–18.
[16]	Matsuda, M. First order algebraic differential equations, Lecture Notes in Math.,804, Springer, Berln, 1980.
[17]	Mokhon’ko, A. Z., and Kolyasa, L. I., Some properties of meromorphic solutions of linear differential equation with meromorphic coefficients, Mat. Stud., 52 (2) (2019), 166–172.
[18]	Zhang, R. R. and Huang, Z. B., Entire solutions of delay differential equations of Malmquist type, J. Appl. Anal. Comput., 10 (5) (2020), 1720–1740.

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Rong Liao, Zhibo Huang. (2020). General Solutions of Some Complex Third-order Differential Equations. American Journal of Applied Mathematics, 8(6), 319-326. https://doi.org/10.11648/j.ajam.20200806.14

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Rong Liao; Zhibo Huang. General Solutions of Some Complex Third-order Differential Equations. Am. J. Appl. Math. 2020, 8(6), 319-326. doi: 10.11648/j.ajam.20200806.14

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AMA Style

Rong Liao, Zhibo Huang. General Solutions of Some Complex Third-order Differential Equations. Am J Appl Math. 2020;8(6):319-326. doi: 10.11648/j.ajam.20200806.14

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@article{10.11648/j.ajam.20200806.14,
  author = {Rong Liao and Zhibo Huang},
  title = {General Solutions of Some Complex Third-order Differential Equations},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {319-326},
  doi = {10.11648/j.ajam.20200806.14},
  url = {https://doi.org/10.11648/j.ajam.20200806.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.14},
  abstract = {According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation  with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs  with entire coefficient  in the neighborhood of z0.},
 year = {2020}
}

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TY  - JOUR
T1  - General Solutions of Some Complex Third-order Differential Equations
AU  - Rong Liao
AU  - Zhibo Huang
Y1  - 2020/12/08
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.14
DO  - 10.11648/j.ajam.20200806.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 319
EP  - 326
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.14
AB  - According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation  with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs  with entire coefficient  in the neighborhood of z0.
VL  - 8
IS  - 6
ER  -

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Author Information

Rong Liao

School of Mathematical Sciences, South China Normal University, Guangzhou, China; Lihu Experimental School of Shenzhen University Town, Shenzhen, China
Zhibo Huang

School of Mathematical Sciences, South China Normal University, Guangzhou, China

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APA Style

Rong Liao, Zhibo Huang. (2020). General Solutions of Some Complex Third-order Differential Equations. American Journal of Applied Mathematics, 8(6), 319-326. https://doi.org/10.11648/j.ajam.20200806.14

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ACS Style

Rong Liao; Zhibo Huang. General Solutions of Some Complex Third-order Differential Equations. Am. J. Appl. Math. 2020, 8(6), 319-326. doi: 10.11648/j.ajam.20200806.14

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AMA Style

Rong Liao, Zhibo Huang. General Solutions of Some Complex Third-order Differential Equations. Am J Appl Math. 2020;8(6):319-326. doi: 10.11648/j.ajam.20200806.14

Copy | Download

@article{10.11648/j.ajam.20200806.14,
  author = {Rong Liao and Zhibo Huang},
  title = {General Solutions of Some Complex Third-order Differential Equations},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {319-326},
  doi = {10.11648/j.ajam.20200806.14},
  url = {https://doi.org/10.11648/j.ajam.20200806.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.14},
  abstract = {According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation  with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs  with entire coefficient  in the neighborhood of z0.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - General Solutions of Some Complex Third-order Differential Equations
AU  - Rong Liao
AU  - Zhibo Huang
Y1  - 2020/12/08
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.14
DO  - 10.11648/j.ajam.20200806.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 319
EP  - 326
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.14
AB  - According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation  with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs  with entire coefficient  in the neighborhood of z0.
VL  - 8
IS  - 6
ER  -

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