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General Solutions of Some Complex Third-order Differential Equations

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), 2020. 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.
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[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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    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

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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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    T1  - General Solutions of Some Complex Third-order Differential Equations
    AU  - Rong Liao
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    Y1  - 2020/12/08
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    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    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
  • School of Mathematical Sciences, South China Normal University, Guangzhou, China

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

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