Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 581-593

ON MEROMORPHIC SOLUTIONS OF CERTAIN TYPE OF NON-LINEAR DIFFERENTIAL EQUATIONS

Liang-Wen Liao, Chung-Chun Yang and Jian-Jun Zhang

Nanjing University, Department of Mathematics
Nanjing, 210093, P.R. China; maliao 'at' nju.edu.cn

China University of Petroleum, Department of Mathematics
Huangdao, P.R. China; chungchun.yang 'at' gmail.com

Jiangsu Institute of Education, Mathematics and Information Technology School
Nanjing, 210013, P.R. China; zhangjianjun1982 'at' 163.com

Abstract. We consider meromorphic solutions of non-linear differential equation of the form

fⁿ + Q_d(z,f) = p₁(z)e^\alpha_1(z) + p₂(z)e^\alpha_2(z),

where Q_d(z,f) is a differential polynomial in f of degree d \le n - 2 with rational functions as its coefficients, p₁, p₂ are rational functions and \alpha_1, \alpha_2 are polynomials. More precisely and mainly we have shown the conditions concerning \alpha_1'/\alpha_2' that will ensure the existence and forms of the possible meromorphic solutions of the above equation. These results have extended and improved some known results obtained most recently.

2010 Mathematics Subject Classification: Primary 30D35; Secondary 33E30, 30D30.

Key words: Nevanlinna's value distribution theory, nonlinear differential equation, differential polynomial.

Reference to this article: L.-W. Liao, C.-C. Yang and J.-J. Zhang: On meromorphic solutions of certain type of non-linear differential equations. Ann. Acad. Sci. Fenn. Math. 38 (2013), 581-593.

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doi:10.5186/aasfm.2013.3840