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On the Non-existence of Meromorphic Solutions of Certain Types of Non-linear Differential Equations

On the Non-existence of Meromorphic Solutions of Certain Types of Non-linear Differential Equations In this paper, we study meromorphic solutions of non-linear differential equations of the form $$f^n+P_d(f)=p_1e^{\alpha _1(z)}+p_2e^{\alpha _2(z)}$$ f n + P d ( f ) = p 1 e α 1 ( z ) + p 2 e α 2 ( z ) , where $$\alpha _1,\alpha _2$$ α 1 , α 2 are polynomials of degree $$k(\ge 1)$$ k ( ≥ 1 ) , $$p_1$$ p 1 , $$p_2$$ p 2 are small meromorphic functions of $$e^{z^k}$$ e z k , $$P_\mathrm{d}(f)$$ P d ( f ) is a differential polynomial in f of degree d with small meromorphic functions of f as its coefficients. Some sufficient conditions on the non-existence of meromorphic solutions of such equations are provided. Our results complement some previous results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On the Non-existence of Meromorphic Solutions of Certain Types of Non-linear Differential Equations

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00272-1
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study meromorphic solutions of non-linear differential equations of the form $$f^n+P_d(f)=p_1e^{\alpha _1(z)}+p_2e^{\alpha _2(z)}$$ f n + P d ( f ) = p 1 e α 1 ( z ) + p 2 e α 2 ( z ) , where $$\alpha _1,\alpha _2$$ α 1 , α 2 are polynomials of degree $$k(\ge 1)$$ k ( ≥ 1 ) , $$p_1$$ p 1 , $$p_2$$ p 2 are small meromorphic functions of $$e^{z^k}$$ e z k , $$P_\mathrm{d}(f)$$ P d ( f ) is a differential polynomial in f of degree d with small meromorphic functions of f as its coefficients. Some sufficient conditions on the non-existence of meromorphic solutions of such equations are provided. Our results complement some previous results.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jun 6, 2019

References