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ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

Part of: Entire and meromorphic functions, and related topics Differential equations in the complex domain

Published online by Cambridge University Press:  24 September 2014

LIANG-WEN LIAO  and
ZHUAN YE
LIANG-WEN LIAO*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, PR China email [email protected]
ZHUAN YE
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA email [email protected] Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, PR China
*
✉[email protected]
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Abstract

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We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then

$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$
With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

Keywords

nonlinear differential equationsentire solutionsmeromorphic solutionsNevanlinna theorynormal familyPicard values

MSC classification

Secondary: 34M05: Entire and meromorphic solutions 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 3 , December 2014 , pp. 391 - 403
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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