The Growth of Solutions of Higher Order Differential Equations with Coefficients Having the Same Order
Received:June 17, 2014  Revised:May 04, 2015
Key Words: order of growth   hyper-order   exponent of convergence of zero sequence   differential equation  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11301232; 11171119) and the Natural Science Foundation of Jiangxi Province (Grant No.20132BAB211009).
Author NameAffiliation
Yanyan ZHAN College of Mathematics and Information Science, Jiangxi Normal University, Jiangxi 330022, P. R. China 
Lipeng XIAO College of Mathematics and Information Science, Jiangxi Normal University, Jiangxi 330022, P. R. China 
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Abstract:
      In this paper, we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations. It is proved that under some conditions for entire functions $F,A_{ji}$ and polynomials $P_j(z),Q_j(z)~(j=0,1,\ldots,k-1;i=1,2)$ with degree $n\geq 1$, the equation $f^{(k)}+(A_{k-1,1}(z)e^{P_{k-1}(z)}+A_{k-1,2}(z)e^{Q_{k-1}(z)})f^{(k-1)}+\cdots+(A_{0,1}(z)e^{P_{0}(z)}+A_{0,2}(z)e^{Q_{0}(z)})f= F,$ where $k\geq2$, satisfies the properties: When $F\equiv 0$, all the non-zero solutions are of infinite order; when $F\not\equiv 0$, there exists at most one exceptional solution $f_0$ with finite order, and all other solutions satisfy $\overline{\lambda}(f)=\lambda(f)=\sigma(f)=\infty$.
Citation:
DOI:10.3770/j.issn:2095-2651.2015.04.004
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