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
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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). |
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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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