The Growth of Solutions of Higher Order Differential Equations with Coefficients Having the Same Order

DOI：10.3770/j.issn:2095-2651.2015.04.004

 作者 单位 占燕燕 江西师范大学数学与信息科学学院, 江西 南昌 330022 肖丽鹏 江西师范大学数学与信息科学学院, 江西 南昌 330022

本文研究了某类高阶齐次与非齐次微分方程解的增长性.对于方程$$f^{(k)}+(A_{k-1,1}(z)e^{P_{k-1}(z)}+A_{k-1,2}(z)e^{Q_{k-1}(z)})f^{(k-1)}+\cdots+(A_{0,1}(z)e^{P_{0}(z)}+A_{0,2}(z)e^{Q_{0}(z)})f = F,$$ 其中$F,A_{ji}$是整函数, $P_j(z),Q_j(z)(j=0,1,\ldots,k-1;i=1,2)$是次数为$n(\geq1)$的多项式, $k\geq2,$我们得到如下结果:若$F\equiv0,$方程的所有非零解的增长级为无穷大;若$F\not\equiv0,$方程至多有一个有限级解,其余解的级均满足$\overline{\lambda}(f)=\lambda(f)=\sigma(f)=\infty.$

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 $\overline{\lambda}(f)=\lambda(f)=\sigma(f)=\infty$.