On the growth of solutions of $w^{\left( n\right)}+e^{-z}w^{^{\prime }}+Q\left( z\right) w=0$ and some related extensions

Abstract

In this paper, we show that if $Q\left( z\right) $ is a nonconstant polynomial, then every solution $w\not\equiv 0$ of the differential equation $w^{\left( n\right) }+e^{-z}w^{^{\prime }}+Q\left( z\right) w=0,$ has infinite order and we give an extension of this result. We will also show that if the equation $w^{\left( n\right) }+e^{-z}w^{^{\prime }}+cw=0$, where $c\neq 0$ is a complex constant, possesses a solution $w\not\equiv 0$ of finite order, then $c=-k^{n}$ where $% k$ is a positive integer. In the end, by study more general, we investigate the problem when $\sigma \left( Q\right) =1.$