Mathematics

CLC number: O 174.5

On Entire Solutions of Two Certain Types of Non-Linear Differential-Difference Equations

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009 , Jiangsu, China

^† To whom correspondence should be addressed. E-mail: alexehuang@sina.com

Received: 20 January 2022

Abstract

In this paper, we mainly investigate entire solutions of the following two non-linear differential-difference equations $f^n(z)+\omega f^{n-\mathrm{1}}(z)f\text{'}(z)+f^{\left(k\right)}(z+c)=p_{\mathrm{1}}{\mathrm{e}}^{{\alpha }_{\mathrm{1}}z}+p_{\mathrm{2}}{\mathrm{e}}^{{\alpha }_{\mathrm{2}}z},n\ge \mathrm{5}$ and $f^n(z)+\omega f^{n-\mathrm{1}}(z)$ $f\text{'}(z)+q(z)f^{\left(k\right)}(z+c){\mathrm{e}}^{Q\left(z\right)}=p_{\mathrm{1}}{\mathrm{e}}^{{\alpha }_{\mathrm{1}}z}+p_{\mathrm{2}}{\mathrm{e}}^{{\alpha }_{\mathrm{2}}z},n\ge \mathrm{4}$, where $k\ge \mathrm{0}$ is an integer, $c,\omega ,p_{\mathrm{1}},p_{\mathrm{2}},{\alpha }_{\mathrm{1}},{\alpha }_{\mathrm{2}}$ are non-zero constants, $q(z)$ is a non-vanishing polynomial and $Q(z)$ is a non-constant polynomial. Under some additional hypotheses, we analyze the existence and expressions of transcendental entire solutions of the above equations.