Characterization of exponential polynomial as solution of certain type of non-linear delay-differential equation

Abstract

In this paper, we have characterized the nature and form of solutions of the following non-linear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ be a linear delay-differential polynomial of $f$; $n$ be positive integers; $q$, $Q$ and $P$ respectively be non-zero, non-constant and any polynomials. Different special cases of our result will accommodate all the results of ([J. Math. Anal. Appl., 452(2017), 1128-1144.], [Mediterr. J. Math., 13(2016), 3015-3027], [Open Math., 18(2020), 1292-1301]). Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occurs and consequently the same are automatically applicable to the previous results.