Study of solutions of certain type of non-linear differential-difference equations

TL;DR

This paper investigates solutions to specific nonlinear differential-difference equations involving polynomials and exponential functions, analyzing their properties and solution behaviors.

Contribution

It provides a detailed analysis of solutions to a class of nonlinear differential-difference equations with polynomial and exponential terms, highlighting conditions for solutions.

Findings

Solutions exist under certain polynomial and exponential conditions.

The equations exhibit unique solution behaviors depending on parameter constraints.

The analysis characterizes the nature of solutions for the given equations.

Abstract

In this paper, we analyze the solutions of the following non-linear differential-difference equations f^n(z) +\omega f^(n-1)f'(z) +p(z)f(z+c) = p_1e^{\alpha}_1z +p_2e^{\alpha}_2z and f^n(z)f'(z) +q(z)e^Q(z)f(z+c) = p_1e^{\alpha}_1z +p_2e^{\alpha}_2z, where n is a positive integer,\omega, p1, p2,{\alpha}1 & {\alpha}2 are non-zero constants satisfying {\alpha}1 not equal to {\alpha}2, {\alpha}1/{\alpha}2 not equal to (n)^+-1, q(z) is a non-vanishing polynomial and Q(z) is a non-constant polynomial.