Analysis of the Differential-Difference Equation $y(x+1/2)-y(x-1/2) = y'(x)$

TL;DR

This paper investigates solution methods for a specific differential-difference equation, establishing conditions for initial functions to ensure unique continuous solutions, thereby advancing understanding of such equations.

Contribution

It introduces solution techniques for the differential-difference equation and provides criteria for initial functions to guarantee unique solutions.

Findings

Identified sufficient conditions for initial functions to produce unique solutions.

Developed solution methods for the differential-difference equation.

Analyzed the existence and uniqueness of solutions under various initial conditions.

Abstract

In this paper we study some solution techniques of differential-difference equation $$ y'(x) = y(x + 1/2)- y(x- 1/2),$$ first without an initial condition and then with some initial function $h$ defined on the unit interval $ [-1/2, 1/2]$. We show some sufficient conditions that an initial function $h$ is admissible, i.e., it yields a unique continuous solution on some symmetric interval about $0$.