Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

TL;DR

This paper analyzes the symmetry properties of linear first-order delay ordinary differential systems, identifying classes with additional symmetries and using these to construct exact solutions.

Contribution

It classifies linear first-order delay ODEs with extra symmetries beyond linearity and demonstrates how to use these symmetries for solution construction.

Findings

Linear DODSs have infinite-dimensional symmetry groups.

Identified classes with additional symmetries beyond linear superposition.

Constructed exact solutions using symmetry reduction.

Abstract

A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional {sub}algebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone. We present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.