# Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

## TL;DR

This paper applies Lie group theory to second-order delay ordinary differential equations, revealing their symmetry structures and deriving exact solutions, including a traffic flow model example.

## Contribution

It extends symmetry analysis to second-order delay differential equations, identifying their symmetry algebra dimensions and providing a method to find exact solutions.

## Key findings

- Symmetry algebra dimension ranges from 0 to 6 for nonlinear DODSs.
- Linear or linearizable DODSs have infinite-dimensional symmetry algebras.
- Exact solutions for a traffic flow DODS model are obtained.

## Abstract

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0 \leq n \leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.