Coefficient characterization of linear differential equations with maximal symmetries

TL;DR

This paper characterizes linear differential equations with maximal symmetries using simple coefficient conditions, revealing their dependence on only two functions and connecting them to known canonical forms.

Contribution

It provides a new set of conditions for identifying equations with maximal symmetry and simplifies the process of finding their symmetry generators.

Findings

Characterization conditions relate coefficients to maximal symmetry.

Equations can be expressed with only two arbitrary functions.

A simplifying ansatz for symmetry generator determination.

Abstract

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre-Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most general canonical forms of the equation.