On point transformations of linear equations of maximal symmetry

TL;DR

This paper presents a method to transform linear equations with maximal symmetry into a normal form, enabling explicit solutions and revealing new classes of solvable equations through point transformations.

Contribution

It introduces a general normal form for maximally symmetric linear equations and derives explicit point transformations and solutions, including new solvable classes.

Findings

Explicit point transformation formulas to reduce equations to canonical form

New expressions for general solutions of maximally symmetric equations

Identification of new solvable classes parameterized by arbitrary functions

Abstract

An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of third highest order. As a result, an explicit expression for the point transformation reducing the equation to its canonical form is obtained, and a simple formula for the expression of the general solution in terms of those of the second-order source equation is recovered. New expressions for the general solution are also obtained, as well as a direct proof of the fact that a linear equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by arbitrary functions are also derived, together with simple algebraic expressions for the corresponding general solution.