Conditional Linearizability Criteria for Scalar Fourth Order Semi-Linear Ordinary Differential Equations

TL;DR

This paper develops geometric criteria for linearizing scalar fourth order semi-linear ordinary differential equations, extending methods from lower orders and highlighting the complexity of higher order cases.

Contribution

It introduces new geometric linearizability criteria for fourth order semi-linear ODEs, extending previous second and third order methods, and discusses potential for higher order extension.

Findings

Derived linearizability criteria for fourth order semi-linear ODEs.

Extended geometric methods to higher order equations.

Highlighted limitations of standard Lie approaches for these equations.

Abstract

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating the third order equation. This yields criteria for linearizability of a class of fourth order semi-linear ordinary differential equations, which have not been discussed in the literature previously. It is shown that the procedure can be extended to higher order. Though the results for the higher orders are complicated, they are doable by algebraic computing. The standard Lie approach, as developed at present does not seem to be amenable to giving results that can be handled even by algebraic computing.