Invariants of third-order ordinary differential equations $y'''=f(x,y,y',y'')$ via fiber preserving transformations

TL;DR

This paper investigates the differential invariants of the general third-order ODEs under fiber preserving transformations, providing explicit conditions for equivalence to canonical forms using Lie's infinitesimal method.

Contribution

It determines all differential invariants and invariant differentiation operators for third-order ODEs under fiber preserving transformations, extending previous quadratic cases.

Findings

All third-order differential invariants identified.

Explicit conditions for equivalence to canonical forms derived.

Applications demonstrating the theoretical results included.

Abstract

Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in the second-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under pseudo-group of fiber preserving equivalence transformations $\bar{x}=\phi(x), \bar{y}=\psi(x,y)$. As a result, all third-order differential invariants of this group and the invariant differentiation operators are determined. This leads to simple necessary explicit conditions for a third-order ODE to be equivalent to the respective canonical form under the considered group of transformations. Applications motivated by the literature are presented.