Integration of differential equations by $\mathcal{C}^{\infty}$-structures

TL;DR

This paper introduces a novel integration method for differential equations using $ ext{C}^ ext{infty}$-structures, enabling solutions for equations lacking traditional symmetries, demonstrated on a Lotka-Volterra model.

Contribution

It presents the concept of $ ext{C}^ extinfty$-structures as a new tool for integrating complex differential equations beyond classical symmetry methods.

Findings

Successfully integrated a Lotka-Volterra model using $ ext{C}^ extinfty$-structures

Solved differential equations without sufficient Lie point symmetries

Extended integrability techniques to broader classes of differential equations

Abstract

Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated with $\mathcal{C}^{\infty}$-structures is used to integrate to a Lotka-Volterra model and several differential equations that lack sufficient Lie point symmetries and cannot be solved using conventional methods.