Qualitative investigation of the solutions to differential equations. (Application of the skew-symmetric differential forms)

TL;DR

This paper discusses how skew-symmetric differential forms, a mathematical tool originally developed by Cartan, can be used to analyze the consistency and integrability conditions of solutions to differential equations.

Contribution

It highlights the role of skew-symmetric differential forms in the qualitative analysis of differential equations, emphasizing their utility in determining solution properties and consistency conditions.

Findings

Skew-symmetric differential forms help identify solution consistency conditions.

The approach clarifies the role of conjugacy in differential equations.

It demonstrates the application of Cartan's analysis to modern differential equation problems.

Abstract

The presented method of investigating the solutions to differential equations is not new. Such an approach was developed by Cartan in his analysis of the integrability of differential equations. Here this approach is outlined to demonstrate the role of skew-symmetric differential forms. The role of skew-symmetric differential forms in a qualitative investigation of the solutions to differential equations is conditioned by the fact that the mathematical apparatus of these forms enables one to determine the conditions of consistency for various elements of differential equations or for the system of differential equations. This enables one, for example, to define the consistency of the partial derivatives in the partial differential equations, the consistency of the differential equations in the system of differential equations, the conjugacy of the function derivatives and of the initial data derivatives in ordinary differential equations and so on. The functional properties of the solutions to differential equations are just depend on whether or not the conjugacy conditions are satisfied.