# Two problems in the theory of differential equations

## TL;DR

This paper explores the supersymmetry inherent in differential equations when viewed through exterior differential forms and questions the practical use of formal integrability criteria.

## Contribution

It introduces the concept of supersymmetry in differential equations via superalgebra and discusses the gap between theoretical criteria and practical applications.

## Key findings

- Differential equations can be associated with supersymmetry through supervarieties.
- Formal integrability criteria are rarely applied in practice.
- Superalgebra provides a new perspective on classical differential equations.

## Abstract

1) The differential equation considered in terms of exterior differential forms, as \'E.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice?