Construction of a universal ordinary differential equation $C^{\infty}$ of order 3

TL;DR

This paper constructs a smooth third-order universal ordinary differential equation capable of approximating any continuous function on a real segment with arbitrary accuracy, demonstrating a significant advance in differential equations and approximation theory.

Contribution

It introduces a novel $C^{ abla}$-smooth third-order universal ODE that can approximate all continuous functions uniformly.

Findings

The constructed ODE is $C^{ abla}$ smooth of order 3.

Any continuous function can be approximated arbitrarily closely by solutions.

The approach advances the understanding of universal differential equations.

Abstract

A universal ordinary differential equation $C^{\infty}$ of order 3 is constructed here. The equation is universal in the sense that any continuous function on a real segment can be approximated by a solution of this equation with an arbitrary accuracy in uniform norm.