Existence and uniqueness theorem for ODE: an overview

TL;DR

This paper reviews the historical development and key concepts of existence and uniqueness theorems for ordinary differential equations, emphasizing the role of Lipschitz continuity in solution theory.

Contribution

It provides an overview of the evolution of existence and uniqueness results for ODEs, highlighting differences between linear and nonlinear cases and the importance of Lipschitz conditions.

Findings

Historical overview of ODE solution theory

Differences between linear and nonlinear ODEs

Significance of Lipschitz continuity in uniqueness

Abstract

The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years after the development of the theory of differential equation. In the article, we shall discuss briefly the differences between linear and nonlinear first order ODE in context of existence and uniqueness of solutions. Special emphasis is given on the Lipschitz continuous functions in the discussion.