On a Side Condition for Wronskian-Involving Differential Equations

TL;DR

This paper explores a special side condition linking differential geometry and differential equations, introducing a method to solve a class of nonlinear ODEs involving Wronskians, with applications to space curve equations.

Contribution

It proposes a novel side condition for Wronskian-involving differential equations and provides a solution method, including new solutions for the Tzitzeica curve equation.

Findings

A new method for solving Wronskian-involving differential equations.

Application to Tzitzeica curve yields new solutions.

Establishes connections between geometry and differential equations.

Abstract

The purpose of this paper is to make a few connections among specific concepts occurring in differential geometry and the theory of differential equations with the aim of identifying an intriguing class of undetermined nonlinear ordinary differential equations whose solutions satisfy a specific side condition consisting in a homogeneous third-order linear ordinary differential equation. A method for solving this class of Wronskian-involving differential equations based on the proposed side condition is presented. The Tzitzeica curve equation arising in the theory of space curves is considered as an example, and new closed and integral-form solutions for this equation are obtained.