Exact Differential Equations and Harmonic Functions

TL;DR

This paper explores the relationship between exact differential equations and harmonic functions, providing conditions for harmonic solutions and analyzing orthogonal trajectories using Cauchy-Riemann equations.

Contribution

It establishes necessary and sufficient conditions for exact equations to have harmonic solutions and links harmonic functions with orthogonal trajectories via Cauchy-Riemann equations.

Findings

Conditions for harmonic solutions of exact differential equations.

Cauchy-Riemann equations ensure orthogonality of trajectories.

Non-vanishing derivatives are key for orthogonal trajectories.

Abstract

In this work, we investigate some connections between exact differential equations and harmonic functions and in particular, we obtain necessary and sufficient conditions for which exact equations admit harmonic solutions. As an application, we consider the orthogonal trajectories of harmonic functions, and among other results we obtain that the Cauchy-Riemann equations and the non-vanishing of the first partial derivatives are sufficient for any two curves to be orthogonal trajectories of each other. All curves throughout the work are restricted to the xy-plane.