Disintegrated order one differential equations and algebraic general solutions

TL;DR

This paper extends Rosenlicht's results to characterize when order one differential equations with rational functions are orthogonal to constants, linking algebraic solutions and orthogonality concepts.

Contribution

It provides a necessary and sufficient condition for orthogonality to constants in rational order one differential equations, expanding the theoretical understanding.

Findings

Established a criterion for orthogonality to constants in rational differential equations

Connected algebraic general solutions with weak orthogonality to constants

Included joint discussion with Nagloo and Vo in the appendix

Abstract

We generalize results of Rosenlicht to give a necessary and sufficient condition for when order one differential equations of the form $D(x) = f(x)$ where $f$ is a rational function is orthogonal to the constants. Following the main results of the paper, we also explain the connection between algebraic general solutions and weak orthogonality to the constants; a portion of this discussion is given in the appendix, written jointly with Joel Nagloo and Ngoc Thieu Vo.