A note on linear differential equations with variable coefficients

TL;DR

This paper explores specific homogeneous first-order linear systems with variable coefficients, providing qualitative insights, a method for Floquet coefficients when coefficients are periodic, and new interpretations of coefficient averages.

Contribution

It introduces a simple method for obtaining Floquet coefficients and offers a novel interpretation of coefficient averages in variable coefficient systems.

Findings

Established qualitative properties of solutions.

Developed a new method for Floquet coefficient calculation.

Compared new results with previous studies.

Abstract

In this manuscript, we deal with some particular type of homogeneous first order linear systems with variable coefficients, in which we provide qualitative properties of the solution. When the coefficients of the indeterminate functions are periodic with the same period, $T$, we obtain a simple method so as to obtain the Floquet coefficients. We give a new interpretation of the averages on the interval $(0,T)$ of the matrix of the coefficients. We discuss some examples and made some comparison with previous results.