Integration of an arbitrary linear ODE

TL;DR

This paper introduces the multex integral operator, a novel mathematical tool that enables explicit integration of arbitrary linear ODEs with variable coefficients, addressing a key gap in classical ODE theory.

Contribution

The paper presents the multex integral operator, generalizing the exponential primitive, to explicitly solve any linear ODE with variable coefficients.

Findings

The multex operator provides explicit solutions to linear ODEs.

It generalizes the exponential primitive operator.

Addresses a fundamental gap in ODE theory.

Abstract

The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses this shortcoming in the basic theory. We introduce the multex integral operator, generalizing to several input functions the standard exponential primitive operator that is inverse to the logarithmic derivative. The multex operator serves to integrate in explicit form an arbitrary linear ordinary differential equation.