A new closed-form expression for the solution of ODEs in a ring of distributions and its connection with the matrix algebra

TL;DR

This paper introduces a novel closed-form solution for homogeneous linear ODEs using a generalized Volterra composition, linking it to infinite matrix inversion and algebraic structures.

Contribution

It extends previous work by connecting the solution to inverting an infinite matrix within a specific algebraic framework.

Findings

New closed-form expression for ODEs based on matrix inversion

Connection established between algebraic structures and differential equation solutions

Extension of Volterra composition to infinite-dimensional settings

Abstract

A new expression for solving homogeneous linear ODEs based on a generalization of the Volterra composition was recently introduced. In this work, we extend such an expression, showing that it corresponds to inverting an infinite matrix. This is done by studying a particular subring and connecting it with a subalgebra of infinite matrices.