Analytical solution of linear ordinary differential equations by differential transfer matrix method

TL;DR

This paper introduces a novel analytical method using differential transfer matrices for solving homogeneous linear ordinary differential equations of arbitrary order with variable coefficients, simplifying the solution process.

Contribution

The paper presents a new transfer matrix-based approach that reduces high-order differential equations to first-order systems and employs perturbation techniques for exact solutions.

Findings

Method successfully solves equations with arbitrary order and variable coefficients.

Approach recovers classical theorems like Abel-Liouville-Ostogradski.

Validated through multiple analytical examples.

Abstract

We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the $n$th-order differential equation to a system of $n$ linear differential equations with unity order. The full analytical solution is then found by the perturbation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the applicability of the method. We show that the Abel-Liouville-Ostogradski theorem can be easily recovered through this approach.