The Parametric Solution of Underdetermined linear ODEs

TL;DR

This paper introduces a new parametric algorithm for solving underdetermined linear ODEs with arbitrary analytic coefficients, offering an alternative to Groebner basis methods and comparing their strengths.

Contribution

It presents a novel dual algorithm to Euclid's for solving underdetermined linear ODEs, expanding computational tools in differential equations.

Findings

The new algorithm effectively solves underdetermined linear ODEs with arbitrary coefficients.

Comparison shows complementary strengths of Euclid's and the dual algorithm.

Implementation is accessible online for practical use.

Abstract

The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are arbitrary analytic functions in the independent variable. A second conceptual aim is to present an algorithm that is in some sense dual to the fundamental Euclids algorithm, and thus an alternative to the special case of a Groebner basis algorithm as it is used for solving linear ODE-systems. In the paper Euclids algorithm and the new `dual version' are compared and their complementary strengths are analysed on the task of solving underdetermined ODEs. An implementation of the described algorithm is interactively accessible under http://lie.math.brocku.ca/crack/demo.