Solving Linear Differential Equations: A Novel Approach

TL;DR

This paper introduces a new method for solving linear differential equations by linking solutions to monomials and revealing underlying symmetries, applicable to many special functions and complex systems.

Contribution

It presents a novel approach that connects solutions to monomials, clarifies symmetries, and applies to a broad class of differential equations and many-particle systems.

Findings

Unified structure of solutions for Hermite, Laguerre, Bessel, and hypergeometric equations

Effective for developing approximate solutions

Applicable to many-particle interacting systems

Abstract

We explicate a procedure to solve general linear differential equations, which connects the desired solutions to monomials x^m of an appropriate degree m. In the process the underlying symmetry of the equations under study, as well as that of the solutions are made transparent. We demonstrate the efficacy of the method by showing the common structure of the solution space of a wide variety of differential equations viz. Hermite, Laguerre, Jocobi, Bessel and hypergeometric etc. We also illustrate the use of the procedure to develop approximate solutions, as well as in finding solutions of many particle interacting systems.