Note on the Application of Divergent Series for Finding a Particular Solution to a Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients

TL;DR

This paper explores how to find particular solutions to nonhomogeneous linear ODEs with constant coefficients using divergent series and summation methods, extending traditional techniques that require convergence.

Contribution

It introduces a novel approach to handle divergent series in the context of differential equations, expanding the applicability of series methods.

Findings

Demonstrates the use of divergent series for particular solutions

Provides a method to sum divergent series in differential equations

Extends existing techniques to non-convergent series cases

Abstract

There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator method are generally known. The latter mentioned method sometimes uses the Maclaurin expansion of an inverse differential operator but only in the case when the obtained series is convergent. The present work deals also with how to find a particular solution if the corresponding infinite series is divergent, using only the terms of that series and the method of summation of divergent series.