Series solution of second-order linear homogeneous ordinary differential equations via complex integration

W Robin

doi:10.12988/imf.2014.4492

Abstract

A method is presented, with standard examples, based on an elementary complex integral expression, for developing, in particular, series solutions for second-order linear homogeneous ordinary differential equations. Straightforward to apply, the method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method eliminates the need to manipulate power series and balance powers, which is a characteristic of the usual approach. The method originated with Herrera [3], but was applied to the solution of certain classes of nonlinear ordinary differential equations by him. Mathematics Subject Classification: 30B10, 30E20 34A25, 34A30

Highlights

One of the commonest means of seeking a solution of a linear homogeneous ordinary differential equation (ODE) is to attempt to find an infinite series solution.This is a well understood process, but can still be a messy business when attempting to develop the recurrence relation after substituting the assumed form of the infinite series into the ODE [7, 8, 9]
In this paper we will introduce a method for finding power series solutions to ODE by direct integration in the complex plane. (All contour integrals that occur below are assumed evaluated in the counter-clockwise direction.) The basis of this integration method, due to Herrera [3], is the elementary result [4] that, if n is an integer
We will consider the problem to be solved on obtaining the recurrence relation for the series solutions; the equations involved being well-known, the process of solving the recurrence relations is widely available in many textbooks and monographs

Summary

Introduction

One of the commonest means of seeking a solution of a linear homogeneous ordinary differential equation (ODE) is to attempt to find an infinite series solution This is a well understood process, but can still be a messy business when attempting to develop the recurrence relation after substituting the assumed form of the infinite series into the ODE [7, 8, 9].

Derivation of the Basic Formula

The Solution of Some Standard Second-Order Equations

Conclusions and Discussion