On the procedure for the series solution of certain general-order homogeneous linear differential equations via the complex integration method

W Robin

doi:10.12988/nade.2014.4712

Abstract

The theory of series solutions for two important classes of the general higherorder linear homogeneous ordinary differential equation is developed ab initio, using an elementary complex integral expression derived and applied in previous papers [10, 11], based on the original work of Herrera [5]. As well as producing general expressions for the recurrence relations for higher-order equations with analytic coefficients or the general-order Fuchs’ equation, the complex integral method is straight-forward to apply as an algorithm on its own. ‘Benchmark’ examples from the general mathematic literature, are presented and a brief discussion of ‘logarithmic’ solutions is included.

Highlights

We continue the project [10, 11] of developing power series and Frobenius series solutions of ordinary differential equations (ODE) using a particular complex integration procedure
As before we find that the technique reduces the solution of the original ODE, through the complex integral transformation [5, 10, 11], to a system of simple equations for the indices of the series coefficients that define the series recurrence relation
Given the basic Frobenius series solution(s), all (!) that is necessary to find further independent particular solutions to an ODE, when required, is for us to re-express the formalism of the previous section in such a manner that we bring it into line with the standard formalism presented in textbooks, in particular references [6] and [9]; but see reference [12]

Summary

Introduction

We continue the project [10, 11] of developing power series and Frobenius series solutions of ordinary differential equations (ODE) using a particular complex integration procedure. Not related to the complex integral method we consider the standard procedure [6, 9] for procuring further solutions and apply it, in part, to the full solution of the fourth-order Bessel-type equation [2] again; the problem [7] from fluid mechanics is mentioned in passing. The paper finishes-off, in section 5, with some general remarks and conclusions

Ordinary Points

Regular Singular Points

Frobenius Series

General Remarks and Conclusions