A Generalization of the Frobenius Method for Ordinary Differential Equations with Regular Singular Points

Raylin Wu,Jung-Hua Chou

doi:10.3844/jmssp.2005.3.7

Abstract

The conventional Frobenius method for second order differential equations with regular singular points is extended to differential equations of higher and lower orders. The conditions of a point being regular singular are addressed. It is also shown that Cauchy-Euler differential equations are a special case of ordinary differential equations with regular singular points.

Highlights

The series solution for a regular singular point of an ordinary differential equation with non-constant coefficients is typically addressed for second order linear ordinary equations as: d 2y dx 2
A natural question is what happens to ordinary differential equations of other orders? what are the conditions for x0 to be regular singular for linear ordinary differential equations of orders other than 2? This study explores these issues and provides concrete solutions
The Frobenius method and the conditions for a regular singular point of second order differential equations can be deduced by setting m = 2

Summary

INTRODUCTION

The series solution for a regular singular point of an ordinary differential equation with non-constant coefficients is typically addressed for second order linear ordinary equations as:. A singular point x0 is regular singular if both (x − x0 ) p(x) and (x − x0 ) q(x) are analytic at x0. The series solution of equation (1) about x0 can be assumed as:. The description given above can be seen in books on ordinary differential equations, e.g., [1,2] or advanced engineering mathematics, such as references [3,4,5,6]. A natural question is what happens to ordinary differential equations of other orders? What are the conditions for x0 to be regular singular for linear ordinary differential equations of orders other than 2? A natural question is what happens to ordinary differential equations of other orders? what are the conditions for x0 to be regular singular for linear ordinary differential equations of orders other than 2? This study explores these issues and provides concrete solutions

A Simple Starting Point

CONCLUSION