On series solutions of y ″ − f ( x ) y = 0 and applications

Pappur N Shivakumar,Yang Zhang

doi:10.1186/1687-1847-2013-47

Abstract

In this paper, we develop explicit expressions for the Taylor series coefficients in the formal Taylor series solution of the second-order linear differential equation for a given arbitrary function in terms of initial conditions. As applications, we apply our results to , and Airy’s equation and give explicit formulas for the Taylor coefficients of , which is a long-standing question.

Highlights

The classical second-order linear differential equation y – f (x)y = ( . )has been the subject of an innumerable amount of papers and of many classical mathematicians
We present a constructive approach which yields explicit expressions for Taylor series solutions in terms of initial conditions
Our method can be used for evaluating precisely and very up to any number of terms of the series as well as for establishing convergence depending on f (x) and its derivatives. This method is new and novel and has a number of unexpected applications including the relationship between the zeros of a Taylor series and the Taylor coefficients, which is not included to keep the length of the paper to the permitted maximum length

Summary

Introduction

Since no general formula solutions have been established before, one had to use certain indirect ways including numerical approaches (see, for example, [ , ] and [ ]). We present a constructive approach which yields explicit expressions for Taylor series solutions in terms of initial conditions. To the best of our knowledge, this is the first explicit formula giving solutions for differential equations of the type

Results

Conclusion