Research

# Power series method and approximate linear differential equations of second order

Soon-Mo Jung1 and Hamdullah Şevli2*

Author Affiliations

1 Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea

2 Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University, Uskudar, Istanbul, 34672, Turkey

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Advances in Difference Equations 2013, 2013:76  doi:10.1186/1687-1847-2013-76

 Received: 14 December 2012 Accepted: 4 March 2013 Published: 26 March 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC: 34A05, 39B82, 26D10, 34A40.

##### Keywords:
power series method; approximate linear differential equation; simple harmonic oscillator equation; Hyers-Ulam stability; approximation

### 1 Introduction

Let X be a normed space over a scalar field , and let be an open interval, where denotes either ℝ or ℂ. Assume that and are given continuous functions. If for every n times continuously differentiable function satisfying the inequality

for all and for a given , there exists an n times continuously differentiable solution of the differential equation

such that for any , where is an expression of ε with , then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [1-8].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9,10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation . It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation (see [12] and also [13-15]).

Moreover, Miura et al.[16] investigated the Hyers-Ulam stability of an nth-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [17-25]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [26-34]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form

(1)

for which is an ordinary point, has the general solution , where is a constant with and the coefficients are analytic at 0 and have power series expansions

for all . Since is an ordinary point of (1), we remark that .

### 2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form

(2)

under the assumption that is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1Assume that the radius of convergence of power seriesisand that there exists a sequencesatisfying the recurrence relation

(3)

for any. Letbe the radius of convergence of power seriesand let, whereis the domain of the general solution to (1). Then every solutionof the linear inhomogeneous differential equation (2) can be expressed by

for all, whereis a solution of the linear homogeneous differential equation (1).

Proof Since is an ordinary point, we can substitute for in (2) and use the formal multiplication of power series and consider (3) to get

for all . That is, is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution of (2) can be expressed by

where is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions , , and of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2Let, , andbe polynomials of degree at most. In particular, letbe the degree of. Assume that the radius of convergence of power seriesisand that there exists a sequencesatisfying the recurrence formula

(4)

for any, where. If the sequencesatisfies the following conditions:

(i) ,

(ii) there exists a complex numberLsuch thatand,

then every solutionof the linear inhomogeneous differential equation (2) can be expressed by

for all, whereandis a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since , and , if we substitute for k in (4), then we have

By (i) and (ii), we have

which implies that the radius of convergence of the power series is . The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that in (1). For this case, we obtain the following corollary.

Corollary 2.3Letbe a distance between the origin 0 and the closest one among singular points of, , orin a complex variablez. If there exists a sequencesatisfying the recurrence relation

(5)

for any, then every solutionof the linear inhomogeneous differential equation

(6)

can be expressed by

for all, whereis a solution of the linear homogeneous differential equation (1) with.

Proof If we put and for each , then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution of (6) in a form of power series in x whose radius of convergence is at least . Moreover, since is a solution of (6), it can be expressed as a sum of both and a solution of the homogeneous equation (1) with . Hence, the radius of convergence of is at least .

Now, every solution of (6) can be expressed by

where is a solution of the linear differential equation (1) with . □

### 3 Approximate differential equation

In this section, let be a constant. We denote by the set of all functions with the following properties:

(a) is expressible by a power series whose radius of convergence is at least ;

(b) There exists a constant such that for any , where

for all and .

Lemma 3.1Given a sequence, letbe a sequence satisfying the recurrence formula (3) for all. Ifand, thenis a linear combination of, , and.

Proof We apply induction on n. Since , if we set in (3), then

i.e., is a linear combination of , , and . Assume now that n is an integer not less than 2 and is a linear combination of , , for all , namely,

where , , are complex numbers. If we replace m in (3) with , then

which implies

where , , are complex numbers. That is, is a linear combination of , , , which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since is an ordinary point of (1), we remark that .

Theorem 3.2Letbe a sequence of complex numbers satisfying the recurrence relation (3) for all, where (b) is referred for the value of, and letbe the radius of convergence of the power series. Define, whereis the domain of the general solution to (1). Assume thatis an arbitrary function belonging toand satisfying the differential inequality

(7)

for alland for some. Let, , be the complex numbers satisfying

(8)

for any integer. If there exists a constantsuch that

(9)

for all integers, then there exists a solutionof the linear homogeneous differential equation (1) such that

for all, whereKis the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with instead of , we have

(10)

for all . In view of (b), there exists a constant such that

(11)

for all .

Moreover, by using (7), (10), and (11), we get

for any . (That is, the radius of convergence of power series is at least .)

According to Theorem 2.1 and (10), can be written as

(12)

for all , where is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the can be expressed by a linear combination of the form (8) for each integer .

Since is a particular solution of (2), if we set , then it follows from (8), (9), and (12) that

for all . □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors declare that this paper is their original paper. All authors read and approved the final manuscript.

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

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