# 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 K , and let I R be an open interval, where K denotes either ℝ or ℂ. Assume that a 0 , a 1 , , a n : I K and g : I X are given continuous functions. If for every n times continuously differentiable function y : I X satisfying the inequality

a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) ε

for all x I and for a given ε > 0 , there exists an n times continuously differentiable solution y 0 : I X of the differential equation

a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) = 0

such that y ( x ) y 0 ( x ) K ( ε ) for any x I , where K ( ε ) is an expression of ε with lim ε 0 K ( ε ) = 0 , 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 y ( x ) = y ( x ) . It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation y ( x ) = λ y ( x ) (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

p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = 0 , (1)

for which x = 0 is an ordinary point, has the general solution y h : ( ρ 0 , ρ 0 ) C , where ρ 0 is a constant with 0 < ρ 0 and the coefficients p , q , r : ( ρ 0 , ρ 0 ) C are analytic at 0 and have power series expansions

p ( x ) = m = 0 p m x m , q ( x ) = m = 0 q m x m and r ( x ) = m = 0 r m x m

for all x ( ρ 0 , ρ 0 ) . Since x = 0 is an ordinary point of (1), we remark that p 0 0 .

### 2 Inhomogeneous differential equation

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

p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m (2)

under the assumption that x = 0 is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence relation

k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m (3)

for any m N 0 . Let ρ 2 be the radius of convergence of power series m = 0 c m x m and let ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by

y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Since x = 0 is an ordinary point, we can substitute m = 0 c m x m for y ( x ) in (2) and use the formal multiplication of power series and consider (3) to get

p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 k = 0 m p m k ( k + 2 ) ( k + 1 ) c k + 2 x m + m = 0 k = 0 m q m k ( k + 1 ) c k + 1 x m + m = 0 k = 0 m r m k c k x m = m = 0 k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] x m = m = 0 a m x m

for all x ( ρ 3 , ρ 3 ) . That is, m = 0 c m x m is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution y : ( ρ 3 , ρ 3 ) C of (2) can be expressed by

y ( x ) = y h ( x ) + m = 0 c m x m ,

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

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

Corollary 2.2Let p ( x ) , q ( x ) , and r ( x ) be polynomials of degree at most d 0 . In particular, let d 0 be the degree of p ( x ) . Assume that the radius of convergence of power series m = 0 a m x m is ρ 1 > 0 and that there exists a sequence { c m } satisfying the recurrence formula

k = m 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m (4)

for any m N 0 , where m 0 = max { 0 , m d } . If the sequence { c m } satisfies the following conditions:

(i) lim m c m 1 / m c m = 0 ,

(ii) there exists a complex numberLsuch that lim m c m / c m 1 = L and p d 0 + L p d 0 1 + + L d 0 1 p 1 + L d 0 p 0 0 ,

then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation (2) can be expressed by

y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where ρ 3 = min { ρ 0 , ρ 1 } and y h ( x ) is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since p d + 1 = p d + 2 = = 0 , q d + 1 = q d + 2 = = 0 and r d + 1 = r d + 2 = = 0 , if we substitute m d + k for k in (4), then we have

a m = k = 0 d [ ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k + ( m d + k + 1 ) c m d + k + 1 q d k + c m d + k r d k ] .

By (i) and (ii), we have

lim sup m | a m | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 × ( p d k + q d k ( m d + k + 2 ) c m d + k + 1 c m d + k + 2 + r d k ( m d + k + 2 ) ( m d + k + 1 ) c m d + k c m d + k + 1 c m d + k + 1 c m d + k + 2 ) | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | k = d d 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | ( m d 0 + 2 ) ( m d 0 + 1 ) c m d 0 + 2 ( p d 0 + L p d 0 1 + + L d 0 p 0 ) | 1 / m = lim sup m | ( p d 0 + L p d 0 1 + + L d 0 p 0 ) ( m d 0 + 2 ) ( m d 0 + 1 ) | 1 / m × ( | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ) ( m d 0 + 2 ) / m = lim sup m | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ,

which implies that the radius of convergence of the power series m = 0 c m x m is ρ 1 . The rest of this corollary immediately follows from Theorem 2.1. □

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

Corollary 2.3Let ρ 3 be a distance between the origin 0 and the closest one among singular points of q ( z ) , r ( z ) , or m = 0 a m z m in a complex variablez. If there exists a sequence { c m } satisfying the recurrence relation

( m + 2 ) ( m + 1 ) c m + 2 + k = 0 m [ ( k + 1 ) c k + 1 q m k + c k r m k ] = a m (5)

for any m N 0 , then every solution y : ( ρ 3 , ρ 3 ) C of the linear inhomogeneous differential equation

y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m (6)

can be expressed by

y ( x ) = y h ( x ) + m = 0 c m x m

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the linear homogeneous differential equation (1) with p ( x ) 1 .

Proof If we put p 0 = 1 and p i = 0 for each i N , then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that m = 0 c m x m 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 y 0 ( x ) of (6) in a form of power series in x whose radius of convergence is at least ρ 3 . Moreover, since m = 0 c m x m is a solution of (6), it can be expressed as a sum of both y 0 ( x ) and a solution of the homogeneous equation (1) with p ( x ) 1 . Hence, the radius of convergence of m = 0 c m x m is at least ρ 3 .

Now, every solution y : ( ρ 3 , ρ 3 ) C of (6) can be expressed by

y ( x ) = y h ( x ) + m = 0 c m x m ,

where y h ( x ) is a solution of the linear differential equation (1) with p ( x ) 1 . □

### 3 Approximate differential equation

In this section, let ρ 1 > 0 be a constant. We denote by C the set of all functions y : ( ρ 1 , ρ 1 ) C with the following properties:

(a) y ( x ) is expressible by a power series m = 0 b m x m whose radius of convergence is at least ρ 1 ;

(b) There exists a constant K 0 such that m = 0 | a m x m | K | m = 0 a m x m | for any x ( ρ 1 , ρ 1 ) , where

a m = k = 0 m [ ( k + 2 ) ( k + 1 ) b k + 2 p m k + ( k + 1 ) b k + 1 q m k + b k r m k ]

for all m N 0 and p 0 0 .

Lemma 3.1Given a sequence { a m } , let { c m } be a sequence satisfying the recurrence formula (3) for all m N 0 . If p 0 0 and n 2 , then c n is a linear combination of a 0 , a 1 , , a n 2 , c 0 , and c 1 .

Proof We apply induction on n. Since p 0 0 , if we set m = 0 in (3), then

c 2 = 1 2 p 0 a 0 r 0 2 p 0 c 0 q 0 2 p 0 c 1 ,

i.e., c 2 is a linear combination of a 0 , c 0 , and c 1 . Assume now that n is an integer not less than 2 and c i is a linear combination of a 0 , , a i 2 , c 0 , c 1 for all i { 2 , 3 , , n } , namely,

c i = α i 0 a 0 + α i 1 a 1 + + α i i 2 a i 2 + β i c 0 + γ i c 1 ,

where α i 0 , , α i i 2 , β i , γ i are complex numbers. If we replace m in (3) with n 1 , then

a n 1 = 2 c 2 p n 1 + c 1 q n 1 + c 0 r n 1 + 6 c 3 p n 2 + 2 c 2 q n 2 + c 1 r n 2 + + n ( n 1 ) c n p 1 + ( n 1 ) c n 1 q 1 + c n 2 r 1 + ( n + 1 ) n c n + 1 p 0 + n c n q 0 + c n 1 r 0 = ( n + 1 ) n p 0 c n + 1 + [ n ( n 1 ) p 1 + n q 0 ] c n + + ( 2 p n 1 + 2 q n 2 + r n 3 ) c 2 + ( q n 1 + r n 2 ) c 1 + r n 1 c 0 ,

which implies

c n + 1 = 1 ( n + 1 ) n p 0 a n 1 n ( n 1 ) p 1 + n q 0 ( n + 1 ) n p 0 c n 2 p n 1 + 2 q n 2 + r n 3 ( n + 1 ) n p 0 c 2 q n 1 + r n 2 ( n + 1 ) n p 0 c 1 r n 1 ( n + 1 ) n p 0 c 0 = α n + 1 0 a 0 + α n + 1 1 a 1 + + α n + 1 n 1 a n 1 + β n + 1 c 0 + γ n + 1 c 1 ,

where α n + 1 0 , , α n + 1 n 1 , β n + 1 , γ n + 1 are complex numbers. That is, c n + 1 is a linear combination of a 0 , a 1 , , a n 1 , c 0 , c 1 , 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 x = 0 is an ordinary point of (1), we remark that p 0 0 .

Theorem 3.2Let { c m } be a sequence of complex numbers satisfying the recurrence relation (3) for all m N 0 , where (b) is referred for the value of a m , and let ρ 2 be the radius of convergence of the power series m = 0 c m x m . Define ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } , where ( ρ 0 , ρ 0 ) is the domain of the general solution to (1). Assume that y : ( ρ 1 , ρ 1 ) C is an arbitrary function belonging to C and satisfying the differential inequality

| p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) | ε (7)

for all x ( ρ 3 , ρ 3 ) and for some ε > 0 . Let α n 0 , α n 1 , , α n n 2 , β n , γ n be the complex numbers satisfying

c n = α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 + β n c 0 + γ n c 1 (8)

for any integer n 2 . If there exists a constant C > 0 such that

| α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 | C | a n | (9)

for all integers n 2 , then there exists a solution y h : ( ρ 3 , ρ 3 ) C of the linear homogeneous differential equation (1) such that

| y ( x ) y h ( x ) | C K ε

for all x ( ρ 3 , ρ 3 ) , whereKis the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with m = 0 b m x m instead of m = 0 c m x m , we have

p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m (10)

for all x ( ρ 3 , ρ 3 ) . In view of (b), there exists a constant K 0 such that

m = 0 | a m x m | K | m = 0 a m x m | (11)

for all x ( ρ 1 , ρ 1 ) .

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

m = 0 | a m x m | K | m = 0 a m x m | K ε

for any x ( ρ 3 , ρ 3 ) . (That is, the radius of convergence of power series m = 0 a m x m is at least ρ 3 .)

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

y ( x ) = y h ( x ) + n = 0 c n x n (12)

for all x ( ρ 3 , ρ 3 ) , where y h ( x ) is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the c n can be expressed by a linear combination of the form (8) for each integer n 2 .

Since n = 0 c n x n is a particular solution of (2), if we set c 0 = c 1 = 0 , then it follows from (8), (9), and (12) that

| y ( x ) y h ( x ) | n = 0 | c n x n | C K ε

for all x ( ρ 3 , ρ 3 ) . □

### 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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