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

- Soon-Mo Jung
^{1}and - Hamdullah Şevli
^{2}Email author

**2013**:76

https://doi.org/10.1186/1687-1847-2013-76

© Jung and Şevli; licensee Springer. 2013

**Received: **14 December 2012

**Accepted: **4 March 2013

**Published: **26 March 2013

## 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

## 1 Introduction

*X*be a normed space over a scalar field $\mathbb{K}$, and let $I\subset \mathbb{R}$ be an open interval, where $\mathbb{K}$ denotes either ℝ or ℂ. Assume that ${a}_{0},{a}_{1},\dots ,{a}_{n}:I\to \mathbb{K}$ and $g:I\to X$ are given continuous functions. If for every

*n*times continuously differentiable function $y:I\to X$ satisfying the inequality

*n*times continuously differentiable solution ${y}_{0}:I\to X$ of the differential equation

such that $\parallel y(x)-{y}_{0}(x)\parallel \le K(\epsilon )$ for any $x\in I$, where $K(\epsilon )$ is an expression of *ε* with ${lim}_{\epsilon \to 0}K(\epsilon )=0$, 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 ${y}^{\prime}(x)=y(x)$. It was further proved by Takahasi *et al.* that the Hyers-Ulam stability holds for the Banach space valued differential equation ${y}^{\prime}(x)=\lambda y(x)$ (see [12] and also [13–15]).

Moreover, Miura *et al.* [16] investigated the Hyers-Ulam stability of an *n* th-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.

for all $x\in (-{\rho}_{0},{\rho}_{0})$. Since $x=0$ is an ordinary point of (1), we remark that ${p}_{0}\ne 0$.

## 2 Inhomogeneous differential equation

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

**Theorem 2.1**

*Assume that the radius of convergence of power series*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$

*is*${\rho}_{1}>0$

*and that there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence relation*

*for any*$m\in {\mathbb{N}}_{0}$.

*Let*${\rho}_{2}$

*be the radius of convergence of power series*${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$

*and let*${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1},{\rho}_{2}\}$,

*where*$(-{\rho}_{0},{\rho}_{0})$

*is the domain of the general solution to*(1).

*Then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*(2)

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *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 ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ for $y(x)$ in (2) and use the formal multiplication of power series and consider (3) to get

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 2.2**

*Let*$p(x)$, $q(x)$,

*and*$r(x)$

*be polynomials of degree at most*$d\ge 0$.

*In particular*,

*let*${d}_{0}$

*be the degree of*$p(x)$.

*Assume that the radius of convergence of power series*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$

*is*${\rho}_{1}>0$

*and that there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence formula*

*for any*$m\in {\mathbb{N}}_{0}$,

*where*${m}_{0}=max\{0,m-d\}$.

*If the sequence*$\{{c}_{m}\}$

*satisfies the following conditions*:

- (i)
${lim}_{m\to \mathrm{\infty}}{c}_{m-1}/m{c}_{m}=0$,

- (ii)
*there exists a complex number**L**such that*${lim}_{m\to \mathrm{\infty}}{c}_{m}/{c}_{m-1}=L$*and*${p}_{{d}_{0}}+L{p}_{{d}_{0}-1}+\cdots +{L}^{{d}_{0}-1}{p}_{1}+{L}^{{d}_{0}}{p}_{0}\ne 0$,

*then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*(2)

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* ${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1}\}$ *and* ${y}_{h}(x)$ *is a solution of the linear homogeneous differential equation* (1).

*Proof*Let

*m*be any sufficiently large integer. Since ${p}_{d+1}={p}_{d+2}=\cdots =0$, ${q}_{d+1}={q}_{d+2}=\cdots =0$ and ${r}_{d+1}={r}_{d+2}=\cdots =0$, if we substitute $m-d+k$ for

*k*in (4), then we have

which implies that the radius of convergence of the power series ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ is ${\rho}_{1}$. The rest of this corollary immediately follows from Theorem 2.1. □

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

**Corollary 2.3**

*Let*${\rho}_{3}$

*be a distance between the origin*0

*and the closest one among singular points of*$q(z)$, $r(z)$,

*or*${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{z}^{m}$

*in a complex variable*

*z*.

*If there exists a sequence*$\{{c}_{m}\}$

*satisfying the recurrence relation*

*for any*$m\in {\mathbb{N}}_{0}$,

*then every solution*$y:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear inhomogeneous differential equation*

*can be expressed by*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* ${y}_{h}(x)$ *is a solution of the linear homogeneous differential equation* (1) *with* $p(x)\equiv 1$.

*Proof* If we put ${p}_{0}=1$ and ${p}_{i}=0$ for each $i\in \mathbb{N}$, then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ 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 ${\rho}_{3}$. Moreover, since ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ 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 $p(x)\equiv 1$. Hence, the radius of convergence of ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$ is at least ${\rho}_{3}$.

where ${y}_{h}(x)$ is a solution of the linear differential equation (1) with $p(x)\equiv 1$. □

## 3 Approximate differential equation

- (a)
$y(x)$ is expressible by a power series ${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ whose radius of convergence is at least ${\rho}_{1}$;

- (b)There exists a constant $K\ge 0$ such that ${\sum}_{m=0}^{\mathrm{\infty}}|{a}_{m}{x}^{m}|\le K|{\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}|$ for any $x\in (-{\rho}_{1},{\rho}_{1})$, where${a}_{m}=\sum _{k=0}^{m}[(k+2)(k+1){b}_{k+2}{p}_{m-k}+(k+1){b}_{k+1}{q}_{m-k}+{b}_{k}{r}_{m-k}]$

for all $m\in {\mathbb{N}}_{0}$ and ${p}_{0}\ne 0$.

**Lemma 3.1** *Given a sequence* $\{{a}_{m}\}$, *let* $\{{c}_{m}\}$ *be a sequence satisfying the recurrence formula* (3) *for all* $m\in {\mathbb{N}}_{0}$. *If* ${p}_{0}\ne 0$ *and* $n\ge 2$, *then* ${c}_{n}$ *is a linear combination of* ${a}_{0},{a}_{1},\dots ,{a}_{n-2}$, ${c}_{0}$, *and* ${c}_{1}$.

*Proof*We apply induction on

*n*. Since ${p}_{0}\ne 0$, if we set $m=0$ in (3), then

*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 ${a}_{0},\dots ,{a}_{i-2}$, ${c}_{0}$, ${c}_{1}$ for all $i\in \{2,3,\dots ,n\}$, namely,

*m*in (3) with $n-1$, then

where ${\alpha}_{n+1}^{0},\dots ,{\alpha}_{n+1}^{n-1}$, ${\beta}_{n+1}$, ${\gamma}_{n+1}$ are complex numbers. That is, ${c}_{n+1}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n-1}$, ${c}_{0}$, ${c}_{1}$, 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 ${p}_{0}\ne 0$.

**Theorem 3.2**

*Let*$\{{c}_{m}\}$

*be a sequence of complex numbers satisfying the recurrence relation*(3)

*for all*$m\in {\mathbb{N}}_{0}$,

*where*(b)

*is referred for the value of*${a}_{m}$,

*and let*${\rho}_{2}$

*be the radius of convergence of the power series*${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$.

*Define*${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1},{\rho}_{2}\}$,

*where*$(-{\rho}_{0},{\rho}_{0})$

*is the domain of the general solution to*(1).

*Assume that*$y:(-{\rho}_{1},{\rho}_{1})\to \mathbb{C}$

*is an arbitrary function belonging to*$\mathcal{C}$

*and satisfying the differential inequality*

*for all*$x\in (-{\rho}_{3},{\rho}_{3})$

*and for some*$\epsilon >0$.

*Let*${\alpha}_{n}^{0},{\alpha}_{n}^{1},\dots ,{\alpha}_{n}^{n-2}$, ${\beta}_{n}$, ${\gamma}_{n}$

*be the complex numbers satisfying*

*for any integer*$n\ge 2$.

*If there exists a constant*$C>0$

*such that*

*for all integers*$n\ge 2$,

*then there exists a solution*${y}_{h}:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$

*of the linear homogeneous differential equation*(1)

*such that*

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* *K* *is the constant determined in* (b).

*Proof*By the same argument presented in the proof of Theorem 2.1 with ${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ instead of ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$, we have

for all $x\in (-{\rho}_{1},{\rho}_{1})$.

for any $x\in (-{\rho}_{3},{\rho}_{3})$. (That is, the radius of convergence of power series ${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$ is at least ${\rho}_{3}$.)

for all $x\in (-{\rho}_{3},{\rho}_{3})$, 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 $n\ge 2$.

for all $x\in (-{\rho}_{3},{\rho}_{3})$. □

## Declarations

### 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.

## Authors’ Affiliations

## References

- Brillouet-Belluot N, Brzdȩk J, Cieplinski K: On some recent developments in Ulam’s type stability.
*Abstr. Appl. Anal.*2012., 2012: Article ID 716936Google Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge; 2002.View ArticleGoogle Scholar - Hyers DH: On the stability of the linear functional equation.
*Proc. Natl. Acad. Sci. USA*1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Hyers DH, Isac G, Rassias TM:
*Stability of Functional Equations in Several Variables*. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar - Hyers DH, Rassias TM: Approximate homomorphisms.
*Aequ. Math.*1992, 44: 125–153. 10.1007/BF01830975MathSciNetView ArticleGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis*. Springer, New York; 2011.View ArticleGoogle Scholar - Rassias TM: On the stability of the linear mapping in Banach spaces.
*Proc. Am. Math. Soc.*1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar - Ulam SM:
*Problems in Modern Mathematics*. Wiley, New York; 1964.Google Scholar - Obłoza M: Hyers stability of the linear differential equation.
*Rocznik Nauk.-Dydakt. Prace Mat.*1993, 13: 259–270.Google Scholar - Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations.
*Rocznik Nauk.-Dydakt. Prace Mat.*1997, 14: 141–146.Google Scholar - Alsina C, Ger R: On some inequalities and stability results related to the exponential function.
*J. Inequal. Appl.*1998, 2: 373–380.MathSciNetGoogle Scholar - Takahasi S-E, Miura T, Miyajima S:On the Hyers-Ulam stability of the Banach space-valued differential equation ${y}^{\prime}=\lambda y$.
*Bull. Korean Math. Soc.*2002, 39: 309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView ArticleGoogle Scholar - Miura T: On the Hyers-Ulam stability of a differentiable map.
*Sci. Math. Jpn.*2002, 55: 17–24.MathSciNetGoogle Scholar - Miura T, Jung S-M, Takahasi S-E:Hyers-Ulam-Rassias stability of the Banach space valued differential equations ${y}^{\prime}=\lambda y$.
*J. Korean Math. Soc.*2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleGoogle Scholar - Miura T, Miyajima S, Takahasi S-E: A characterization of Hyers-Ulam stability of first order linear differential operators.
*J. Math. Anal. Appl.*2003, 286: 136–146. 10.1016/S0022-247X(03)00458-XMathSciNetView ArticleGoogle Scholar - Miura T, Miyajima S, Takahasi S-E: Hyers-Ulam stability of linear differential operator with constant coefficients.
*Math. Nachr.*2003, 258: 90–96. 10.1002/mana.200310088MathSciNetView ArticleGoogle Scholar - Cimpean DS, Popa D: On the stability of the linear differential equation of higher order with constant coefficients.
*Appl. Math. Comput.*2010, 217(8):4141–4146. 10.1016/j.amc.2010.09.062MathSciNetView ArticleGoogle Scholar - Cimpean DS, Popa D: Hyers-Ulam stability of Euler’s equation.
*Appl. Math. Lett.*2011, 24(9):1539–1543. 10.1016/j.aml.2011.03.042MathSciNetView ArticleGoogle Scholar - Jung S-M: Hyers-Ulam stability of linear differential equations of first order.
*Appl. Math. Lett.*2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004MathSciNetView ArticleGoogle Scholar - Jung S-M: Hyers-Ulam stability of linear differential equations of first order, II.
*Appl. Math. Lett.*2006, 19: 854–858. 10.1016/j.aml.2005.11.004MathSciNetView ArticleGoogle Scholar - Jung S-M: Hyers-Ulam stability of linear differential equations of first order, III.
*J. Math. Anal. Appl.*2005, 311: 139–146. 10.1016/j.jmaa.2005.02.025MathSciNetView ArticleGoogle Scholar - Jung S-M: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients.
*J. Math. Anal. Appl.*2006, 320: 549–561. 10.1016/j.jmaa.2005.07.032MathSciNetView ArticleGoogle Scholar - Lungu N, Popa D: On the Hyers-Ulam stability of a first order partial differential equation.
*Carpath. J. Math.*2012, 28(1):77–82.MathSciNetGoogle Scholar - Lungu N, Popa D: Hyers-Ulam stability of a first order partial differential equation.
*J. Math. Anal. Appl.*2012, 385(1):86–91. 10.1016/j.jmaa.2011.06.025MathSciNetView ArticleGoogle Scholar - Popa D, Rasa I: The Frechet functional equation with application to the stability of certain operators.
*J. Approx. Theory*2012, 164(1):138–144. 10.1016/j.jat.2011.09.009MathSciNetView ArticleGoogle Scholar - Jung S-M: Legendre’s differential equation and its Hyers-Ulam stability.
*Abstr. Appl. Anal.*2007., 2007: Article ID 56419. doi:10.1155/2007/56419Google Scholar - Jung S-M: Approximation of analytic functions by Airy functions.
*Integral Transforms Spec. Funct.*2008, 19(12):885–891. 10.1080/10652460802321287MathSciNetView ArticleGoogle Scholar - Jung S-M: Approximation of analytic functions by Hermite functions.
*Bull. Sci. Math.*2009, 133: 756–764. 10.1016/j.bulsci.2007.11.001MathSciNetView ArticleGoogle Scholar - Jung S-M: Approximation of analytic functions by Legendre functions.
*Nonlinear Anal.*2009, 71(12):e103-e108. 10.1016/j.na.2008.10.007View ArticleGoogle Scholar - Jung S-M:Hyers-Ulam stability of differential equation ${y}^{\u2033}+2x{y}^{\prime}-2ny=0$.
*J. Inequal. Appl.*2010., 2010: Article ID 793197. doi:10.1155/2010/793197Google Scholar - Jung S-M: Approximation of analytic functions by Kummer functions.
*J. Inequal. Appl.*2010., 2010: Article ID 898274. doi:10.1155/2010/898274Google Scholar - Jung S-M: Approximation of analytic functions by Laguerre functions.
*Appl. Math. Comput.*2011, 218(3):832–835. doi:10.1016/j.amc.2011.01.086 10.1016/j.amc.2011.01.086MathSciNetView ArticleGoogle Scholar - Jung S-M, Rassias TM: Approximation of analytic functions by Chebyshev functions.
*Abstr. Appl. Anal.*2011., 2011: Article ID 432961. doi:10.1155/2011/432961Google Scholar - Kim B, Jung S-M: Bessel’s differential equation and its Hyers-Ulam stability.
*J. Inequal. Appl.*2007., 2007: Article ID 21640. doi:10.1155/2007/21640Google Scholar - Ross CC:
*Differential Equations - An Introduction with Mathematica*. Springer, New York; 1995.Google Scholar - Kreyszig E:
*Advanced Engineering Mathematics*. 9th edition. Wiley, New York; 2006.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.