# Modified Integral Homotopy Expansive Method to Find Power Series Solutions of Linear Ordinary Differential Equations about Ordinary Points

Authors: Uriel Filobello-Nino; Hector Vazquez-Leal; Jesus Huerta-Chua; Victor Manuel Jimenez-Fernandez; Agustin L. Herrera-May; Darwin Mayorga-Cruz
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This article presents the Modified Integral Homotopy Expansive Method (MIHEM) which is utilized to find power series solutions for linear ordinary differential equations about ordinary points. This method is a modification of the integral homotopy expansive method. The proposal consists in providing a versatile, easy to employ and systematic method. Thus, we will see that MIHEM requires only of elementary integrations and that the initial function will be always the same for the linear ordinary differential equations of the same order which contributes to ease the procedure. Therefore, it is expected that this article contributes to change the idea that an effective method has to be long and difficult, such as it is the case of Power Series Method (PSM). This method expresses a differential equation as an integral equation, and the integrand of the equation in terms of a homotopy. We will see along this work the convenience of this procedure.


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## Body

## 1. Introduction

The subject of nonlinear differential equations is important because many physical phenomena are described by this type of equations; however, finding solutions for such equations usually is a difficult task. Hence the importance of developing approximate methods to find their solutions such as variational approaches [1, 2], tanh method [3], exp-function [4, 5], Adomian’s decomposition method [6, 7], parameter expansion [8], homotopy perturbation method [9–12], homotopy analysis method [13, 14], perturbation method [15–17], integral homotopy expansive method [18], among many others. In general, these equations are still an open research topic.In contrast, finding solutions of linear differential equations that contain variable coefficients is considered a subject which is closed because their theory was established a long time ago. Most of the times, the methods to find solutions for these equations are presented in terms of power series, because it is difficult to find exact solutions. We will see that those solutions are established about both ordinary points and singular points. As a matter of fact, the case of the so called, singular regular points is important because there always exist the Frobenius series solution for these equations, despite the difficulties that singular points carry on [19–21]. Most of the times, the search for solutions of linear differential equations is concerned rather with the obtaining of approximate solutions than with an alternative proposal in order to obtain power series solutions as an alternative to basic PSM. Thus, [22] obtained an exact solution for two cases which corresponds the values n=0, and n=1 of a parameter of the denominated Lane-Emden singular equation (LEE) which is related with a great deal of phenomena in physics such as stellar structure among others. We note that for the above mentioned cases LEE is a linear ordinary differential equation with variable coefficients. Although originally [22] was conceived to find approximations for nonlinear differential equations, the above mentioned article obtained solutions about the regular singular point x=0. We note that for cases different from n=0, 1, and 5, the Lane-Emden equation is nonlinear. On the other hand [23] proposed Adomian decomposition method and differential transform method in order to solve once more the Lane-Emden equation for the same values n=0, and n=1. Although these methods solved again the problem, they were long in comparison with [22]. As a matter of fact, Adomian decomposition method and homotopy analysis method (HAM) are usually long and cumbersome although effective. On the other hand [13] proposed HAM method in order to find analytical solutions but for the case of linear systems of partial differential equations with good precision. Besides [24] found an analytical approximate solution for the important Bessel linear differential equation of order zero by using the exponentially fitted collocation approximation method. On the other hand [25] proposed the spectral method in order to solve linear second order differential equations with constant coefficients, and [9] solved among others the important case of linear Euler-Lagrange equations. In a sequence, [12] solved also variational problems by using Laplace Transform Homotopy Perturbation Method, besides of nonlinear problems it found solutions for linear problems about singular regular points. In all these cases, the goal was obtaining a solution for the differential equation to solve, but not to provide an alternative method to get power series solutions.In particular, this article is concerned with the obtaining of the general solution for a linear ODE about an ordinary point through the use of power series. Although [26] employed HPM and nonlinearities distribution HPM with the objective to find power series solutions for linear ODES, that article was focused for the case of initial value problems due to the nature of HPM. It is well known that basic HPM method requires the knowledge of the initial conditions of the problem to solve. This work not only seeks to provide a general solution for linear equations about regular points, but will provide a proof that its series solutions are equivalent to that, obtained for PSM. This point is important because as it is well known, although PSM is usually long and cumbersome there are not alternatives that simplify the procedure. This work proposes an alternative method, the Modified Integral Homotopy Expansive Method (MIHEM). We will see that MIHEM is relatively easy to use and provides an adequate alternative method based in the solution of elementary integrals with the end to find general solutions in series for the case of ordinary points.Paper organization is as follows. Section2, introduces brief review of Linear Differential Equations with Variable Coefficients. Section 3, presents a basic idea of the proposed method MIHEM. In Section 4, we will apply MIHEM with the purpose to find power series solutions to linear differential equations about ordinary points. The main results obtained in this work are discussed in Section 5. Section 6 has to do with future work as a continuation of this method and the conclusions. Finally, Appendix presents the equivalence of PSM and MIHEM.


## 2. Linear Differential Equations

As it is found in the literature, linear differential equations with constant coefficients have exact analytical solutions [19, 20]; in contrast, the methods to find solutions for linear differential equations of variable coefficients are based on infinite series expansions.To start, consider a differential equation of the form:(1)y″x+pxy′x+qxyx=0.If the functionspx and qx are analytic at x=x0 (that is, they can be represented by means of a powers series of x−x0, with positive convergence radius [19, 20]) then x=x0 is an ordinary point of the differential equation; otherwise, the point x=x0 can be considered as a singular point [19]. This let us to present two methods for the solution of (1).


### 2.1. Power Series

The simplest method occurs when the solutions for (1) are expressed in the neighbourhood of an ordinary point x0 [19, 20]. For this case, the solutions are found in the power series form(2)y=∑n=0cnx−x0n,where cn are the unknown coefficients, which will be determined by substituting (2) into the equation that is to be solved. This work is concerned with the case of ordinary points.


### 2.2. Frobenius Series

Singular points can be classified into regular and irregular [19]. The Frobenius method, for the case of regular singular points, allow us to find power series solutions of the form(3)y=∑n=0cnx−x0n+r,where r is a parameter to be determined besides of coefficients cn.In accordance to the theory of linear differential equations, the general solution of (1), for the case of ordinary points, can be expressed in terms of the superposition of two linearly independent series of the form (2), and for the simpler case of regular singular points, in the form of (3)An important result that accounts the nature of solutions near of ordinary points is contained in the following fundamental theorem [20].Theorem 1.
Letx0 be an ordinary point of the differential (1) and a0, a1 arbitrary constants. Then, there is a unique function yx that is analytic at x0 and is a solution of (1) in a certain neighbourhood of this point, and also satisfies the initial conditions yx0=a0, y′x0=a1. Besides, if the powers series expansions of Px and Qx are valid in an interval x−x0<R, R>0, then the power series expansion of this solution is also valid on the same interval [20].
We will see later the importance of this theorem and its relation with the results obtained for this article.
Next, we establish the following fundamental theorem for non-homogeneous second order linear differential equations which will result useful in the next section [20].Theorem 2.
LetPx, Qx and Rx be continuous functions on an interval a≤x≤b. If x0 is any point in this interval, and y0 and y0′ are any numbers whatever, then the initial value problem.
y″x+Pxy′x+Qvyx=Rx, yx0=y0, and y′x0=y0, has one and only one solution y=yx on the interval a≤x≤b [20].


## 2.1. Power Series

The simplest method occurs when the solutions for (1) are expressed in the neighbourhood of an ordinary point x0 [19, 20]. For this case, the solutions are found in the power series form(2)y=∑n=0cnx−x0n,where cn are the unknown coefficients, which will be determined by substituting (2) into the equation that is to be solved. This work is concerned with the case of ordinary points.


## 2.2. Frobenius Series

Singular points can be classified into regular and irregular [19]. The Frobenius method, for the case of regular singular points, allow us to find power series solutions of the form(3)y=∑n=0cnx−x0n+r,where r is a parameter to be determined besides of coefficients cn.In accordance to the theory of linear differential equations, the general solution of (1), for the case of ordinary points, can be expressed in terms of the superposition of two linearly independent series of the form (2), and for the simpler case of regular singular points, in the form of (3)An important result that accounts the nature of solutions near of ordinary points is contained in the following fundamental theorem [20].Theorem 1.
Letx0 be an ordinary point of the differential (1) and a0, a1 arbitrary constants. Then, there is a unique function yx that is analytic at x0 and is a solution of (1) in a certain neighbourhood of this point, and also satisfies the initial conditions yx0=a0, y′x0=a1. Besides, if the powers series expansions of Px and Qx are valid in an interval x−x0<R, R>0, then the power series expansion of this solution is also valid on the same interval [20].
We will see later the importance of this theorem and its relation with the results obtained for this article.
Next, we establish the following fundamental theorem for non-homogeneous second order linear differential equations which will result useful in the next section [20].Theorem 2.
LetPx, Qx and Rx be continuous functions on an interval a≤x≤b. If x0 is any point in this interval, and y0 and y0′ are any numbers whatever, then the initial value problem.
y″x+Pxy′x+Qvyx=Rx, yx0=y0, and y′x0=y0, has one and only one solution y=yx on the interval a≤x≤b [20].


## 3. MIHEM Method

The contribution of this work is focused to the case of ordinary points, but employing a novel method, the Modified Integral Homotopy Expansive Method (MIHEM). We will see that MIHEM method is easy to use and provide an adequate method in order to find general solutions in series for the case of solutions for which the solutions of (1) are expressed in the neighbourhood of an ordinary point x0. Although initially, the integral homotopy expansive method (IHEM) [18] was conceived above all as a method to get approximate and exact solutions for ordinary differential equations. This work will show the manner as MIHEM modify to IHEM in order to widen the application of Integral Homotopy Expansive Method with the end to solve linear differential equations. In summary MIHEM is expressed as follows. Given a linear ordinary differential equation, MIHEM expresses it as an integral equation, which somehow is solved for y. At this step, the integrand of the equation is expressed in terms of a homotopy and it is assumed that y is expressed as a power series of the homotopy parameter p [10, 11]. In this point, unlike of IHEM, we will assume zero the initial conditions of the problem (in case the problem provided said conditions) and will propose as initial function y0x=A+Bx, where A and B will result to be the arbitrary constants for the general solution. The rest of the procedure is the same of IHEM method [18], equating in both sides of the integral equation the identical powers of p terms, it is possible to determine a sequence of unknown functions equation which will determine the MIHEM solution.To understand how MIHEM works, consider a general nonlinear differential equation, which can be expressed as [10, 11].(4)Lu+fx=0.x∈Ω,with the following boundary conditions(5)Fu,∂u∂n=0,x∈Γ,where F, is a boundary operator; fx, is a given analytical function; Γ, is the domain boundary for Ω and L is a linear operator.From (4), we will solve for u in terms of the integral equation:(6)ux=A+Bx+∫x0x∫x0t−a1su′s−a0sus−fsdsdt.In (6) it is assumed that L is a second order linear operator corresponding to the general linear differential equation of the form:(7)u″x+a1xu′x+a0xux+fx=0.According to the method, a homotopy is introduced assuming that the incognita functionu is expressed as a power series of homotopy parameter p.(8)ux=v0x+v1xp+v2xp2+….Besides, another assumption of the proposed method is, unlike IHEM method [18], that the integral version of the problem does not contain the initial conditions out of the integral so that (6) is rewritten as(9)ux=∫x0x∫x0t−a1su′s−a0sus−fsdsdt,from all the above, we propose the following iterative process(10)∑n=0∞pnνx=∫x0x∫x0t1−pws+p−a1s∑n=0∞pnνn′s−a0s∑n=0∞pnνns−fsdsdt,where p is the homotopy parameter belonging to the interval 0,1; wx is a function introduced using the flexibility of the homotopy method. Although (10) could be useful, the final version of MIHEM is obtained assuming that wr=0, in such a way that(11)∑n=0∞pnνnx=∫x0x∫x0tp−a1s∑n=0∞pnνn′s−a0s∑n=0∞pnνns−fsdsdt.In accordance with the proposed method, we propose as initial function(12)y0=A+Bx,where A and B will result to be the arbitrary constants of the general solution.Then, after equating identicalp power terms, the values for the sequence ν0, ν1, ν2 can be found by solving in a systematic way the integrals arising from the different orders.(13)ν0x=A+Bx,ν1x=∫x0x∫x0t−a1sν0′s−a0sν0s−fsdsdt,ν2x=∫x0x∫x0t−a1sν1′s−a0sν1sdsdt,ν3x=∫x0x∫x0t−a1sν2′s−a0sν2sdsdt,…νjx=∫x0x∫x0t−a1sνj−1′s−a0sνj−1sdsdt.…In this way, to get an approximate or exact solution for (7), the results of (13) are substituted into (8), taking the limit p⟶1, the following solution is obtained [10, 11].(14)Ux=v0x+v1x+v2x+v3x….For the case of first order linear differential equations the procedure is very similar but instead of (12) the initial function y0=A.With the end to prove that (14) indeed is a solution of (9) we have to show that function Ux is continuous. We will appeal to the following argument based on Theorem 2 already mentioned. Next, we restrict our considerations to the conditions imposed for this theorem. We will assume that the coefficient functions a1x, a0x and fx from (7) are continuous on an interval a≤x≤b. Then, given a point in this interval x0 and any numbers y0 and y0′, the ODE (7) subject to yx0=y0 and y′x0=y0′ has one and only one solution y=yx in a≤x≤b (we note that many times the above mentioned interval can be extended to −∞,∞ [20]. The proposed method MIHEM express (7) in terms of the integral (9) and the solution of the problem is expressed through the iterative process proposed by (10). We begin from the initial function (12) where A and B correspond to the initial conditions yx0=A and y′x0=B. Next, the first iteration of the process corresponds to ν1x (see (13)) is given by integrals of a1x, a0x⋅y0x and fx. Since y0x is a polynomial function, in principle, these integrals provide continuous functions. In other way, given that a1xa0x and fx are supposed to be continuous in a≤x≤b, then it is possible to express them in terms of a power series of x−x0 assuming that the above mentioned functions are analytic at x0 with positive convergence radius R. From the properties of these series in its convergence interval, the operations above mentioned, even the integrals involved are valid for x−x0<R. As a consequence, it is possible to make the necessary rearrangements to express function ν1x as a convergent series for the values x−x0<R and for the same reason ν1x is continuous. On the other hand, similar arguments apply to the other integrals expressed by ν2x, ν3x,…, Thus, for instance, ν2x is obtained by integrating the products of series that emanate from a1x⋅ν1′x and a0x⋅ν1x.From the above argument, these operations are valid ina≤x≤b. As a matter of fact differentiation ν1′x is a valid operation inside the convergence interval and the products of convergent series above mentioned are performed applying the distributive property and grouping the like terms in such a way that result convergent series for x−x0<R, even after performing the required integrations for MIHEM. As a matter of fact, ν1x is represented for a convergent series in x−x0<R and for the same reason is a continuous function in the same interval. It is possible to follow in this way and note that (14) (after applying the limit p⟶1) is expressed for a sum of convergent series in this interval and for the same reason we can rearrange their terms added term-wise. Thus, we finally get a convergent series valid in x−x0<R which represents a continuous function. Given that, in accordance with Theorem 2, the posed second order inhomogeneous problem possesses a unique solution, then the MIHEM solution (14) is a continuous function that indeed represents the solution of the proposed problem with the conditions established in Theorem 2 and for the same reason also is a solution for (9) assuming valid the conditions of the aforementioned theorem.To ease the application of MIHEM, the homotopy technique allows to introduce the homotopy parameterp in the coefficient functions a1x,a0x and fx, that is a1px, a0px and fpx in the iterative process given by (10). This procedure is especially useful when some of these functions is not a polynomial [19]. Example 4 shows this procedure.


## 4. Application of MIHEM to Find Power Series Solutions to Linear Differential Equations for the Case of Ordinary Points

Next, we exemplify the use of MIHEM method with to end to solve linear differential equations for the case of ordinary points.Example 1.
Obtain the general solution for the following linear first order differential equation.(15)y′+2xy=0.
This example compares in detail the MIHEM method and Power Series Method (PSM).
Since the pointx0=0 is an ordinary point of (15), the Power Series Method is appropriate in order to obtain the solution for (15).


### 4.1. Power Series Method

In accordance with PSM we will assume a solution of the form (see (2)) [19].(16)y=∑n=0∞cnxn,next, we substitute (16) into (15) to obtain.(17)∑n=1∞ncnxn−1+∑n=0∞2cnxn+1=0,with the purpose to add the sums we have to rewrite (17) in the following manner(18)c1+∑n=2∞ncnxn−1+∑n=0∞2cnxn+1=0.Following the power series algorithm, we change the mute variable as follows:for the first sum we substitutek=n−1, and for the second one k=n+1, in such way that (18) is written as(19)c1+∑k=1∞k+1ck+1xk+∑k=1∞2ck−1xk=0.After adding the sums we rewrite (19) in the following compact way(20)c1+∑k=1∞k+1ck+1+2ck−1xk=0,thus, equating to zero the coefficients of the different powers of x, we get(21)c1=0,(22)k+1ck+1+2ck−1=0,for k=1,2,3,…(22) is denominated recursive relation, which is employed with the purpose to determine the ck coefficients. Given that k+1≠0, then (22) can be written as(23)ck+1=−2ck−1k+1,for k=1,2,3,…After iterating (23), we obtain the values:(24)k=1,c2=−c0,k=2,c3=0,k=3,c4=−2c24=c02!,k=4,c5=0,k=5,c6=−2c46=−c03!,k=6,c7=0,k=7,c8=−2c68=c04!,…Therefore, after substituting (24) into (16) we get(25)yx=c0+c1x+c2x2+c3x3+c4x4+c5x5+c6x6+…,yx=c0+0−c0x2+0+c02!x4+0−c03!x6+c04!x8−….or(26)yx=c01−x2+12!x4−13!x6+14!x8−…,where c0 is arbitrary and for the same reason we have found the general solution for (15).As a matter of fact, we recognize the series (26) as e−x2, therefore the general solution can be written as(27)yx=c0e−x2.We note that this procedure is even longer and cumbersome for the case of linear differential equations of first order.


### 4.2. MIHEM Method

In order to employ the proposed method we express (15) as an integral equation:(28)yx=−∫0x2sysds.In accordance with MIHEM we express (29) as follows(29)∑n=0∞pnνnx=−2p∫0xx∑n=0∞pnνnsds.Given that the differential equation is of first order, we propose as initial function the constant(30)ν0x=A.In such a way that iterating (31) we get(31)ν1x=−2∫0xsν0sds=−2∫0xsAds.thus,(32)ν1x=−Ax2,ν2x=−2∫0xsν1sds=2∫0xsAs2ds.therefore,(33)ν2x=Ax42,ν3x=−2∫0xsν2sds=2∫0xsAs42ds,ν3x=−Ax66,ν4x=−2∫0xsν3sds=2∫0xsAs66ds,integrating(34)ν4x=Ax824.After substituting equations (31)–(34) into (14) we get(35)yx=A1−x2+12!x4−13!x6+14!x8−…,given that A is arbitrary, then (35) is a general solution of (15), as a matter of fact this result is the same obtained for Power Series Method (27). We emphasize the ease of MIHEM to obtain (35).Example 2.
Obtain the general solution for the following linear second order differential equation.(36)y″−1+xy=0.
This example will compare in detail the MIHEM and Power Series Method [19].
Given that the pointx0=0 is an ordinary point of (36), we will employ the Power Series Method with the end to obtain a solution.
In accordance with PSM we will assume a solution of the form (16)(37)y=∑n=0∞cnxn.
Since PSM method is extremely long, we will provide a summary of the followed procedure.
By substituting (37) into (36) we get on the one hand(38)c2=c02,on the other hand the following recurrence relation(39)ck+2=ck+ck−1k+1k+2,k=1,2,3,…,the procedure that yields in (38) and (39) is similar but more complicated than the one that yielded in (21) and (23).
We note that unlike (23), (39) is a recurrence relation with three terms.
By iterating (39) and choosing c1=0 we obtain the coefficients(40)c1=0,c2=c02,c3=c06,c4=c024,c5=c030,…,which values are substituted into (37), to get(41)y1x=c01+12x2+16x3+124x4+130x5+….
Iterating a second time by choosingc2=0 we get the coefficients(42)c0=0,c2=0,c3=c16,c4=c112,c5=c1120,…,substituting these coefficients into (37) we obtain(43)y2x=c1x+16x3+112x4+1120x5+….
From (41) and (43), the general solution of (36) is expressed as(44)yx=c01+12x2+16x3+124x4+130x5+…+c1x+16x3+112x4+1120x5+….
In accordance with Theorem1, series (41) and (43) converge for all x.
As a matter of fact, this process is long and hard for most of applications.


### 4.3. MIHEM Method

In order to employ the proposed method we express (36) as an integral equation:(45)yx=∫0x∫0ts+1ysdsdt.In accordance with MIHEM we express (45) as follows(46)∑n=0∞pnνnx=p∫0x∫0t1+s∑n=0∞pnνnsdsdt.Given that the differential equation to solve is of second order, we propose as initial function(47)y0=A+Bx.In such a way that after iterating (46)(48)ν1x=∫0x∫0t1+xν0sdsdt,thus, by substituting (47) into (48) we get(49)ν1x=∫0x∫0t1+sA+Bsdsdt.After perform the elementary successive integrals in (49) we obtain:(50)ν1x=Ax22+A+Bx36+Bx412.On the other hand(51)ν2x=∫0x∫0t1+xν1sdsdt.By substituting (50) into (51) and after performing basic integrals we get(52)ν2x=Ax424+x530+x6180+…+Bx5120+x6120+x7504+….After substituting (47), (50), and (52) into (14), we get(53)yx=A1+12x2+16x3+124x4+130x5+…+Bx+16x3+112x4+1120x5+….We note that (44) and (53) are the same. Nevertheless we emphasize the ease of MIHEM to obtain the same result but employing only two iterations and elementary integrals in a systematic way.Example 3.
This example considers Hermite differential equation which is important for its applications in physics [21].
This example will employ the MIHEM method, not only for solving the Hermite differential equation but to provide a deeper analysis of this problem.
The differential equation to solve is:(54)y′′−2xy′+2αy=0,where α is a constant.
In order to employ the proposed method we express (54) as an integral equation:(55)yx=∫0x∫0t2sy′s−2αysdsdt.
In accordance with MIHEM we express (55) as follows(56)∑n=0∞pnνnx=p∫0x∫0t2s∑n=0∞pnνn′s−2α∑n=0∞pnνnsdsdt.
Given that (54) is a second order linear differential equation we propose as initial function(57)y0=A+Bx.
So that, iterating (56) we get the following elementary integral:(58)ν1x=∫0x∫0t2xν0′s−2αν0sdsdt,thus, by substituting (57) into (58) and performing the indicated integrations we get(59)ν1x=2B1−αx36−αx2A.
On the other hand the second iteration results in(60)ν2x=∫0x∫0t2sν1′s−2αν1sdsdt,from (59) and (60), after performing some elementary operations we obtain(61)ν2x=A22αα−24!x4+B22α−1α−35!x5.
On the other hand(62)ν3x=∫0x∫0t2sν2′s−2αν2sdsdt,substituting (61) into (62) yields in(63)ν3x=−A23αα−2α−46!x6−B23α−1α−3α−57!x7.
Therefore, the general solution of (54) is obtained substituting (57), (59), (61), and (63) into (14).(64)yx=Ay1x+By2x,where(65)y1x=1−2α2!x2+22αα−24!x4−23αα−2α−46!x6+…,(66)y2x=x−2α−13!x3+22α−1α−35!x5−23α−1α−3α−57!x7+…,
Given that this same result it is obtained for Power Series Method [20], then in accordance with Theorem 1, both series (65) and (66) converge for all x.
The case of non-negative integerα is particularly relevant for the following.
For eachα value, one of these series ends and results a polynomial while the other is an infinite series. From (65) and (66) it is clear that y1x will be a polynomial if α is even and y2x will result a polynomial if α is odd. Taking successively the values α=0,1,2,3,4,5, we obtain the polynomials:(67)p0x=1,p1x=x,p2x=1−2x2,p3x=x−23x3,p4x=1−4x2+43x4,p5x=1−43x3+415x5,
The so called Hermite polynomials are obtained considering constant multiples of the polynomials (67) with the property that the terms containing the highest powers of x are of the form 2nxn.
From (67) it is possible to obtain some Hermite polynomials (denoted Hnx) as follows [21].(68)H0x=1,H1x=2x,H2x=4x2−2,H3x=8x3−12x,H4x=16x4−48x2+12,H5x=32x5−160x3+120x….
In this case MIHEM not only provided a general solution for (54) but provided the adequate forms (65) and (66) from which easily let to identify the important Hermite polynomials (68)
Next, we compare PSM and MIHEM methods with the end to study the following linear differential equation with non-polynomial coefficients.Example 4.
Obtain the general solution for the following linear second order differential equation(69)y″+e−xy=0.
Given that the pointx0=0 is an ordinary point of (69), we will employ the Power Series Method with the end to obtain a solution.
In accordance with PSM we will assume a solution of the form(70)y=∑n=0∞anxn,after differentiating (70) we get(71)y′′x=∑n=2∞annn−1xn−2.
On the other hand is well known the Taylor series ofe−x(72)e−x=1−x+12x2−16x3+124x4−1120x5+1720x6−….
We will get the first six terms of (70).
The substitution of results (70)–(72) into the left hand side of (69) yields in(73)∑n=2∞annn−1xn−2+a5+a32−a26+a124−a0120−a4x5+a4+a22−a16+a024−a3x4+a3+a12−a06−a2x3+a2+a02−a1x2+a1−a0x+a0=0.
After developing the sum in (73) and grouping in powers of x we get(74)a0+2a2x0+6a3+a1−a0x1+12a4+a2+a02−a1x2+20a5+a3+a12−a06−a2x3+30a6+a4+a22−a16+a024−a3x4+42a7+a5+a32−a26+a124−a0120−a4+…=0.
After setting each power ofx equal to zero, we obtain.(75)a2=−a02,a3=16a0−a1,a4=a112,a5=120−a02−a13,….
Therefore, after substituting (75) into (70) we obtain the general solution(76)yx=a01−x22+x36−x540+…+a1x−x36+x412−x560+….
Although we simplified several steps, the above process is long and cumbersome for most applications.


### 4.4. MIHEM Method

In order to employ the proposed method we express (69) as an integral equation:(77)yx=−∫0x∫0te−sysdsdt.In accordance with MIHEM we express (77) as follows(78)∑n=0∞pnνnx=−p∫0x∫0te−ps∑n=0∞pnνnsdsdt.In (78) we introduced the homotopy parameter into the non polynomial factor with the purpose to distribute the exponential function in the different iterations of the proposed method to ease the application of MIHEM.After substituting(79)e−px=1−px+12p2x2−16p3x3+124p4x4−1120p5x5+1720p6x6−…,into (78) and after equating the coefficients of identical powers of p, we get the following relations:(80)ν0x=A+Bx.(initial function).(81)ν1x=−∫0x∫0tν0sdsdt,thus, by substituting (80) into (81) we get(82)ν1x=−Ax22−Bx36,(83)ν2x=−∫0x∫0t−sν0s+ν1sdsdt,substituting (80) and (82) into (83) and after performing elementary integrals we get(84)ν2x=Ax36+B+A2x412+Bx5120.On the other hand the following iteration results in(85)ν3x=−∫0x∫0tν2s−sν1s+s22ν0sdsdt,from (80), (82), (84) and (85) we get(86)ν3x=−Ax424−x5202A3+B2−x630B4+A24−Bx75040.Iterating (78) we get the following elementary integral:(87)ν4x=−∫0x∫0tν3s−sν2s+s22ν1s−s36ν0sdsdt.Taking into account that we have proposed to obtain the first six terms of the series solution (70), we note that only the last integral of (87) contributes to this part of the solution, whereby we write (87) as follows(88)ν4x=∫0x∫0ts36ν0sdsdt+….After substituting (80) into (88) we obtain(89)ν4x=Ax5120+Bx6180+….Substituting (80), (82), (84), (86), and (89) into (14) we get(90)yx=A1−x22+x36−x540+…+Bx−x36+x412−x560+….We note that we obtained the same results (76) (PSM) and (90) (MIHEM) as it should be.On the other hand, we note that the fifth iteration of PSM provides, until the fifth power of the solution while the fourth iteration of MIHEM provides the same information and more of the following powers.Example 5.
This example provides the general solution for the following linear second order inhomogeneous differential equation by using MIHEM.(91)y″−2+4x2y+x2+4x2=0,
To employ the proposed method, we express the following integral equation in terms of(92)yx=∫0x∫0t2+4s2ys−s2+4s2dsdt.
In accordance with the proposed method we express (92) as follows(93)∑n=0∞pnνnx=p∫0x∫0t2+4s2∑n=0∞pnνns−s2+4s2dsdt.
Next, we propose as initial function(94)y0=A+Bx,so that we obtain from (93).(95)ν1x=∫0x∫0t2+4s2ν0s−s2+4s2dsdt.
Thus, after substituting (94) in the above expression we get(96)ν1x=∫0x∫0t2+4s2A+Bs−2s+4s3dsdt.
After performing elementary operations we obtain(97)ν1x=Ax2+B−13x3+A3x4+B−15x5.
In the same way,(98)ν2x=∫0x∫0t2+4s2ν1sdsdt.
After substituting (97) into (98) we get(99)ν2x=∫0x∫0t2+4s2As2+B−13s3+A3s4+B−15s5dsdt.
After performing elementary operations we get(100)ν2x=Ax46+B−130x5+2A3+4Ax630+2B−15+4B−13x742+A3x814+B−190x9….
From the above results, we get an approximate solution for (91) in the following way(101)yx=A1+x2+x42+745x6+142x8+…+Bx+B−13x3+7B−130x5+13B−1315x7+B−190x9+….


## 4.1. Power Series Method

In accordance with PSM we will assume a solution of the form (see (2)) [19].(16)y=∑n=0∞cnxn,next, we substitute (16) into (15) to obtain.(17)∑n=1∞ncnxn−1+∑n=0∞2cnxn+1=0,with the purpose to add the sums we have to rewrite (17) in the following manner(18)c1+∑n=2∞ncnxn−1+∑n=0∞2cnxn+1=0.Following the power series algorithm, we change the mute variable as follows:for the first sum we substitutek=n−1, and for the second one k=n+1, in such way that (18) is written as(19)c1+∑k=1∞k+1ck+1xk+∑k=1∞2ck−1xk=0.After adding the sums we rewrite (19) in the following compact way(20)c1+∑k=1∞k+1ck+1+2ck−1xk=0,thus, equating to zero the coefficients of the different powers of x, we get(21)c1=0,(22)k+1ck+1+2ck−1=0,for k=1,2,3,…(22) is denominated recursive relation, which is employed with the purpose to determine the ck coefficients. Given that k+1≠0, then (22) can be written as(23)ck+1=−2ck−1k+1,for k=1,2,3,…After iterating (23), we obtain the values:(24)k=1,c2=−c0,k=2,c3=0,k=3,c4=−2c24=c02!,k=4,c5=0,k=5,c6=−2c46=−c03!,k=6,c7=0,k=7,c8=−2c68=c04!,…Therefore, after substituting (24) into (16) we get(25)yx=c0+c1x+c2x2+c3x3+c4x4+c5x5+c6x6+…,yx=c0+0−c0x2+0+c02!x4+0−c03!x6+c04!x8−….or(26)yx=c01−x2+12!x4−13!x6+14!x8−…,where c0 is arbitrary and for the same reason we have found the general solution for (15).As a matter of fact, we recognize the series (26) as e−x2, therefore the general solution can be written as(27)yx=c0e−x2.We note that this procedure is even longer and cumbersome for the case of linear differential equations of first order.


## 4.2. MIHEM Method

In order to employ the proposed method we express (15) as an integral equation:(28)yx=−∫0x2sysds.In accordance with MIHEM we express (29) as follows(29)∑n=0∞pnνnx=−2p∫0xx∑n=0∞pnνnsds.Given that the differential equation is of first order, we propose as initial function the constant(30)ν0x=A.In such a way that iterating (31) we get(31)ν1x=−2∫0xsν0sds=−2∫0xsAds.thus,(32)ν1x=−Ax2,ν2x=−2∫0xsν1sds=2∫0xsAs2ds.therefore,(33)ν2x=Ax42,ν3x=−2∫0xsν2sds=2∫0xsAs42ds,ν3x=−Ax66,ν4x=−2∫0xsν3sds=2∫0xsAs66ds,integrating(34)ν4x=Ax824.After substituting equations (31)–(34) into (14) we get(35)yx=A1−x2+12!x4−13!x6+14!x8−…,given that A is arbitrary, then (35) is a general solution of (15), as a matter of fact this result is the same obtained for Power Series Method (27). We emphasize the ease of MIHEM to obtain (35).Example 2.
Obtain the general solution for the following linear second order differential equation.(36)y″−1+xy=0.
This example will compare in detail the MIHEM and Power Series Method [19].
Given that the pointx0=0 is an ordinary point of (36), we will employ the Power Series Method with the end to obtain a solution.
In accordance with PSM we will assume a solution of the form (16)(37)y=∑n=0∞cnxn.
Since PSM method is extremely long, we will provide a summary of the followed procedure.
By substituting (37) into (36) we get on the one hand(38)c2=c02,on the other hand the following recurrence relation(39)ck+2=ck+ck−1k+1k+2,k=1,2,3,…,the procedure that yields in (38) and (39) is similar but more complicated than the one that yielded in (21) and (23).
We note that unlike (23), (39) is a recurrence relation with three terms.
By iterating (39) and choosing c1=0 we obtain the coefficients(40)c1=0,c2=c02,c3=c06,c4=c024,c5=c030,…,which values are substituted into (37), to get(41)y1x=c01+12x2+16x3+124x4+130x5+….
Iterating a second time by choosingc2=0 we get the coefficients(42)c0=0,c2=0,c3=c16,c4=c112,c5=c1120,…,substituting these coefficients into (37) we obtain(43)y2x=c1x+16x3+112x4+1120x5+….
From (41) and (43), the general solution of (36) is expressed as(44)yx=c01+12x2+16x3+124x4+130x5+…+c1x+16x3+112x4+1120x5+….
In accordance with Theorem1, series (41) and (43) converge for all x.
As a matter of fact, this process is long and hard for most of applications.


## 4.3. MIHEM Method

In order to employ the proposed method we express (36) as an integral equation:(45)yx=∫0x∫0ts+1ysdsdt.In accordance with MIHEM we express (45) as follows(46)∑n=0∞pnνnx=p∫0x∫0t1+s∑n=0∞pnνnsdsdt.Given that the differential equation to solve is of second order, we propose as initial function(47)y0=A+Bx.In such a way that after iterating (46)(48)ν1x=∫0x∫0t1+xν0sdsdt,thus, by substituting (47) into (48) we get(49)ν1x=∫0x∫0t1+sA+Bsdsdt.After perform the elementary successive integrals in (49) we obtain:(50)ν1x=Ax22+A+Bx36+Bx412.On the other hand(51)ν2x=∫0x∫0t1+xν1sdsdt.By substituting (50) into (51) and after performing basic integrals we get(52)ν2x=Ax424+x530+x6180+…+Bx5120+x6120+x7504+….After substituting (47), (50), and (52) into (14), we get(53)yx=A1+12x2+16x3+124x4+130x5+…+Bx+16x3+112x4+1120x5+….We note that (44) and (53) are the same. Nevertheless we emphasize the ease of MIHEM to obtain the same result but employing only two iterations and elementary integrals in a systematic way.Example 3.
This example considers Hermite differential equation which is important for its applications in physics [21].
This example will employ the MIHEM method, not only for solving the Hermite differential equation but to provide a deeper analysis of this problem.
The differential equation to solve is:(54)y′′−2xy′+2αy=0,where α is a constant.
In order to employ the proposed method we express (54) as an integral equation:(55)yx=∫0x∫0t2sy′s−2αysdsdt.
In accordance with MIHEM we express (55) as follows(56)∑n=0∞pnνnx=p∫0x∫0t2s∑n=0∞pnνn′s−2α∑n=0∞pnνnsdsdt.
Given that (54) is a second order linear differential equation we propose as initial function(57)y0=A+Bx.
So that, iterating (56) we get the following elementary integral:(58)ν1x=∫0x∫0t2xν0′s−2αν0sdsdt,thus, by substituting (57) into (58) and performing the indicated integrations we get(59)ν1x=2B1−αx36−αx2A.
On the other hand the second iteration results in(60)ν2x=∫0x∫0t2sν1′s−2αν1sdsdt,from (59) and (60), after performing some elementary operations we obtain(61)ν2x=A22αα−24!x4+B22α−1α−35!x5.
On the other hand(62)ν3x=∫0x∫0t2sν2′s−2αν2sdsdt,substituting (61) into (62) yields in(63)ν3x=−A23αα−2α−46!x6−B23α−1α−3α−57!x7.
Therefore, the general solution of (54) is obtained substituting (57), (59), (61), and (63) into (14).(64)yx=Ay1x+By2x,where(65)y1x=1−2α2!x2+22αα−24!x4−23αα−2α−46!x6+…,(66)y2x=x−2α−13!x3+22α−1α−35!x5−23α−1α−3α−57!x7+…,
Given that this same result it is obtained for Power Series Method [20], then in accordance with Theorem 1, both series (65) and (66) converge for all x.
The case of non-negative integerα is particularly relevant for the following.
For eachα value, one of these series ends and results a polynomial while the other is an infinite series. From (65) and (66) it is clear that y1x will be a polynomial if α is even and y2x will result a polynomial if α is odd. Taking successively the values α=0,1,2,3,4,5, we obtain the polynomials:(67)p0x=1,p1x=x,p2x=1−2x2,p3x=x−23x3,p4x=1−4x2+43x4,p5x=1−43x3+415x5,
The so called Hermite polynomials are obtained considering constant multiples of the polynomials (67) with the property that the terms containing the highest powers of x are of the form 2nxn.
From (67) it is possible to obtain some Hermite polynomials (denoted Hnx) as follows [21].(68)H0x=1,H1x=2x,H2x=4x2−2,H3x=8x3−12x,H4x=16x4−48x2+12,H5x=32x5−160x3+120x….
In this case MIHEM not only provided a general solution for (54) but provided the adequate forms (65) and (66) from which easily let to identify the important Hermite polynomials (68)
Next, we compare PSM and MIHEM methods with the end to study the following linear differential equation with non-polynomial coefficients.Example 4.
Obtain the general solution for the following linear second order differential equation(69)y″+e−xy=0.
Given that the pointx0=0 is an ordinary point of (69), we will employ the Power Series Method with the end to obtain a solution.
In accordance with PSM we will assume a solution of the form(70)y=∑n=0∞anxn,after differentiating (70) we get(71)y′′x=∑n=2∞annn−1xn−2.
On the other hand is well known the Taylor series ofe−x(72)e−x=1−x+12x2−16x3+124x4−1120x5+1720x6−….
We will get the first six terms of (70).
The substitution of results (70)–(72) into the left hand side of (69) yields in(73)∑n=2∞annn−1xn−2+a5+a32−a26+a124−a0120−a4x5+a4+a22−a16+a024−a3x4+a3+a12−a06−a2x3+a2+a02−a1x2+a1−a0x+a0=0.
After developing the sum in (73) and grouping in powers of x we get(74)a0+2a2x0+6a3+a1−a0x1+12a4+a2+a02−a1x2+20a5+a3+a12−a06−a2x3+30a6+a4+a22−a16+a024−a3x4+42a7+a5+a32−a26+a124−a0120−a4+…=0.
After setting each power ofx equal to zero, we obtain.(75)a2=−a02,a3=16a0−a1,a4=a112,a5=120−a02−a13,….
Therefore, after substituting (75) into (70) we obtain the general solution(76)yx=a01−x22+x36−x540+…+a1x−x36+x412−x560+….
Although we simplified several steps, the above process is long and cumbersome for most applications.


## 4.4. MIHEM Method

In order to employ the proposed method we express (69) as an integral equation:(77)yx=−∫0x∫0te−sysdsdt.In accordance with MIHEM we express (77) as follows(78)∑n=0∞pnνnx=−p∫0x∫0te−ps∑n=0∞pnνnsdsdt.In (78) we introduced the homotopy parameter into the non polynomial factor with the purpose to distribute the exponential function in the different iterations of the proposed method to ease the application of MIHEM.After substituting(79)e−px=1−px+12p2x2−16p3x3+124p4x4−1120p5x5+1720p6x6−…,into (78) and after equating the coefficients of identical powers of p, we get the following relations:(80)ν0x=A+Bx.(initial function).(81)ν1x=−∫0x∫0tν0sdsdt,thus, by substituting (80) into (81) we get(82)ν1x=−Ax22−Bx36,(83)ν2x=−∫0x∫0t−sν0s+ν1sdsdt,substituting (80) and (82) into (83) and after performing elementary integrals we get(84)ν2x=Ax36+B+A2x412+Bx5120.On the other hand the following iteration results in(85)ν3x=−∫0x∫0tν2s−sν1s+s22ν0sdsdt,from (80), (82), (84) and (85) we get(86)ν3x=−Ax424−x5202A3+B2−x630B4+A24−Bx75040.Iterating (78) we get the following elementary integral:(87)ν4x=−∫0x∫0tν3s−sν2s+s22ν1s−s36ν0sdsdt.Taking into account that we have proposed to obtain the first six terms of the series solution (70), we note that only the last integral of (87) contributes to this part of the solution, whereby we write (87) as follows(88)ν4x=∫0x∫0ts36ν0sdsdt+….After substituting (80) into (88) we obtain(89)ν4x=Ax5120+Bx6180+….Substituting (80), (82), (84), (86), and (89) into (14) we get(90)yx=A1−x22+x36−x540+…+Bx−x36+x412−x560+….We note that we obtained the same results (76) (PSM) and (90) (MIHEM) as it should be.On the other hand, we note that the fifth iteration of PSM provides, until the fifth power of the solution while the fourth iteration of MIHEM provides the same information and more of the following powers.Example 5.
This example provides the general solution for the following linear second order inhomogeneous differential equation by using MIHEM.(91)y″−2+4x2y+x2+4x2=0,
To employ the proposed method, we express the following integral equation in terms of(92)yx=∫0x∫0t2+4s2ys−s2+4s2dsdt.
In accordance with the proposed method we express (92) as follows(93)∑n=0∞pnνnx=p∫0x∫0t2+4s2∑n=0∞pnνns−s2+4s2dsdt.
Next, we propose as initial function(94)y0=A+Bx,so that we obtain from (93).(95)ν1x=∫0x∫0t2+4s2ν0s−s2+4s2dsdt.
Thus, after substituting (94) in the above expression we get(96)ν1x=∫0x∫0t2+4s2A+Bs−2s+4s3dsdt.
After performing elementary operations we obtain(97)ν1x=Ax2+B−13x3+A3x4+B−15x5.
In the same way,(98)ν2x=∫0x∫0t2+4s2ν1sdsdt.
After substituting (97) into (98) we get(99)ν2x=∫0x∫0t2+4s2As2+B−13s3+A3s4+B−15s5dsdt.
After performing elementary operations we get(100)ν2x=Ax46+B−130x5+2A3+4Ax630+2B−15+4B−13x742+A3x814+B−190x9….
From the above results, we get an approximate solution for (91) in the following way(101)yx=A1+x2+x42+745x6+142x8+…+Bx+B−13x3+7B−130x5+13B−1315x7+B−190x9+….


## 5. Discussion

This work proposed the MIHEM method, as a useful tool in order to find power series solutions of linear ordinary differential equations for the case of ordinary points.Although our examples sustain that Power Series Method and the proposed method MIHEM provided the same results under the conditions above mentioned, appendix, will be dedicated to show the equivalence of two methods for solving the general case (1) of linear differential equations assuming that coefficient functionspx and qx are analytic at x=x0 (this equivalence can be inferred from the theoretical discussion established just below of (14)). Despite Power Series Method is the classical method to solve linear equations about ordinary points, we emphasize that PSM usually is long and cumbersome and it requires concentration to avoid errors at the handle of sums and its index. Unlike it, MIHEM was introduced with the same objective, but we noted that it is a method systematically based in elementary integrals, beginning always of the same initial function for the linear differential equations of the same order. As a matter of fact, from the proposed problems it is clear that, besides elementary algebra, the other systematically employed mathematical result is the basic integral ∫xndx=xn+1/n+1.Example1 solved a linear first order differential equation, after comparing both methods, it resulted clear that MIHEM is more direct and easier to use in comparison with the traditional PSM method. In fact, we noted that PSM method is most of the times cumbersome and difficult to use. Thus, the Power Series Method algorithm implies among other things, to change the mute variable, adding of sums, obtaining the so-called recursive relation, which is employed with the purpose to determinate the ck coefficients, and so on. Conversely, we emphasized the ease of MIHEM to obtain the same results but with less effort, its procedure is systematically based in the solution of basic integrals. Example 2 solved a second order differential equation by using again both methods. We note that for this case PSM resulted even more cumbersome and long in comparison with the first case study; thus, we noted that its iterations were performed using a recurrence relation with three terms. In general terms, PSM requires more work as it is employed to solve higher order differential equations. The third case study considered the solution of Hermite differential equation which is important for its applications in quantum mechanics. This example employed the MIHEM method, not only for solving the Hermite differential equation but to provide a deeper analysis of this problem. Its application was useful to obtain the so called Hermite polynomials which are important functions of the so called mathematical-physics branch [19, 20]. Example 4 provided a case of non-polynomial coefficients. PSM directly multiplied series (70) and (72) in order to get (74) and a equations system whose solution is given by (75). Although we simplify several steps, the above process is long and cumbersome for most of applications. On the other hand, the homotopy parameter was introduced into the non-polynomial factor e−x in (78) with the purpose to distribute the exponential function in the different iterations of the proposed method to ease the application of MIHEM. We note that the rest of the procedure is similar to the one used in the previous examples that is, based in basic integrals. Given that PSM and MIHEM provide the same results, the proposal of this work is taking advantage of the ease of the proposed method in the search of solutions for linear differential equations for the case of ordinary points. Finally, Example 5 shows the application of MIHEM, for the case of an inhomogeneous problem. The obtained approximate solution shows the ease with which MIHEM handles these problems.We emphasize the relevance of the proposed method MIHEM for the series solution of linear ordinary differential equations. The method is convenient for the applications, not only because this is based in the solution of elementary integrals but, it is wholly systematic. For instance, the method always begins with the same initial function for all equations of the same order. As a matter of fact, given a linear problem MIHEM always expresses the problem in terms of the integral (6) and express it in terms of the iterative process (10). Our examples show that this procedure is straightforward and simple. The case of PSM usually is cumbersome and in general requires to get the so called recurrence relation from which are calculated the coefficients of the series solution. From (22) and (39) and the example (4), we note that this procedure many times cannot be established without waiting for the implementation of the recurrence formula to have no relevant changes. Thus, we note that recurrence (22) directly relate two c coefficients while (39) required to relate three of them. What is more, example (4) did not even require a recurrence relation.From the mentioned above, it is clear that MIHEM is a powerful method which indeed ease the obtaining of series solutions about ordinary points for ordinary linear differential equations, but [18] essentially considered the application of the proposed method, although slightly modified, with the purpose to get exact and approximate solutions for the case of nonlinear differential equations. In a sequence, the next section will show, as a future work, the possibility of employing MIHEM with the end to solve linear problems about singular points. Thus, the proposed method is not only a powerful tool for research, but is a method widely recommended to be implemented in university courses from the undergraduate level onwards.


## 6. Concluding Remarks

Although MIHEM was introduced for obtaining power series solutions of linear ordinary differential equations about of ordinary points, a natural continuation or as future work, would be to adapt this method with the purpose to solve linear ordinary differential equations about singular points, at least for the case of regular singular points (see Section3 and references [19, 20]). In spite of that, we are not yet in the position to justify the use of MIHEM to solve problems about singular points, we will present two case studies in which the proposed method works satisfactory although the conditions to ensure the validity of MIHEM are not satisfied.Example 6.
Obtain the general solution for the following linear first order differential equation.(102)xy′−y=0,about the singular point x=0.
Given that (102) can be rewritten as(103)y′=yx,then, in accordance with MIHEM algorithm we convert (103) as the following integral equation(104)y=∫0xyssds.
Introducing the handy process described for Section4, we get(105)∑n=0∞pnνnx=p∫0x∑n=0∞pnνnssds.
In accordance with the proposed method, we will propose as initial function(106)y0x=A.but said choice would yield in a solution with a singular point in x = 0 due to the the presence of lnx in the result. In order to avoid this result we propose the following initial function(107)ν0x=Ax.in such a way that(108)ν1x=∫0xν0ssds.
After substituting (107) into (108) we obtain(109)ν1x=Ax.in the same way:(110)ν2x=Ax,(111)ν3x=Ax.thus, after n iterations we get(112)νnx=Ax.
By substituting (107) and (109)–(112) into (14) we obtain:(113)yx=Bx,where we defined the arbitrary constant B=nA.
By mere substitution, we note that (113) is the general solution for linear (102).Example 7.
Obtain the general solution for the following linear second order inhomogeneous differential equation.(114)y′′−2x2y−8x=0.about the regular singular point x=0.
As usual, we express (114) in terms of the following integral equation:(115)yx=∫0x∫0t2s2ys+8sdsdt.
In agreement with the proposed method we express (115) as(116)∑n=0∞pnνnx=p∫0x∫0t2∑n=0∞pnνnss2+8sdsdt.
In the same way we will avoid to get a singular solution forx=0, considering the initial function (instead of (12))(117)ν0x=Ax2+Bx3,so that from (116) we obtain:(118)ν1x=∫0x∫0t2ν0ss2+8sdsdt.
Thus, from (117) and (118) we get, after performing elementary operations(119)ν1x=Ax2+2B+86x3.
In the same way it is straightforward to show that(120)ν2x=Ax2+2B+836x3.as a matter of fact, the only difference of the successive iterations from (119), is a factor acting on the cubic term, so the nth iteration can be expressed from the above iterations as follows:(121)yx=nAx3+B1+118+836+⋯x3.
Expressing the above result in terms of just two constants(122)yx=A′x2+Cx3,where A′=nA and C=B+2B/6+2B/36+…+8/6+8/36+….
Since that is a linear equation, the general solution only has to have one arbitrary constant [19, 20], to understand how this occurs, it is enough to substitute (122) into the differential equation to obtain C=2.
After substituting this value into (122) we obtain(123)yx=A′x2+2x3.
It is easy to verify thatyhx=A′x2 is the solution of the homogeneous part of the (114) and (123) is the solution for (114). We note that the general solution for a homogeneous linear differential equation has to contain two arbitrary constants and (123) only shows yhx=A′x2, it results that the manner in choosing the initial function (117) eliminates a singular solution. To see it, we know from the theory of linear differential equations that given a known solution for linear (1) it is possible to get a second solution [19, 20]. Therefore if y1=x2, then a second solution is given by: y2=x2∫dx/x4, or y2=−1/3x and the general solution for the homogeneous part is yhx=A′x2−D/3x for other arbitrary constant D. Thus the obtained solution omitted the singular part D=0 because we proposed an initial function with the purpose to avoid singularity solutions but it is clear that always it is possible to recover the total solution. From these examples, it is conjectured that MIHEM method has potentiality in order to handle in general singular points.
Although MIHEM is not sufficiently proved as a general tool to solve linear equations about singular points, it is noted that conversely, PSM is totally inadequate for getting solutions of (102) and (114) about x=0 because for these cases it is a singular point. In fact, instead of PSM, it should be employed the already explained Frobenius method [19, 20], but this is even more cumbersome and long than PSM.
As conclusion, this work introduced the Modified Integral Homotopy Expansive Method (MIHEM) which showed potential in order to find power series solutions for linear ordinary differential equations for the case of ordinary points. A relevance difference with PSM is the versatility of the proposed method which is shown in five case studies. The first advantage of the method is that MIHEM requires only of elementary integrals. The second is that the initial function for the case of linear second order differential equations is alwaysA+Bx, and A for the case of linear first order differential equations. This contributes to systematize the procedure. Finally, once the series (14) is obtained, we proceed to factorize A and B in order to get two linearly independent solutions and for the same a general solution.
Therefore, we note that the simplicity of the proposed method does not make it less effective, but on the contrary, we emphasize its convenience in practical applications. The proposal of MIHEM is that of an effective method and easy to use for linear ordinary differential equations for the case of ordinary points. As a matter of fact, we noted that the proposed MIHEM is a potential useful method to solve also ordinary differential equations about singular points, although in general, this subject should be part of a future work.



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*Source: 1016251-2022-04-28.xml*