Abstract and Applied Analysis

Volume 2016 (2016), Article ID 1796316, 9 pages

http://dx.doi.org/10.1155/2016/1796316

## Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

Department of Mathematics and Applied Mathematics, University of Venda, P/B X5050, Thohoyandou, Limpopo 0950, South Africa

Received 13 June 2016; Revised 25 October 2016; Accepted 7 November 2016

Academic Editor: Jaume Giné

Copyright © 2016 J. C. Ndogmo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.

#### 1. Introduction

Linear ordinary differential equations (LODEs) are quite probably the most common type of differential equations that occur in physics and in many other mathematically based fields of science. However, their most important properties such as their transformation properties, their general solutions, and even their symmetry properties remain largely unknown.

In a short paper published by Krause and Michel [1] in 1988 certain specific properties of LODEs of maximal symmetry were established. In particular, the said paper shows that such equations are precisely the iterative ones, and equivalently those which can be reduced by an invertible point transformation to the trivial equation , which we shall refer to as the canonical form. However that short paper left a number of important questions unanswered. It does not provide, for instance, any expression for the point transformation mapping a given equation of maximal symmetry to the canonical form.

Almost at the same time the problem of generating LODEs of maximal symmetry was considered by Mahomed and Leach [2] who found an algorithm for obtaining expressions for the most general normal form of these equations based on the direct computation of the symmetry algebra. Computations with this algorithm remain however quite tedious and the authors managed to provide a general expression for the LODEs of maximal symmetry only up to the order eight. In fact the expression of the corresponding eight-order equation of maximal symmetry found in that paper is incorrect.

A more direct algorithm for generating this class of equations based on the simple fact that they are iterative was proposed recently in [3]. Nevertheless, some of the main results of the latter paper concerning in particular the generation and the point transformations of the class of LODEs of maximal symmetry still have room for improvements.

It is worthwhile to mention that in recent years the study of ordinary differential equations (ODEs) of maximal symmetry algebra has given rise to a considerable number of research papers in the scientific literature. Examples of such studies include the determination of various types of symmetry subalgebras and some of their applications for systems of second-order ODEs [4, 5]. Momoniat and collaborators also studied the algebraic properties of first integrals of scalar second-order and third-order ODEs as well as for systems of second-order ODEs of maximal symmetry [6, 7]. On the other hand a solution algorithm for second-order and third-order ODEs of maximal symmetry based on Janet bases and Loewy decompositions was obtained in [8, 9]. A similar study of second-order LODEs of maximal Lie point symmetry and third-order LODEs of maximal contact symmetry was carried out in [10] for equations in canonical form ( and ) to investigate amongst others the properties of their first integrals and exceptional symmetries.

In this paper, we provide a much simpler differential operator than that found in [3] for generating linear iterative equations of a general order. This gives rise to a simple algorithm for testing LODEs for maximal symmetry based solely on their coefficients. The operator thus found corrects the wrong one obtained in [2, Equations (3.20) and (3.21)] as one of the main conclusions of that paper. On the other hand, we give a more direct proof than that of Krause and Michel [1] to the fact that LODEs reducible by an invertible point transformation to the canonical form are precisely those which are iterative.

We also establish several results concerning the solutions of this class of equations and in particular their transformation to canonical form. In contrast to the very well-known paper by Ermakov [11] who managed to find only some very specific cases from a restricted class for which the second-order source equation is solvable, we provide large families of second-order equations for which the general solution is given by simple algebraic formulas. All such families are parameterized by an entirely arbitrary nonzero function and the general solutions thus found for the second-order source equation yield through a very simple quadratic formula that for the whole corresponding class of equations of maximal symmetry of a general order.

#### 2. Iterations of Linear Equations

Let and be two smooth functions of , and consider the differential operator . We shall often denote by a differential function of the variables . Linear iterative equations are the iterations of the first-order LODE , given bywhere is the identity operator. A linear iterative equation of a general order thus has the formSettingand applying (1) show that the coefficients in the general expression (2) of an iterative equation satisfy the recurrence relationMoreover, setting or in (4) shows by induction on thatand applying (4) recursively and using the conventions set in (3) give the new recurrence relationWe note that (6) provides an algorithm for the computation of the coefficients in terms of the parameters and of the source equation and the operator , and the resulting formula has effectively been obtained in [3, Theorem 2.2]. Moreover, it is of course also possible to compute directly in terms of and the parameters and , and for one finds thatIf we divide through the general th order linear iterative equation in (2) by , it takes the formIt is clear that this equation represents the standard form of the general linear iterative equation with leading coefficient one. Moreover, the well-known change of the dependent variable maps (8a) into its normal form in which the coefficient of the term of second highest order has vanished. This transformation however simply amounts to choosing and such that ; that is, . Therefore, for given parameters and of the operator , an th order LODE in normal formis iterative if and only ifwhere is given by (6). It follows from (7a) that the requirement that holds is equivalent to havingand this shows in particular that any iterative equation in normal form can be expressed in terms of the parameter alone; that is, it depends on a single arbitrary function. Clearly, the coefficients can also be expressed solely in terms of , , and the derivatives of . For instance, by setting for any given function ,it follows from (7b) that in (9a)-(9b) we haveIn fact, as already noted in [2, 3], the coefficients depend only on the function and its derivatives. For simplicity, it will often be convenient to denote the coefficient of the term of third highest order in (9a) simply by .

Having noted that the coefficients of every LODE of maximal symmetry in normal form depend solely on and its derivatives, an important problem considered in [2] was to find a linear ordinary differential operator * depending solely on * and its derivatives, and which generates the most general form of the linear th order equation of maximal symmetry in the dependent variable . In a recent paper [3], it was established that, for an arbitrary parameter of the source equation, the operatorgenerates the linear iterative equation of an arbitrary order in normal form and in its most general form (9a)-(9b). Therefore, although the operator (13) and the equation it generates depend explicitly on (and not on ) and its derivatives, on the basis of the result of [1] identifying linear iterative equations with LODEs of maximal symmetry, the two equations and should always be the same for all , although such an operator * has not yet been found*. This is due in part to the fact that a general expression for is not available for all and . Nonetheless, we show here that it is naturally possible to make use of the differential operator to generate directly a LODE of maximal symmetry of the general form (9a) in which the coefficients depend only on and its derivatives.

Indeed, for any value of it follows from (6) and (9a), (9b), (11), and (12) and a simple induction on that each coefficient in (9a)-(9b) depends linearly on . Moreover, it follows from (11)-(12) thatwhere . Therefore, applying the substitution (14) to yields the desired equation, that is, the LODE of the form (9a)-(9b) in which depends only on and its derivatives. More formally, the resulting differential operator can be represented as For example, for or , evaluating yields the following expressions directly in terms of and its derivatives alone.However, if in addition we also apply to these expressions for the substitution (14), which amounts to applying directly the operator to we obtainComparing this with the known expressions for LODEs of maximal symmetry expressed solely in terms of and its derivatives [2, 3] shows that and yield indeed the indicated expressions.

Another important observation made in this paper is that if we set for a certain nonzero function , the expression for in (11) is much simpler and reduces to . Setting is thus equivalent to letting be a solution of the equationwhich is referred to as the second-order source equation for (9a)-(9b). Thus, if we express with replaced by , the substitution rule for similar to that given for in (14) takes the much simpler form

Denote by the operator in which is replaced with and to which the substitution (19) is applied. That is,In other words is just as in (13), in which the source parameter has been replaced by and to which the substitution (19) is then applied. Then for the same reasons that generates the most general form of linear iterative equations, also does the same. However, in view of the much simpler substitution rule (19), the computational cost for generating these equations is much lower using rather than . Indeed for arbitrary values of , generating equations of maximal symmetry of order greater than ten has been up to now a very tedious task using the original algorithm of [2], but it is now easy to generate such equations at much higher orders in most standard computers using computing systems such as MATHEMATICA. For instance, the coefficient of which is the largest expression in size in the equation is given by

Of course, if one can generate a LODE of maximal symmetry of a general order and in its most general form, then the identification of any given LODE as a member or not in the class of linear equations of maximal symmetry is also achieved. This is due to the fact that, in their normal form, th order LODEs (9a) of maximal symmetry are completely and uniquely determined by the coefficients . Indeed, suppose thatis a given LODE in normal form that we wish to test for the maximality of its symmetry algebra, where are given functions of If this equation is indeed of maximal symmetry, then by (12) the coefficient of the corresponding second-order source equation should satisfy . Consequently, is of maximal symmetry if and only if it coincides with the corresponding generated equation .

Using the operator , we shall give in the next section a much simpler and direct proof than that of [1] to the fact that a LODE is iterative if and only if it is reducible by a point transformation to the canonical form. To close this section, we note that by Abel’s Identity the Wronskian of any two linearly independent solutions and of the source equation (18) is a nonzero constant and will be normalized to one. It should be noted that linearly independent solutions of the second-order equation (18) are not known for arbitrary values of the coefficient .

There are of course many other obvious criteria that can be used to identify expressions which do not correspond to LODEs of maximal symmetry. For example, the coefficients are in fact differential polynomials in . This implies in particular that if, for instance, is polynomial (in ), then all must be polynomials, and if it is a sine function, then the must all be polynomials of sine and cosine functions.

#### 3. Reduction to Canonical Form and General Solution

The equivalence group of linear th order equations in normal form (9a) is well known to be given by invertible point transformations of the formwhere is an arbitrary locally invertible function and an arbitrary nonzero constant [9, 12, 13]. In other words, an equation of the form (9a) is reducible by a point transformation to the canonical form if and only if there exists a transformation of the form (23) that maps such an equation to the canonical form.

Let us denote bythe Schwarzian derivative of the function . By studying the expression of the source parameter of the transformed equation under equivalence transformations, a simple characterization of the point transformation that maps an iterative equation to the canonical form was found in [3, Theorem 4.3]. This result states that a point transformation reduces a given iterative equation, which without loss of generality may be assumed to be of the form (9a), to the canonical form if and only if it is of the form (23), where is the inverse of the function satisfyingon account of (12), assuming that is the source parameter of the equation. By making use of this result of [3], we can now provide a more direct proof than that given in [1] for the following result.

Theorem 1. *A linear ordinary differential equation is iterative if and only if it can be reduced to the canonical form by an invertible point transformation.*

*Proof. *As usual one may assume that the iterative equation is in its normal form and has source parameter . On the basis of the above stated result from Theorem 4.3 of [3], it follows that the transformation (23) where is the inverse of the function maps the iterative equation to its canonical form, as the latter expression for solves (25). More directly, one can prove that the class of iterative equations and that of equations reducible by an invertible point transformation to the canonical form are the same. Indeed, let represent the element of the equivalence group of linear th order ODEs in normal form, acting on the space of independent variable and dependent variable , and given by (23). Denoting by the transformed equation, it follows that we have exactlywhereIn other words, generating a linear iterative equation in for any given value of using the differential operator and transforming the canonical equation in the new variables and using the point transformation operator yield exactly the same equations, and this proves the result.

In [1] Krause and Michel proved this theorem indirectly as a consequence of equivalence relations, by proving the equivalence between equations of maximal symmetry and iterative equations on one hand and between equations reducible by invertible point transformations and equations of maximal symmetry on the other hand. The part in that proof stating that an ODE (linear or nonlinear) is reducible to the canonical form if and only if it has maximal symmetry was however already proved by Lie [14–16]. The direct proof given here will allow us to construct various point transformations of practical importance for iterative equations as well as various forms of their general solutions.

It should first be noted that the right hand side of (26a) gives other methods for generating linear iterative equations with source parameter In practice, however, linear iterative equations arise generally with no reference to any source parameter but are expressed solely in terms of the coefficient and its derivatives. In this case, on the basis of a remark made in the previous section, a solution to the equation in (25) is simply given by , where is a solution of the second-order source equation (18). Consequently, if we denote by the transformation operator given bywhere as in (19), then and generate exactly the same linear iterative equation expressed solely in terms of and its derivatives.

On the other hand, since the transformation operator maps the trivial equation to the most general form (9a) of the iterative equation, its explicit expression can be used to derive the general solution of (9a). Indeed, under this operator we haveConsequently, linearly independent solutions to the general th order iterative equation (9a) are given byIn particular, if and are two linearly independent solutions of the source equation (18), then . Consequently, in terms of and , the linearly independent solutions to (9a) in (29) above can be rewritten as

Formula (30) is well known and was cited without proof in [1], and it is an important result for which we have not been able to find the proof in the recent literature. We naturally also found formula (28) only in this paper. Moreover, formula (29) established above provides a much simpler result by showing that linearly independent solutions of (9a) can be expressed solely in terms of a single nonzero solution of the second-order source equation. Indeed, the solutions in (29) are clearly linearly independent as their Wronskian equals the nonzero constant .

Theorem 2. *A linear ordinary differential equation is iterative if and only if it has linearly independent solutions of the form (30), where and are two linearly independent solutions of the corresponding second-order source equation (18).*

*Proof. *The fact that a linear iterative equation has linearly independent solutions of the stated form is established in (30). Conversely, if a LODE has linearly independent solutions of the form (30), then since the second-order source equation is reducible to the canonical form by a point transformation, without loss of generality we may assume that such a transformation reduces to 1 and to . Consequently, the corresponding linearly independent solutions of the LODE are polynomials of degree at most , and thus the transformed equation is in canonical form. It then clearly follows from Theorem 1 that the equation is iterative.

Recall that the th symmetric power of a LODE is the LODE of minimal order such that for every set of solutions of the product is a solution of . Moreover, it is well known that the th symmetric power of a second-order LODE has order (see [17] and the references therein). It thus follows from Theorem 2 that for an th order LODE is of maximal symmetry if and only if it is the symmetric power of a second-order LODE. It has in fact been established in [17, Theorems 1 and 2] that the symmetric power of a LODE is an iterative LODE.

#### 4. Solvability of Equations of Maximal Symmetry

The results obtained thus far in this paper will be used in this section to show amongst others that LODEs of maximal symmetry are highly solvable and that it is indeed the case for the second-order source equation (18) whose solutions completely determine those of the corresponding LODEs of maximal symmetry. In a very popular paper published in Russian in 1880 and recently translated into English [11], Ermakov stated that the majority of second-order linear homogeneous ODEs for which it is possible to find conditions for their solvability are of the formwhere the are some arbitrary constants. He then moved on in the paper to obtain some very specific cases of equations which are solvable from this class, together with their general solutions.

A couple of years later in the same decade Hill [18] considered in the study of lunar stability the most general form of the second-order LODE in normal form, that is, in the form (18), but in which the coefficient is a periodic function. Hill’s equation and its important variants such as Meissner Equation and Mathieu Equation have been amply studied [19], and it is well known that its solutions and their related properties can be described by means of Floquet theory. These solutions can also be expressed in terms of Hill’s determinant [20]. It should be mentioned that, in the context of the present paper, a differential equation is called solvable if the closed form expression of its general solution is known, regardless of the type of functions in terms of which this solution is expressed.

The results of this paper extend the scope of solvability of linear second-order equations obtained by Ermakov, Hill, and others such as Loewy [21] not only to more larger families of the coefficient function in (18), but moreover to all LODEs of maximal symmetry of a general order. This statement holds provided that the parameter of the source equation (18) is known. Indeed, it follows from (29) and the condition relating the parameter of the source equation and the solution of the corresponding second-order equation, that for* arbitrary* values of the source parameter, two linearly independent solutions of the second-order equation of the formare given byTherefore, linearly independent solutions of the corresponding th iteration of (32a) can also be obtained from (29) through the substitution . More directly in terms of , we have the following result which follows immediately from (32a)-(32b) and (30).

Theorem 3. *Let be a given nonzero function considered as the source parameter of the differential operator generating an th order LODE of maximal symmetry and in normal form. Then linearly independent solutions of are given by*

On one hand and on the basis of the results obtained in the previous section, it follows from Theorem 3 that not only can we generate a LODE of maximal symmetry of any order and for any nonzero function , but also the general solution of any such equation is given in closed form in terms of the source parameter by (33). On the other hand, for any given LODE of maximal symmetry, as long as its corresponding source parameter is known, its general solution is given also by (33).

Now, to make a more direct comparison with the class of equations (31) considered by Ermakov, we note that linearly independent solutions can also be found for the standard form of (32a) which in terms of the arbitrary coefficient of takes the formThe two linearly independent solutions of (34) are then given byWe note that as opposed to the very specific cases of solvable equations found by Ermakov from the restricted class of equations (31), not only does (34) depend intrinsically on one* arbitrary function*, but also its general solution is given by the simple formula (35) in terms of the source parameter .

We further clarify the application of Theorem 3 by an example.

*Example 4. *Let be a nonzero real number and consider the particular case where the source equation (18) is given byThen it is not obvious how to solve (36) using standard methods. However, the latter coefficient is an expression of the form as defined in (11) and (12) and more precisely with . Therefore, since we know the source parameter of the differential operator generating (36), it follows that an explicit expression of its general solution is given in closed form by (32b). Moreover, it also follows from Theorem 3 that the closed form expression of the general solution of any equation of arbitrary order generated by the same differential operator with source parameter is given by (33).

In practice, a LODE occurs however with no reference to a source parameter of the differential operator that generated it, but rather in terms of its coefficients. As already indicated any LODE can always be assumed to be of the normal form (9a)-(9b), and even in such a case the results of Theorem 3 can also be used to find the general solution of LODEs of maximal symmetry, in virtually all known solvable cases, in addition to those cases which can be solved only with the proposed method. It follows from Theorem 3 that to find the general solution it suffices to find the parameter of the source equation and by (12) this amounts to solving for the nonlinear second-order equation.where is the coefficient appearing in (9a)-(9b) while is given by (11).

This approach for solving linear differential equations has some similarities with the Loewy decomposition method [21, 22] in which the solution of a linear differential equation is known once a factorization of the differential operator generating the equation has been computed. In this approach, finding the coefficients of each factor in the factorization is also achieved for second- and third-order LODEs with rational coefficients by solving certain Riccati equations. We illustrate the comparison of these two methods by considering a very simple example.

*Example 5. *Consider the second-order equationwhose solution by Loewy decomposition is discussed in [22]. This equation has Loewy decompositionwhere Lclm stands for least common left multiple [22] and the coefficients and given byare rational nonequivalent solutions of the Riccati equationIt then follows from a result of Loewy [22, Lemma 2.4] that a fundamental set of solutions of (38a) is given byGiven that every second-order LODE is of maximal symmetry, to solve (38a) with the method proposed in this paper, we first reduce it to its normal form (although from (34) this is not necessary), through the usual transformation given in Section 2. The reduced equation takes the simpler formThe nonlinear equation (37) for finding the source parameter reduces to , where is given here by (41). It has general solutionfor some constants of integration and . Therefore, by (32b) linearly independent solutions of (41) are given byThe inverse of the transformation mapping (38a)–(38c) to (41) then shows that a fundamental system of solutions of (38a)–(38c) is given bywhich is the same as that obtained in (40).

The drawback with these two methods for solving LODEs is that they require the solutions of nonlinear ODEs which, in general, are more difficult to solve than the original LODE. However, for the method proposed in this paper, equation (37) for finding turns out to be linearizable and the linearized equation might be much easier to solve than the original equation.

As far as ODEs are concerned, the Loewy decomposition method is designed in principle for LODEs of all orders. However, that method does not seem to be accessible because its scope is quite limited, due in particular to the fact that the implementation of its algorithm involves a very high computational cost. The Loewy decomposition theory has also quite a large and very specific vocabulary of its own taken from differential algebra. Moreover, there are too many types of Loewy decompositions for a given linear differential operator, and there are in particular 12 such types for third-order linear operators alone. On the other hand the Loewy decomposition method is limited only to equations with rational coefficients for which the solutions of the corresponding (nonlinear) Riccati equations for finding the coefficients of the factorized operator are available.

Of course the method proposed in this paper for finding solutions of LODEs is limited to the case of equations of maximal symmetry. The determining nonlinear second-order equation (37) for finding the source parameter is however the same for equations of all orders since is an arbitrary coefficient in (37), and it is also linearizable as already indicated. Moreover, as Theorem 3 shows, it is easily applicable to equations of all orders as long as (37) can be solved for .

Nevertheless, it should be emphasized that the most important result obtained in this section is not a method for solving a given lode. It is rather the fact that to any arbitrary smooth function we have associated a LODE of maximal symmetry and arbitrary order and for which the closed form solution is given by (33). This shows that such equations are highly solvable, because traditionally most solvable LODEs are limited to those having rational coefficients.

#### 5. Concluding Remarks

In this paper, we found a cost effective algorithm for generating LODEs of maximal symmetry. This algorithm corrects the wrong one proposed in [2], as quoted in the paper. We have also shown how such an algorithm can be used to easily identify linear ODEs of maximal symmetry and proved some fundamental results concerning the reduction of ODEs of maximal symmetry to canonical form. We also found general and optimal expressions for their general solutions.

Clearly, the source equation (18) is hardly solvable for arbitrary values of the corresponding coefficient . We have however shown that for every given nonzero smooth function , where the function may include in particular any kind of special function, the general solution of any th order LODE of maximal symmetry in normal form generated by the differential operator having as coefficient is given by the closed form expression (33).

On the other hand, the inverse problem of finding for a given value of turns out to be equivalent to solving a different LODE of the form (18), because the nonlinear second-order ODE is linearizable. Nevertheless, it seems quite possible to describe all possible values of the arbitrary coefficient function by an appropriate choice of a source parameter . This in turn would then mean that all linear ODEs of maximal symmetry are exactly solvable.

#### Competing Interests

The author declares that there is no conflict of interests regarding the publication of this article.

#### References

- J. Krause and L. Michel, “Équations différentielles linéaires d'ordre $n>2$ ayant une algèbre de Lie de symétrie de dimension $n+4$,”
*Comptes Rendus de l'Académie des Sciences*, vol. 307, no. 18, pp. 905–910, 1988. View at Google Scholar - F. M. Mahomed and P. G. L. Leach, “Symmetry Lie algebras of
*n*th order ordinary differential equations,”*Journal of Mathematical Analysis and Applications*, vol. 151, no. 1, pp. 80–107, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J.-C. Ndogmo and F. M. Mahomed, “On certain properties of linear iterative equations,”
*Central European Journal of Mathematics*, vol. 12, no. 4, pp. 648–657, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Campoamor-Stursberg, “An alternative approach to systems of second-order ordinary differential equations with maximal symmetry. Realizations $\mathfrak{s}\mathfrak{l}\left(n+2,\mathbb{R}\right)$ by special functions,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 37, pp. 200–211, 2016. View at Publisher · View at Google Scholar - R. Campoamor-Stursberg and J. Guerón, “Linearizing systems of second-order ODEs via symmetry generators spanning a simple subalgebra,”
*Acta Applicandae Mathematicae*, vol. 127, pp. 105–115, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. S. Mahomed and E. Momoniat, “Algebraic properties of first integrals for scalar linear third-order ODEs of maximal symmetry,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 530365, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - K. S. Mahomed and E. Momoniat, “Symmetry classification of first integrals for scalar linearizable second-order ODEs,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 847086, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - F. Schwarz, “Solving second order ordinary differential equations with maximal symmetry group,”
*Computing*, vol. 62, no. 1, pp. 1–10, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - F. Schwarz, “Equivalence classes, symmetries and solutions of linear third-order differential equations,”
*Computing*, vol. 69, no. 2, pp. 141–162, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. G. Leach, R. R. Warne, N. Caister, V. Naicker, and N. Euler, “Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry,”
*Indian Academy of Sciences. Proceedings. Mathematical Sciences*, vol. 120, no. 1, pp. 113–130, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - V. P. Ermakov, “Second-order differential equations: conditions of complete integrability,”
*Applicable Analysis and Discrete Mathematics*, vol. 2, no. 2, pp. 123–145, 2008, Translated from the 1880 Russian original by A. O. Harin, and edited by P. G. L. Leach. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Ndogmo, “Equivalence transformations of the Euler-Bernoulli equation,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 5, pp. 2172–2177, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. C. Ndogmo, “Some results on equivalence groups,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 484805, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. Lie, “Klassification und Integration von gewöhnlichen Differentialgleichungen zwischen x; y; die eine Gruppe von Transformationen gestetten. I,”
*Mathematische Annalen*, vol. 22, pp. 213–253, 1888. View at Google Scholar - P. J. Olver, “Differential invariants and invariant differential equations,”
*Lie Groups and Their Applications*, vol. 1, no. 1, pp. 177–192, 1994. View at Google Scholar · View at MathSciNet - P. J. Olver,
*Equivalence, Invariants, and Symmetry*, Cambridge University Press, Cambridge, UK, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - M. Bronstein, T. Mulders, and J.-A. Weil, “On symmetric powers of differential operators,” in
*Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '97)*, pp. 156–163, ACM, 1997. View at Publisher · View at Google Scholar - G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,”
*Acta Mathematica*, vol. 8, no. 1, pp. 1–36, 1886. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. Teschl,
*Ordinary Differential Equations and Dynamical Systems*, American Mathematical Society, Providence, RI, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - W. Magnus and S. Winkler,
*Hill's Equation*, John Wiley & Sons, New York, NY, USA, 1966. - A. Loewy, “Über vollständig reduzible lineare homogene Differentialgleichungen,”
*Mathematische Annalen*, vol. 62, no. 1, pp. 89–117, 1906. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Schwarz, “Loewy decomposition of linear differential equations,”
*Bulletin of Mathematical Sciences*, vol. 3, no. 1, pp. 19–71, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus