Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2016, Article ID 1796316, 9 pages http://dx.doi.org/10.1155/2016/1796316

Research Article Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

J. C. Ndogmo

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

Correspondence should be addressed to J. C. Ndogmo; [email protected]

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

Academic Editor: Jaume Gine´

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.

An effective method for generating linear ordinary differentialuations eq 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 Computations with this algorithm remain however quite tedious and the authors managed to provide a general ex- Linear ordinary differential equations (lodes) are quite prob- pression for the lodes of maximal symmetry only up to the ablythemostcommontypeofdifferentialequationsthat order eight. In fact the expression of the corresponding eight- occur in physics and in many other mathematically based order equation of maximal symmetry found in that paper is fields of science. However, their most important properties incorrect. such as their transformation properties, their general solu- A more direct algorithm for generating this class of equa- tions, and even their symmetry properties remain largely tions based on the simple fact that they are iterative was pro- unknown. posed recently in [3]. Nevertheless, some of the main results In a short paper published by Krause and Michel [1] in of the latter paper concerning in particular the generation and 1988 certain specific properties of lodes of maximal sym- the point transformations of the class of lodes of maximal metrywereestablished.Inparticular,thesaidpapershows symmetry still have room for improvements. that such equations are precisely the iterative ones, and equiv- Itisworthwhiletomentionthatinrecentyearsthe alentlythosewhichcanbereducedbyaninvertiblepoint study of ordinary differential equations (odes) of maximal (𝑛) transformation to the trivial equation 𝑦 =0,whichwe symmetry algebra has given rise to a considerable number of shall refer to as the canonical form. However that short paper research papers in the scientific literature. Examples of such left a number of important questions unanswered. It does not studies include the determination of various types of symme- provide, for instance, any expression for the point transfor- try subalgebras and some of their applications for systems of mation mapping a given equation of maximal symmetry to second-order odes [4, 5]. Momoniat and collaborators also the canonical form. studied the algebraic properties of first integrals of scalar Almostatthesametimetheproblemofgeneratinglodes second-order and third-order odes as well as for systems of maximal symmetry was considered by Mahomed and of second-order odes of maximal symmetry [6, 7]. On the Leach [2] who found an algorithm for obtaining expres- other hand a solution algorithm for second-order and third- sions for the most general normal form of these equations order odes of maximal symmetry based on Janet bases and based on the direct computation of the symmetry algebra. Loewy decompositions was obtained in [8, 9]. A similar study 2 Abstract and Applied Analysis of second-order lodes of maximal Lie point symmetry and Moreover, setting 𝑗=0or 𝑗=𝑛in (4) shows by induction on third-order lodes of maximal contact symmetry was carried 𝑛 that 󸀠󸀠 out in [10] for equations in canonical form (𝑦 =0and 0 𝑛 󸀠󸀠󸀠 𝐾 =𝑟, 𝑦 =0) to investigate amongst others the properties of their 𝑛 first integrals and exceptional symmetries. 𝐾𝑛 =Ψ𝑛−1 [𝑠] , (5) In this paper, we provide a much simpler differential 𝑛 operator than that found in [3] for generating linear iterative ∀𝑛 ≥ 1, equations of a general order. This gives rise to a simple algorithm for testing lodes for maximal symmetry based and applying (4) recursively and using the conventions set in solely on their coefficients. The operator thus found corrects (3) give the new recurrence relation the wrong one obtained in [2, Equations (3.20) and (3.21)] as 𝑛 𝐾𝑗 = ∑𝑟𝑛−𝑘Ψ𝐾𝑗−1, 𝑗=0,...,𝑛, 𝑛≥1. oneofthemainconclusionsofthatpaper.Ontheotherhand, 𝑛 𝑘−1 for (6) we give a more direct proof than that of Krause and Michel 𝑘=𝑗 [1] to the fact that lodes reducible by an invertible point (𝑛) transformation to the canonical form 𝑦 =0are precisely We note that (6) provides an algorithm for the computation 𝐾𝑗 𝑟 𝑠 those which are iterative. of the coefficients 𝑛 in terms of the parameters and of the Ψ We also establish several results concerning the solutions source equation and the operator , and the resulting formula of this class of equations and in particular their transforma- has effectively been obtained in [3, Theorem 2.2]. Moreover, 𝐾𝑗 tion to canonical form. In contrast to the very well-known it is of course also possible to compute 𝑛 directly in terms of paperbyErmakov[11]whomanagedtofindonlysomevery 𝑛 and the parameters 𝑟 and 𝑠,andfor𝑗=1,2one finds that specific cases from a restricted class for which the second- 𝑛 1 𝑛−1 󸀠 order source equation is solvable, we provide large families 𝐾𝑛 =𝑟 [𝑛𝑠 +( )𝑟 ], (7a) of second-order equations for which the general solution 2 is given by simple algebraic formulas. All such families are 𝑛 parameterized by an entirely arbitrary nonzero function and 𝐾2 =𝑟𝑛−2 [( )Ψ[𝑠] the general solutions thus found for the second-order source 𝑛 2 equation yield through a very simple quadratic formula that (7b) for the whole corresponding class of equations of maximal 𝑛 3𝑛 − 5 +( )(3𝑠𝑟󸀠 +𝑟𝑟󸀠󸀠 + 𝑟󸀠2)] . symmetry of a general order. 3 4 2. Iterations of Linear Equations If we divide through the general 𝑛th order linear iterative 𝑛 0 𝑛 equation Ψ [𝑦] = 0 in (2) by 𝐾𝑛 =𝑟,ittakestheform Let 𝑟 =0̸ and 𝑠 be two smooth functions of 𝑥, and consider Ψ = 𝑟(𝑑/𝑑𝑥) +𝑠 (𝑛) 1 (𝑛−1) 𝑗 (𝑛−𝑗) 𝑛 the differential operator .Weshalloften 0=𝑦 +𝐵𝑛𝑦 +⋅⋅⋅+𝐵𝑛𝑦 +⋅⋅⋅+𝐵𝑛𝑦, (8a) denote by 𝐹[𝑥1,...,𝑥𝑚] a differential function of the vari- 𝑗 ables 𝑥1,...,𝑥𝑚. Linear iterative equations are the iterations 𝑗 𝐾𝑛 𝑛 󸀠 𝐵 = . (8b) Ψ [𝑦] = 0 of the first-order lode Ψ[𝑦] ≡ 𝑟𝑦 +𝑠𝑦=0,given 𝑛 𝑟𝑛 by Itisclearthatthisequationrepresentsthestandardformof 𝑛 𝑛−1 0 the general linear iterative equation with leading coefficient Ψ [𝑦]=Ψ (Ψ [𝑦]) , for 𝑛≥1with Ψ =𝐼, (1) one. Moreover, the well-known change of the dependent vari- 𝑥 1 𝐼 able 𝑦 󳨃→𝑦exp((1/𝑛) ∫ 𝐵𝑛(𝑡)𝑑𝑡) maps (8a) into its normal where is the identity operator. A linear iterative equation of 𝑥0 a general order 𝑛 thus has the form form in which the coefficient of the term of second highest order has vanished. This transformation however simply 𝑛 0 (𝑛) 1 (𝑛−1) 2 (𝑛−2) 𝑛−1 󸀠 𝑟 𝑠 𝐵1 =0 𝐾1 =0 Ψ [𝑦]≡𝐾𝑛𝑦 +𝐾𝑛𝑦 +𝐾𝑛𝑦 +⋅⋅⋅+𝐾𝑛 𝑦 amounts to choosing and such that 𝑛 ;thatis, 𝑛 . (2) Therefore, for given parameters 𝑟 and 𝑠 of the operator Ψ,an +𝐾𝑛𝑦=0. 𝑛 𝑛th order lode in normal form (𝑛) 2 (𝑛−2) 𝑗 (𝑛−𝑗) 𝑛 Setting 𝑦 +𝐴𝑛𝑦 +⋅⋅⋅+𝐴𝑛𝑦 +⋅⋅⋅+𝐴𝑛𝑦=0 (9a)

𝑗 is iterative if and only if 𝐾 =0, for 𝑗<0or 𝑗>𝑚, 𝑚 󵄨 (3) 𝐾𝑗 󵄨 𝐾𝑗 =1, 𝑚=𝑗=0 𝐴𝑗 = 𝑛 󵄨 ,(2≤𝑗≤𝑛), 𝑚 for 𝑛 𝑛 󵄨 (9b) 𝑟 󵄨 1 󵄨𝐾𝑛 =0 𝑗 and applying (1) show that the coefficients 𝐾𝑛 in the general 𝑗 where 𝐾𝑛 is given by (6). It follows from (7a) that the require- expression (2) of an iterative equation satisfy the recurrence 1 ment that 𝐾 =0holdsisequivalenttohaving relation 𝑛 1 󸀠 𝑗 𝑗 j−1 𝑠=− (𝑛−1) 𝑟 , (10) 𝐾𝑛 =𝑟𝐾𝑛−1 +Ψ𝐾𝑛−1, for 0≤𝑗≤𝑛,𝑛≥1. (4) 2 Abstract and Applied Analysis 3

𝑗 𝑗 and this shows in particular that any iterative equation in nor- form (9a)-(9b) in which 𝐴𝑛 =𝐴𝑛[q] depends only on q mal form can be expressed in terms of the parameter 𝑟 alone; and its derivatives. More formally, the resulting differential that is, it depends on a single arbitrary function. Clearly, the operatorcanberepresentedas 𝑗 coefficients 𝐴 can also be expressed solely in terms of 𝑛, 𝑟, 𝑛 1 󵄨 󵄨 and the derivatives of 𝑟. For instance, by setting for any given Φ𝑟 = Ψ𝑛󵄨 =Φ󵄨 . 𝑛 𝑛 󵄨 𝑛󵄨𝑟(𝑗)=𝐹 [𝑟,q], 𝑗≥2 (15) 𝑟 󵄨𝐾1=0, 𝑟(𝑗)=𝐹 [𝑟,q], 𝑗≥2 𝑗 function 𝜉, 𝑛 𝑗 2 𝑛=3 4 Φ [𝑦] [𝜉󸀠 (𝑥)] −2𝜉(𝑥) 𝜉󸀠󸀠 (𝑥) For example, for or ,evaluating 𝑛 yields the A (𝜉)(𝑥) = , 𝑟 2 (11) following expressions directly in terms of and its derivatives 4[𝜉(𝑥)] alone. Φ [𝑦] it follows from (7b) that in (9a)-(9b) we have 3 2 𝑦(𝑟󸀠3 −2𝑟𝑟󸀠𝑟󸀠󸀠 +𝑟2𝑟(3)) 𝑦󸀠 (𝑟󸀠2 −2𝑟𝑟󸀠󸀠) 𝐴2 = A (𝑟) , =− + +𝑦(3), 𝑟3 𝑟2 𝑛+1 (12) 2 Φ [𝑦] and more generally 𝐴 =( ) A (𝑟) . 4 𝑛 3 3𝑦 (27𝑟󸀠4 −68𝑟𝑟󸀠2𝑟󸀠󸀠 +24𝑟2𝑟󸀠𝑟(3) +4𝑟2 (7𝑟󸀠󸀠2 −2𝑟𝑟(4))) (16) = 𝑗 4 In fact, as already noted in [2, 3], the coefficients 𝐴𝑛 depend 16𝑟 A(𝑟) =2 𝐴 only on the function 2 and its derivatives. For 5𝑦󸀠 (𝑟󸀠3 −2𝑟𝑟󸀠𝑟󸀠󸀠 +𝑟2𝑟(3)) 5(𝑟󸀠2 −2𝑟𝑟󸀠󸀠)𝑦󸀠󸀠 simplicity, it will often be convenient to denote the coefficient − + 2 𝑟3 2𝑟2 𝐴2 of the term of third highest order in (9a) simply by q. Having noted that the coefficients of every lode of +𝑦(4). maximal symmetry in normal form depend solely on q and its derivatives, an important problem considered in [2] was However, if in addition we also apply to these expressions Γ𝑛[𝑦] for Φ𝑛[𝑦] the substitution (14), which amounts to applying to find a linear ordinary differential operator depending 𝑟 solely on q and its derivatives, and which generates the most directly the operator Φ𝑛 to 𝑦 we obtain 𝑛 general form of the linear th order equation of maximal 𝑟 󸀠 󸀠 󸀠󸀠󸀠 symmetry in the dependent variable 𝑦=𝑦(𝑥).Inarecent Φ3 [𝑦]=2q 𝑦+4q𝑦 +𝑦 , (17a) paper [3], it was established that, for an arbitrary parameter 𝑟 Φ𝑟 [𝑦] =3𝑦(3q2 + q󸀠󸀠) + 10𝑦󸀠q󸀠 +10q𝑦󸀠󸀠 +𝑦(4). of the source equation, the operator 4 (17b) 1 󵄨 Φ = Ψ𝑛󵄨 Comparing this with the known expressions for lodes of 𝑛 𝑛 󵄨 (13) 𝑟 󵄨𝐾1=0 maximal symmetry expressed solely in terms of q and its 𝑛 𝑟 𝑟 derivatives [2, 3] shows that Φ3 and Φ4 yield indeed the generates the linear iterative equation of an arbitrary order indicated expressions. 𝑛 in normal form and in its most general form (9a)-(9b). Another important observation made in this paper is 2 Therefore, although the operator (13) and the equation it that if we set 𝑟=𝑢 for a certain nonzero function 𝑢,the 2 2 generates depend explicitly on 𝑟 (and not on q =𝐴2)andits expression for A(𝑟) = A(𝑢 ) in (11) is much simpler and 󸀠󸀠 derivatives, on the basis of the result of [1] identifying linear reduces to −𝑢 /𝑢.Settingq = A(𝑟) is thus equivalent to iterative equations with lodes of maximal symmetry, the two letting 𝑢 be a solution of the equation Γ𝑛[𝑦] = 0 Φ𝑛[𝑦] = 0 equations and should always be the 󸀠󸀠 same for all 𝑛≥3, although such an operator Γ𝑛[𝑦] has not 𝑦 + q𝑦=0, (18) yet been found. This is due in part to the fact that a general 𝑗 𝑗 expression for 𝐴 =𝐴[q] is not available for all 𝑛 and 𝑗. which is referred to as the second-order source equation for 𝑛 𝑛 Φ [𝑦] 𝑟 𝑢2 Nonetheless, we show here that it is naturally possible to make (9a)-(9b). Thus, if we express 𝑛 with replaced by ,the 𝑢(𝑗) 𝑟(𝑗) use of the differential operator Φ𝑛 to generate directly a lode substitution rule for similar to that given for in (14) of maximal symmetry of the general form (9a) in which the takes the much simpler form 𝑗 coefficients 𝐴 depend only on q and its derivatives. 𝑛 𝑢(𝑗) =𝐷𝑗−2 (−𝑢q) fl 𝐻 [𝑢, q] , 𝑗≥2. Indeed, for any value of 𝑛≥2it follows from (6) and (9a), 𝑥 𝑗 for (19) 𝑗≥2 (9b), (11), and (12) and a simple induction on that each 𝑢 𝑗 (𝑗) Θ Φ 𝑟 coefficient 𝐴 in (9a)-(9b) depends linearly on 𝑟 .Moreover, Denote by 𝑛 the operator 𝑛 in which is replaced with 𝑛 𝑢2 it follows from (11)-(12) that andtowhichthesubstitution(19)isapplied.Thatis, 󵄨 󸀠2 2 Θ𝑢 =Φ󵄨 . 𝑟 −4q𝑟 𝑛 𝑛󵄨𝑟=𝑢2,𝑢(𝑗)=𝐻 [𝑢,q], 𝑗≥2 (20) 𝑟(𝑗) =𝐷𝑗−2 ( ) fl 𝐹 [𝑟, q] , 𝑗≥2, 𝑗 𝑥 2𝑟 𝑗 for (14) 𝑢 In other words Θ𝑛 is just Φ𝑛 as in (13), in which the source 2 where 𝐷𝑥 =𝑑/𝑑𝑥. Therefore, applying the substitution (14) parameter 𝑟 has been replaced by 𝑢 andtowhichthesub- to Φ𝑛[𝑦] yields the desired equation, that is, the lode of the stitution (19) is then applied. Then for the same reasons that 4 Abstract and Applied Analysis

𝑟 Φ𝑛[𝑦, q] generates the most general form of linear iterative given functions of 𝑥. If this equation is indeed of maximal 𝑢 equations, Θ𝑛[𝑦, q] also does the same. However, in view of symmetry, then by (12) the coefficient q of the corresponding the much simpler substitution rule (19), the computational q =𝐹2/(𝑛+1 ) 𝑢 second-order source equation should satisfy 𝑛 3 . cost for generating these equations is much lower using Θ𝑛 L[𝑦] = 0 𝑟 Consequently, is of maximal symmetry if and only rather than Φ𝑛. Indeed for arbitrary values of q, generating if it coincides with the corresponding generated equation 𝑢 equations of maximal symmetry of order greater than ten Θ𝑛[𝑦, q]=0. 𝑢 has been up to now a very tedious task using the original Using the operator Θ𝑛, we shall give in the next section algorithm of [2], but it is now easy to generate such equations a much simpler and direct proof than that of [1] to the fact at much higher orders in most standard computers using that a lode is iterative if and only if it is reducible by a point computing systems such as mathematica. For instance, the transformation to the canonical form. To close this section, 15 (0) coefficient 𝐴15 of 𝑦=𝑦 which is the largest expression in we note that by Abel’s Identity the Wronskian of any two 𝑢 size in the equation Θ15[𝑦] = 0 is given by linearly independent solutions 𝑢 and V of the source equation (18) is a nonzero constant and will be normalized to one. It 15 6 󸀠 󸀠5 𝐴15 = 14 (2 (52022476800q q + 2132810240q should be noted that linearly independent solutions of the second-order equation (18) are not known for arbitrary values 5 (3) 󸀠3 (4) + 6656237568q q + 341232100q q of the coefficient q. There are of course many other obvious criteria that can + 1024q4 (73853676q󸀠q󸀠󸀠 + 286397q(5)) be used to identify expressions L[𝑦] which do not corre- +32q3 (2646561024q󸀠3 + 207959056q󸀠󸀠q(3) spond to lodes of maximal symmetry. For example, the coef- 𝑗 𝑗 ficients 𝐴𝑛 =𝐴𝑛[q] are in fact differential polynomials in q. + 123346720q󸀠q(4) + 184297q(7))+2q󸀠2 (863911980q󸀠󸀠q(3) This implies in particular that if, for instance, q is polynomial 𝑥 𝐴𝑗 (7) 2 󸀠2 (3) (in ), then all 𝑛 must be polynomials, and if it is a sine + 784597q )+8q (2094143648q q 𝑗 function, then the 𝐴𝑛 must all be polynomials of sine and + 33615550q(3)q(4) + 22652990q󸀠󸀠q(5) cosine functions.

+ q󸀠 (2835404512q󸀠󸀠2 + 9750858q(6)) + 7323q(9)) 3. Reduction to Canonical Form and 3 + 13 (457296 [q(3)] + q(5) (740970q󸀠󸀠2 + 2717q(6)) (21) General Solution + 2068q(4)q(7) +2q(3) (1100540q󸀠󸀠q(4) + 591q(8)) The equivalence group of linear 𝑛th order equations in nor- mal form (9a) is well known to be given by invertible point 󸀠󸀠 (9) 󸀠 󸀠󸀠3 (4) 2 + 491q q )+q (780095196q + 8478360 [q ] transformations of the form

+ 14231306q(3)q(5) + 8279146q󸀠󸀠q(6) + 1793q(10)) 𝑥=𝑓(𝑧) ,

󸀠3 󸀠󸀠 󸀠2 (5) +2q (12696730880q q + 151883460q q (𝑛−1)/2 (23) 𝑦=𝜆[𝑓󸀠 (𝑥)] 𝑤, + 2519682q(4)q(5) +2q(3) (257461378q󸀠󸀠2 + 913529q(6))

+ 937436q󸀠󸀠q(7) + 139q(11) where 𝑓 is an arbitrary locally invertible function and 𝜆 an arbitrary nonzero constant [9, 12, 13]. In other words, an equa- 2 +4q󸀠 (95070193 [q(3)] + 152580285q󸀠󸀠q(4) + 79587q(8)))) tion of the form (9a) is reducible by a point transformation to the canonical form if and only if there exists a transformation + q(13)). of the form (23) that maps such an equation to the canonical form. Of course, if one can generate a lode of maximal Let us denote by symmetry of a general order and in its most general form, then the identification of any given lode as a member or not (−3𝜉󸀠󸀠2 +2𝜉󸀠𝜉(3)) in the class of linear equations of maximal symmetry is also 𝑆 (𝜉)(𝑧) = (24) achieved. This is due to the fact that, in their normal form, 𝑛th 2𝜉󸀠2 order lodes (9a) of maximal symmetry are completely and 2 uniquely determined by the coefficients 𝐴 . Indeed, suppose 𝑛 the Schwarzian derivative of the function 𝜉=𝜉(𝑧).Bystud- that yingtheexpressionofthesourceparameterofthetrans- (𝑛) 2 (𝑛−2) 𝑗 (𝑛−𝑗) 𝑛 formed equation under equivalence transformations, a sim- L [𝑦] ≡ 𝑦 +𝐹𝑛 𝑦 +⋅⋅⋅+𝐹𝑛𝑦 +⋅⋅⋅+𝐹𝑛 𝑦 (22) ple characterization of the point transformation that maps =0 aniterativeequationtothecanonicalformwasfoundin[3, Theorem 4.3]. This result states that a point transformation is a given lode in normal form that we wish to test reduces a given iterative equation, which without loss of 𝑗 for the maximality of its symmetry algebra, where 𝐹𝑛 are generality may be assumed to be of the form (9a), to the Abstract and Applied Analysis 5

(𝑛) canonical form 𝑤 (𝑧) = 0 ifandonlyifitisoftheform(23), Itshouldfirstbenotedthattherighthandsideof(26a) where 𝑓 is the inverse of the function 𝑧=ℎ(𝑥)satisfying gives other methods for generating linear iterative equations with source parameter 𝑟. In practice, however, linear iterative 2 1 equations arise generally with no reference to any source 𝐴2 (𝑥) = 𝑆 (ℎ)(𝑥) , 2 parameter but are expressed solely in terms of the coefficient (25) 2 1 q =𝐴2(𝑥) and its derivatives. In this case, on the basis of A (𝑟)(𝑥) = 𝑆 (ℎ)(𝑥) , ℎ or equivalently, 2 a remark made in the previous section, a solution to the equation q = (1/2)𝑆(ℎ)(𝑥) in (25) is simply given by 2 on account of (12), assuming that 𝑟 is the source parameter ℎ = ∫(𝑑𝑥/𝑢 ),where𝑢 is a solution of the second-order 𝑢 oftheequation.Bymakinguseofthisresultof[3],wecan source equation (18). Consequently, if we denote by Ω𝑛 the now provide a more direct proof than that given in [1] for the transformation operator given by following result. 𝑑𝑥 Ω𝑢 [∫ ,𝜆,𝑧,𝑤] Theorem 1. A linear ordinary differential equation is iterative 𝑛 𝑢2 if and only if it can be reduced to the canonical form by an 󵄨 (27) 𝑑𝑥 󵄨 invertible point transformation. =𝐾 (𝜆, 𝑢2)Ω [∫ ,𝜆,𝑧,𝑤]󵄨 , 𝑛 𝑛 2 󵄨 𝑢 󵄨 (𝑗) 󵄨𝑢 =𝐻𝑗[𝑢,q], 𝑗≥2

Proof. Asusualonemayassumethattheiterativeequationis 𝑗−2 𝑢 in its normal form and has source parameter 𝑟.Onthebasisof where 𝐻𝑗[𝑢, q]=𝐷𝑥 (−𝑢q) as in (19), then Θ𝑛[𝑦] and 𝑢 (𝑛) the above stated result from Theorem 4.3 of [3], it follows that Ω𝑛 ⋅𝑤 (𝑧) generate exactly the same linear iterative equation the transformation (23) where 𝑓 is the inverse of the function expressed solely in terms of q and its derivatives. ℎ = ∫(𝑑𝑥/𝑟) maps the iterative equation to its canonical On the other hand, since the transformation operator ℎ 𝑢 2 form, as the latter expression for solves (25). More directly, Ω𝑛[∫(𝑑𝑥/𝑢 ), 𝜆, 𝑧, 𝑤] maps the trivial equation to the most one can prove that the class of iterative equations and that of general form (9a) of the iterative equation, its explicit expres- equations reducible by an invertible point transformation to sioncanbeusedtoderivethegeneralsolutionof(9a).Indeed, the canonical form are the same. Indeed, let Ω𝑛[𝑓,𝜆,𝑥,𝑦]≡ under this operator we have Ω𝑛[𝑓,𝜆] represent the element of the equivalence group of 1 𝑛−1 linear 𝑛th order odes Δ≡Δ𝑛[𝑦] = 0 in normal form, 𝑦= 𝑢 𝑤, acting on the space of independent variable 𝑥 and dependent 𝜆 (28) variable 𝑦, and given by (23). Denoting by Ω𝑛[𝑓,𝜆] ⋅ Δ =0 𝑑𝑥 𝑧=∫ . the transformed equation, it follows that we have exactly 𝑢2 𝑑𝑥 𝑛 Φ [𝑦]=𝐾 (𝜆,) 𝑟 Ω [∫ ,𝜆,𝑧,𝑤]⋅𝑤(𝑛) (𝑧) , Consequently, linearly independent solutions to the general 𝑛 𝑛 𝑛 𝑟 (26a) 𝑛th order iterative equation (9a) are given by 𝑑𝑥 𝑘 where 𝑦 =𝑢𝑛−1 (∫ ) , 𝑘=0,...,𝑛−1. (29) 𝑘 𝑢2 1 𝐾𝑛 (𝜆,) 𝑟 = . (𝜆 [𝑟 (𝑥)](𝑛+1)/2) (26b) In particular, if 𝑢 and V are two linearly independent solutions 2 ofthesourceequation(18),then∫(𝑑𝑥/𝑢 )=V/𝑢.Con- 𝑢 V 𝑛 In other words, generating a linear iterative equation in 𝑦= sequently, in terms of and ,the linearly independent 𝑦(𝑥) for any given value of 𝑟 using the differential operator solutions to (9a) in (29) above can be rewritten as Φ 𝑤(𝑛)(𝑧) = 0 𝑛−1−𝑘 𝑘 𝑛 and transforming the canonical equation in 𝑦𝑘 =𝑢 V , 𝑘=0,...,𝑛−1. (30) the new variables 𝑥 and 𝑦 using the point transformation operator 𝐾𝑛(𝜆, 𝑟)Ω𝑛[∫(𝑑𝑥/𝑟),𝜆,𝑧,𝑤]yield exactly the same Formula(30)iswellknownandwascitedwithoutproof equations, and this proves the result. 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 In [1] Krause and Michel proved this theorem indirectly found formula (28) only in this paper. Moreover, formula (29) as a consequence of equivalence relations, by proving the established above provides a much simpler result by showing equivalence between equations of maximal symmetry and that linearly independent solutions of (9a) can be expressed iterative equations on one hand and between equations solely in terms of a single nonzero solution of the second- reducible by invertible point transformations and equations order source equation. Indeed, the solutions 𝑦𝑘 in (29) are of maximal symmetry on the other hand. The part in that clearly linearly independent as their Wronskian equals the 𝑛−1 proof stating that an ode (linear or nonlinear) is reducible to nonzero constant ∏𝑗=1 𝑗!. thecanonicalformifandonlyifithasmaximalsymmetrywas howeveralreadyprovedbyLie[14–16].Thedirectproofgiven Theorem 2. A linear ordinary differential equation is iterative here will allow us to construct various point transformations if and only if it has 𝑛 linearly independent solutions 𝑦𝑘 of the of practical importance for iterative equations as well as form (30), where 𝑢 and V are two linearly independent solutions various forms of their general solutions. of the corresponding second-order source equation (18). 6 Abstract and Applied Analysis

Proof. The fact that a linear iterative equation has linearly The results of this paper extend the scope of solvability of independent solutions of the stated form is established linear second-order equations obtained by Ermakov, Hill, and in (30). Conversely, if a lode has 𝑛 linearly independent others such as Loewy [21] not only to more larger families of solutions of the form (30), then since the second-order the coefficient function q in (18), but moreover to all lodes of 󸀠󸀠 source equation is reducible to the canonical form 𝑤 = maximal symmetry of a general order. This statement holds 0 by a point transformation, without loss of generality we provided that the parameter 𝑟 of the source equation (18) may assume that such a transformation reduces 𝑢 to 1 and V is known. Indeed, it follows from (29) and the condition 2 to 𝑧. Consequently, the corresponding linearly independent 𝑟=𝑢 relatingtheparameterofthesourceequationandthe solutions of the lode are polynomials of degree at most 𝑛− solution of the corresponding second-order equation, that for 1, and thus the transformed equation is in canonical form. arbitrary values 𝑟=𝑟(𝑥)of the source parameter, two linearly It then clearly follows from Theorem 1 that the equation is independent solutions of the second-order equation of the iterative. form 󸀠󸀠 Recall that the 𝑚th symmetric power of a lode Δ[𝑦] = 0 𝑦 + A (𝑟) 𝑦=0 (32a) is the lode Ω[𝑦] = 0 of minimal order such that for every set of 𝑚 solutions 𝑦1,...,𝑦𝑚 of Δ[𝑦] = 0 the product 𝑦1 ...𝑦𝑚 are given by is a solution of Ω[𝑦] =.Moreover,itiswellknownthat 0 𝑑𝑥 𝑗 the 𝑚th symmetric power of a second-order lode has order 𝑦𝑗 = √𝑟(∫ ) , for 𝑗 = 0, 1. (32b) 𝑚+1(see [17] and the references therein). It thus follows 𝑟 𝑚≥1 (𝑚 + 1) from Theorem 2 that for an th order lode is 𝑛 of maximal symmetry if and only if it is the symmetric power Therefore, linearly independent solutions of the corre- sponding (𝑛−2)thiterationof(32a)canalsobeobtainedfrom of a second-order lode. It has in fact been established in [17, 2 (29) through the substitution 𝑟=𝑢.Moredirectlyinterms Theorems 1 and 2] that the symmetric power of a lode is an 𝑟 iterative lode. of , we have the following result which follows immediately from (32a)-(32b) and (30).

4. Solvability of Equations of Theorem 3. Let 𝑟 be a given nonzero function considered as Maximal Symmetry the source parameter of the differential operator generating an 𝑛th order lode Δ 𝑛[𝑦] = 0 of maximal symmetry and in The results obtained thus far in this paper will be used in normal form. Then 𝑛 linearly independent solutions 𝑠𝑘 of this section to show amongst others that lodes of maximal Δ 𝑛[𝑦] = 0 are given by symmetry are highly solvable and that it is indeed the case 𝑘 for the second-order source equation (18) whose solutions 𝑛−1 𝑑𝑥 𝑠 =(√𝑟) (∫ ) , 𝑘=0,...,𝑛−1. (33) completely determine those of the corresponding lodes of 𝑘 𝑟 maximal symmetry. In a very popular paper published in Russian in 1880 and recently translated into English [11], On one hand and on the basis of the results obtained in Ermakov stated that the majority of second-order linear the previous section, it follows from Theorem 3 that not only homogeneous odes for which it is possible to find conditions can we generate a lode of maximal symmetry of any order for their solvability are of the form and for any nonzero function 𝑟, but also the general solution of any such equation is given in closed form in terms of the 2 󸀠󸀠 󸀠 (𝛼1𝑥 +𝛼2𝑥+𝛼3)𝑦 +(𝛼4𝑥+𝛼5)𝑦 +𝛼6𝑦=0, (31) source parameter 𝑟 by (33). On the other hand, for any given lode of maximal symmetry, as long as its corresponding where the 𝛼𝑗 are some arbitrary constants. He then moved on source parameter 𝑟 is known, its general solution is given also in the paper to obtain some very specific cases of equations by (33). which are solvable from this class, together with their general Now, to make a more direct comparison with the class of solutions. equations (31) considered by Ermakov, we note that linearly A couple of years later in the same decade Hill [18] independent solutions can also be found for the standard considered in the study of lunar stability the most general form of (32a) which in terms of the arbitrary coefficient 𝐵= 󸀠 form of the second-order lode in normal form, that is, in 𝐵(𝑥) of 𝑦 takes the form the form (18), but in which the coefficient q is a periodic 󸀠󸀠 󸀠 1 2 󸀠 function. Hill’s equation and its important variants such as 𝑦 +𝐵𝑦 + (4A (𝑟) +𝐵 +2𝐵) 𝑦=0. (34) Meissner Equation and Mathieu Equation have been amply 4 studied [19], and it is well known that its solutions and their The two linearly independent solutions of (34) are then given related properties can be described by means of Floquet by theory. These solutions can also be expressed in terms of Hill’s determinant [20]. It should be mentioned that, in the context 𝑑𝑥 𝑗 𝑦 = √𝑟(∫ ) 𝑒−(1/2) ∫ 𝐵(𝑥)𝑑𝑥,𝑗=0,1. (35) of the present paper, a differential equation is called solvable 𝑗 𝑟 if the closed form expression of its general solution is known, regardless of the type of functions in terms of which this We note that as opposed to the very specific cases of solvable solution is expressed. equations found by Ermakov from the restricted class of Abstract and Applied Analysis 7 equations (31), not only does (34) depend intrinsically on one where Lclm stands for least common left multiple [22] and arbitrary function, but also its general solution is given by the the coefficients 𝑎1 and 𝑎2 given by simple formula (35) in terms of the source parameter 𝑟. 2 1 We further clarify the application of Theorem 3 by an 𝑎1 =2+ − , example. 𝑥 𝑥+3/2 (38c) 2 2𝑥 − 2 Example 4. Let 𝛼 be a nonzero real number and consider the 𝑎 = − 2 𝑥 𝑥2 −2𝑥+3/2 particular case where the source equation (18) is given by 𝛼2 +[ (𝑥𝛼)]2 are rational nonequivalent solutions of the Riccati equation 𝑦󸀠󸀠 + q𝑦=0, q = ln . with 2 (36) 4𝑥2 [ln (𝑥𝛼)] 1 4 𝑎󸀠 −𝑎2 +(2+ )𝑎+ =0. 2 (39) Then it is not obvious how to solve (36) using standard meth- 𝑥 𝑥 ods. However, the latter coefficient q is an expression of the It then follows from a result of Loewy [22, Lemma 2.4] that a form A(𝑟) as defined in (11) and (12) and more precisely with 𝛼 fundamental set of solutions of (38a) is given by 𝑟=𝑥ln(𝑥 ). Therefore, since we know the source parameter 𝑟 of the differential operator generating (36), it follows that 2 4 1 𝑦 = − + , an explicit expression of its general solution is given in closed 1 3 3𝑥 𝑥2 form by (32b). Moreover, it also follows from Theorem 3 (40) 𝑒−2𝑥 (3+2𝑥) that the closed form expression of the general solution of 𝑦 = . any equation of arbitrary order 𝑛 generated by the same 2 𝑥2 differential operator with source parameter 𝑟 is given by (33). Given that every second-order lode is of maximal symmetry, In practice, a lode occurs however with no reference to tosolve(38a)withthemethodproposedinthispaper,we a source parameter of the differential operator that generated first reduce it to its normal form (although from (34) this it, but rather in terms of its coefficients. As already indicated is not necessary), through the usual transformation given in anylodecanalwaysbeassumedtobeofthenormalform Section 2. The reduced equation takes the simpler form (9a)-(9b), and even in such a case the results of Theorem 3 [15 + 4𝑥 (1+𝑥)] canalsobeusedtofindthegeneralsolutionoflodesof 0=𝑦󸀠󸀠 + q (𝑥) 𝑦, q (𝑥) =− . with 2 (41) maximal symmetry, in virtually all known solvable cases, in 4𝑥 addition to those cases which can be solved only with the The nonlinear equation (37) for finding the source parameter proposed method. It follows from Theorem 3 that to find 𝑟 reduces to A(𝑟) = q(𝑥),whereq(𝑥) is given here by (41). It the general solution it suffices to find the parameter 𝑟 of the has general solution source equation and by (12) this amounts to solving for 𝑟 the nonlinear second-order equation. −2𝑥 2𝑥 2𝑥 2𝑥 2 2 𝑒 (3𝑒 −4𝑒 𝑥+2𝑒 𝑥 +6𝑘1 +4𝑥𝑘1) 𝑘2 𝑛+1 𝑟= (42) 2 𝑥3 𝐴 =( ) A (𝑟) , (37) 𝑛 3 for some constants of integration 𝑘1 and 𝑘2. Therefore, by 2 where 𝐴𝑛 is the coefficient appearing in (9a)-(9b) while A(𝑟) (32b) linearly independent solutions of (41) are given by is given by (11). 𝑗 This approach for solving linear differential equations 𝑒𝑥 (3+2𝑥(−2 + 𝑥)) 𝑒−2𝑥 (3+2𝑥) 𝑠 = [− ] , has some similarities with the Loewy decomposition method 𝑗 3 √𝑥 8 (3+2𝑥(−2 + 𝑥)) (43) [21, 22] in which the solution of a linear differential equation is known once a factorization of the differential operator 𝑗=0,1. generating the equation has been computed. In this approach, finding the coefficients of each factor in the factorization is The inverse of the transformation mapping (38a)–(38c) to also achieved for second- and third-order lodes with rational (41) then shows that a fundamental system of solutions of coefficients by solving certain Riccati equations. We illustrate (38a)–(38c) is given by the comparison of these two methods by considering a very 3+2𝑥(−2 + 𝑥) simple example. 𝑦 = , 1 𝑥2 (44) Example 5. Consider the second-order equation 𝑒−2𝑥 (3+2𝑥) 1 𝑦 = , 0=𝑦󸀠󸀠 +(2+ )𝑦󸀠 +𝑦, 2 𝑥2 𝑥 (38a) whose solution by Loewy decomposition is discussed in [22]. which is the same as that obtained in (40). This equation has Loewy decomposition The drawback with these two methods for solving lodes 𝑑 𝑑 0= ( +𝑎, +𝑎)[𝑦]=0, is that they require the solutions of nonlinear odes which, in Lclm 𝑑𝑥 2 𝑑𝑥 1 (38b) general, are more difficult to solve than the original lode. 8 Abstract and Applied Analysis

However, for the method proposed in this paper, equation quite possible to describe all possible values of the arbitrary (37) for finding 𝑟 turns out to be linearizable and the coefficient function q by an appropriate choice of a source linearized equation might be much easier to solve than the parameter 𝑟.Thisinturnwouldthenmeanthatalllinearodes original equation. of maximal symmetry are exactly solvable. As far as odes are concerned, the Loewy decomposition method is designed in principle for lodes of all orders. How- Competing Interests ever, that method does not seem to be accessible because its scope is quite limited, due in particular to the fact that the The author declares that there is no conflict of interests implementation of its algorithm involves a very high compu- regarding the publication of this article. tational cost. The Loewy decomposition theory has also quite a large and very specific vocabulary of its own taken from References differential algebra. Moreover, there are too many types of Loewy decompositions for a given linear differential operator, [1] J. Krause and L. Michel, “Equations´ differentielles´ lineaires´ and there are in particular 12 such types for third-order linear d’ordre 𝑛>2ayant une algebre` de Lie de symetrie´ de dimension operators alone. On the other hand the Loewy decomposition 𝑛+4,” ComptesRendusdel’Academie´ des Sciences,vol.307,no. method is limited only to equations with rational coefficients 18, pp. 905–910, 1988. for which the solutions of the corresponding (nonlinear) [2]F.M.MahomedandP.G.L.Leach,“SymmetryLiealgebrasof Riccati equations for finding the coefficients of the factorized nth order ordinary differential equations,” Journal of Mathemat- operator are available. ical Analysis and Applications,vol.151,no.1,pp.80–107,1990. Of course the method proposed in this paper for find- [3] J.-C. Ndogmo and F. M. Mahomed, “On certain properties of ing solutions of lodes is limited to the case of equations linear iterative equations,” CentralEuropeanJournalofMathe- of maximal symmetry. The determining nonlinear second- matics,vol.12,no.4,pp.648–657,2014. order equation (37) for finding the source parameter 𝑟 is [4] R. Campoamor-Stursberg, “An alternative approach to systems 𝐴2 of second-order ordinary differential equations with maximal however the same for equations of all orders since 𝑛 is sl(𝑛 + 2, R) an arbitrary coefficient in (37), and it is also linearizable as symmetry. Realizations by special functions,” Com- munications in Nonlinear Science and Numerical Simulation,vol. already indicated. Moreover, as Theorem 3 shows, it is easily 37, pp. 200–211, 2016. applicable to equations of all orders as long as (37) can be 𝑟 [5] R. Campoamor-Stursberg and J. Gueron,´ “Linearizing systems solved for . of second-order ODEs via symmetry generators spanning a Nevertheless, it should be emphasized that the most im- simple subalgebra,” Acta Applicandae Mathematicae,vol.127, portant result obtained in this section is not a method for pp. 105–115, 2013. solving a given lode. It is rather the fact that to any arbitrary [6] K. S. Mahomed and E. Momoniat, “Algebraic properties of first 𝑟 smooth function we have associated a lode of maximal integrals for scalar linear third-order ODEs of maximal symme- symmetry and arbitrary order and for which the closed form try,” Abstract and Applied Analysis, vol. 2013, Article ID 530365, solution is given by (33). This shows that such equations are 8pages,2013. highly solvable, because traditionally most solvable lodes are [7] K. S. Mahomed and E. Momoniat, “Symmetry classification of limited to those having rational coefficients. first integrals for scalar linearizable second-order ODEs,” Jour- nal of Applied Mathematics,vol.2012,ArticleID847086,14 5. Concluding Remarks pages, 2012. [8] F. Schwarz, “Solving second order ordinary differential equa- In this paper, we found a cost effective algorithm for gener- tions with maximal symmetry group,” Computing,vol.62,no.1, ating lodes of maximal symmetry. This algorithm corrects pp. 1–10, 1999. the wrong one proposed in [2], as quoted in the paper. We [9] F. Schwarz, “Equivalence classes, symmetries and solutions of have also shown how such an algorithm can be used to easily linear third-order differential equations,” Computing,vol.69, identify linear odes of maximal symmetry and proved some no. 2, pp. 141–162, 2002. fundamental results concerning the reduction of odes of [10] P. G. Leach, R. R. Warne, N. Caister, V. Naicker, and N. Euler, maximal symmetry to canonical form. We also found general “Symmetries, integrals and solutions of ordinary differential and optimal expressions for their general solutions. equations of maximal symmetry,” Indian Academy of Sciences. Clearly,thesourceequation(18)ishardlysolvablefor Proceedings. Mathematical Sciences, vol. 120, no. 1, pp. 113–130, 2010. arbitrary values of the corresponding coefficient q.Wehave [11] V.P. Ermakov, “Second-order differential equations: conditions however shown that for every given nonzero smooth function of complete integrability,” Applicable Analysis and Discrete 𝑟,wherethefunction𝑟 may include in particular any kind 𝑛 Mathematics,vol.2,no.2,pp.123–145,2008,Translatedfrom of special function, the general solution of any th order the 1880 Russian original by A. O. Harin, and edited by P. G. L. lode of maximal symmetry in normal form generated by the Leach. 𝑟 differential operator having as coefficient is given by the [12] J. C. Ndogmo, “Equivalence transformations of the Euler- closed form expression (33). Bernoulli equation,” Nonlinear Analysis: Real World Applica- Ontheotherhand,theinverseproblemoffinding𝑟 for tions,vol.13,no.5,pp.2172–2177,2012. agivenvalueofq turns out to be equivalent to solving a [13] J. C. Ndogmo, “Some results on equivalence groups,” Journal different lode of the form (18), because the nonlinear second- of Applied Mathematics, vol. 2012, Article ID 484805, 11 pages, order ode A(𝑟) = q is linearizable. Nevertheless, it seems 2012. Abstract and Applied Analysis 9

[14] S. Lie, “Klassification und Integration von gewohnlichen¨ Dif- ferentialgleichungen zwischen x; y; die eine Gruppe von Trans- formationen gestetten. I,” Mathematische Annalen,vol.22,pp. 213–253, 1888. [15] P. J. Olver, “Differential invariants and invariant differential equations,” Lie Groups and Their Applications,vol.1,no.1,pp. 177–192, 1994. [16] P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK, 1995. [17] M. Bronstein, T.Mulders, and J.-A. Weil, “On symmetric powers of differential operators,” in Proceedings of the International SymposiumonSymbolicandAlgebraicComputation(ISSAC’97), pp.156–163,ACM,1997. [18] G.W.Hill,“Onthepartofthemotionofthelunarperigeewhich is a function of the mean motions of the sun and moon,” Acta Mathematica,vol.8,no.1,pp.1–36,1886. [19] G. Teschl, Ordinary Differential Equations and Dynamical Sys- tems, American Mathematical Society, Providence, RI, USA, 2012. [20] W. Magnus and S. Winkler, Hill’s Equation,JohnWiley&Sons, New York, NY, USA, 1966. [21] A. Loewy, “Uber¨ vollstandig¨ reduzible lineare homogene Dif- ferentialgleichungen,” Mathematische Annalen,vol.62,no.1,pp. 89–117, 1906. [22] F. Schwarz, “Loewy decomposition of linear differential equa- tions,” Bulletin of Mathematical Sciences,vol.3,no.1,pp.19–71, 2013. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Submit your manuscripts at http://www.hindawi.com

International Journal of Advances in Combinatorics Mathematical Physics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

International Journal of Journal of Mathematics and Mathematical Discrete Mathematics Sciences

Journal of International Journal of Journal of Function Spaces Stochastic Analysis Optimization Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014