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 () transformation to the trivial equation =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 =0or =in (4) shows by induction on third-order lodes of maximal contact symmetry was carried that out in [10] for equations in canonical form ( =0and 0 =, =0) to investigate amongst others the properties of their first integrals and exceptional symmetries. =Ψ−1 [] , (5) In this paper, we provide a much simpler differential operator than that found in [3] for generating linear iterative ∀ ≥ 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 = ∑−Ψ−1, =0,...,, ≥1. oneofthemainconclusionsofthatpaper.Ontheotherhand, −1 for (6) 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 =0are precisely We note that (6) provides an algorithm for the computation those which are iterative. of the coefficients 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, tion to canonical form. In contrast to the very well-known it is of course also possible to compute directly in terms of paperbyErmakov[11]whomanagedtofindonlysomevery and the parameters and ,andfor=1,2one finds that specific cases from a restricted class for which the second- 1 −1 order source equation is solvable, we provide large families = [ +( ) ], (7a) of second-order equations for which the general solution 2 is given by simple algebraic formulas. All such families are parameterized by an entirely arbitrary nonzero function and 2 =−2 [( )Ψ[] the general solutions thus found for the second-order source 2 equation yield through a very simple quadratic formula that (7b) for the whole corresponding class of equations of maximal 3 − 5 +( )(3 + + 2)] . symmetry of a general order. 3 4 2. Iterations of Linear Equations If we divide through the general th order linear iterative 0 equation Ψ [] = 0 in (2) by =,ittakestheform Let =0̸ and be two smooth functions of , and consider Ψ = (/) + () 1 (−1) (−) the differential operator .Weshalloften 0= + +⋅⋅⋅+ +⋅⋅⋅+, (8a) denote by [1,...,] a differential function of the vari- ables 1,...,. Linear iterative equations are the iterations = . (8b) Ψ [] = 0 of the first-order lode Ψ[] ≡ +=0,given by Itisclearthatthisequationrepresentsthestandardformof −1 0 the general linear iterative equation with leading coefficient Ψ []=Ψ (Ψ []) , for ≥1with Ψ =, (1) one. Moreover, the well-known change of the dependent vari- 1 able →exp((1/) ∫ ()) maps (8a) into its normal where is the identity operator. A linear iterative equation of 0 a general order thus has the form form in which the coefficient of the term of second highest order has vanished. This transformation however simply 0 () 1 (−1) 2 (−2) −1 1 =0 1 =0 Ψ []≡ + + +⋅⋅⋅+ amounts to choosing and such that ;thatis, . (2) Therefore, for given parameters and of the operator Ψ,an +=0. th order lode in normal form () 2 (−2) (−) Setting + +⋅⋅⋅+ +⋅⋅⋅+=0 (9a) is iterative if and only if =0, for <0or >, (3) =1, ==0 = ,(2≤≤), for (9b) 1 =0 and applying (1) show that the coefficients in the general where is given by (6). It follows from (7a) that the require- expression (2) of an iterative equation satisfy the recurrence 1 ment that =0holdsisequivalenttohaving relation 1 j−1 =− (−1) , (10) =−1 +Ψ−1, for 0≤≤,≥1. (4) 2 Abstract and Applied Analysis 3 and this shows in particular that any iterative equation in nor- form (9a)-(9b) in which =[q] depends only on q mal form can be expressed in terms of the parameter alone; and its derivatives. More formally, the resulting differential that is, it depends on a single arbitrary function. Clearly, the operatorcanberepresentedas coefficients can also be expressed solely in terms of , , 1 and the derivatives of . For instance, by setting for any given Φ = Ψ =Φ . ()= [,q], ≥2 (15) 1=0, ()= [,q], ≥2 function , 2 =3 4 Φ [] [ ()] −2() () For example, for or ,evaluating yields the A ()() = , 2 (11) following expressions directly in terms of and its derivatives 4[()] alone. Φ [] it follows from (7b) that in (9a)-(9b) we have 3 2 (3 −2 +2(3)) (2 −2) 2 = A () , =− + +(3), 3 2 +1 (12) 2 Φ [] and more generally =( ) A () . 4 3 3 (274 −682 +242(3) +42 (72 −2(4))) (16) = 4 In fact, as already noted in [2, 3], the coefficients depend 16 A() = 2 only on the function 2 and its derivatives. For 5 (3 −2 +2(3)) 5(2 −2) simplicity, it will often be convenient to denote the coefficient − + 2 3 22 2 of the term of third highest order in (9a) simply by q. Having noted that the coefficients of every lode of +(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 Γ[] for Φ[] the substitution (14), which amounts to applying to find a linear ordinary differential operator depending solely on q and its derivatives, and which generates the most directly the operator Φ to we obtain general form of the linear th order equation of maximal symmetry in the dependent variable =().Inarecent Φ3 []=2q +4q + , (17a) paper [3], it was established that, for an arbitrary parameter Φ [] =3(3q2 + q) + 10q +10q +(4). of the source equation, the operator 4 (17b) 1 Φ = Ψ Comparing this with the known expressions for lodes of (13) 1=0 maximal symmetry expressed solely in terms of q and its derivatives [2, 3] shows that Φ3 and Φ4 yield indeed the generates the linear iterative equation of an arbitrary order indicated expressions.