Received: 27 June 2023 Revised: 14 October 2023 Accepted: 21 February 2024 DOI: 10.1002/mma.10025 R E S E A R C H A R T I C L E Symmetry algebra classification of scalar nth-order ordinary differential equations Said Waqas Shah1 F. M. Mahomed2 H. Azad1 1Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan 2DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa Correspondence Said Waqas Shah, Abdus Salam School of Mathematical Sciences GC University, Lahore. 68-B, New Muslim Town, Lahore 54600, Pakistan. Email: waqas.shah@sms.edu.pk Communicated by: C. Qu We obtain a complete classification of scalar nth-order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general n. The n = 1, 2, 3 cases are well-known in the literature. Further, it is known that there are three types of nth-order equations depending upon the point symme- try algebra they possess, namely, first-order equations which admit an infinite dimensional Lie algebra of point symmetries, second-order equations possessing the maximum 8-point symmetries, and higher-order, n ≥ 3, admitting the maxi- mum n+ 4 dimensional point symmetry algebra. We show that scalar nth-order equations for n > 5 do not admit maximally an n+3 dimensional real Lie algebra of point symmetries. Moreover, we prove that for n > 4, equations can admit two types of n+2 dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Fur- thermore, we present the types of maximal real n dimensional and higher than n-dimensional point symmetry algebras admissible for equations of order n ≥ 4 and their canonical forms. The types of lower-dimensional point symmetry alge- bras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems. KEYWORDS invariants, Lie symmetry classification of ODEs, symmetry Lie algebras MSC CLASSIFICATION 34C14, 34C20 1 INTRODUCTION We obtain a complete classification of scalar nth-order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general n. The precise results are given in Section 7 as theorems (7.1 to 7.5). Symmetry Lie algebras of scalar nth-order ordinary differential equations (ODEs) have been extensively studied over several years since the initial ground breaking works of Lie [1–3]. Lie [3], inter alia, provided all continuous groups of transformations in the complex plane. He emphasized that this can form the basis of classification as well as reduction of scalar nth-order ODEs which he implicitly performed. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2024 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd. Math. Meth. Appl. Sci. 2024;47:8449–8470. wileyonlinelibrary.com/journal/mma 8449 https://doi.org/10.1002/mma.10025 https://orcid.org/0000-0002-3015-7231 https://orcid.org/0000-0002-6995-5820 http://crossmark.crossref.org/dialog/?doi=10.1002%2Fmma.10025&domain=pdf&date_stamp=2024-04-08 WAQAS SHAH ET AL. The classification of Lie algebras in terms of vector fields is essential in the algebraic study of scalar nth-order ODEs which possess infinitesimal symmetries, both symmetry classification and reduction algorithms. After the works of Lie, there have been much interest in this area. Lie algebras of vector fields in the real plane are completely classified in González-López et al. [4]. Recently, a proof of Lie's classification of solvable Lie algebras of vector fields in the plane is presented (see Azad et al. [5]). Of essence also is the contribution [6] which refers to contemporary works on Lie algebra realizations as well as equivalence of realizations. In recent years, there have been much focus on the Lie algebra classification of ODEs in several works as we refer to henceforth. Equations of order 1 have infinite Lie point symmetries and are equivalent to each other via point transforma- tion. For scalar higher-order ODEs, Lie [1] proved that the maximum dimension of the point symmetry Lie algebra for a scalar second-order ODE is eighth-dimensional and occurs for linear and linearizable by point transformation equations. Lie [2] obtained a complex classification of second-order ODEs in terms of their point symmetry Lie algebras. Mahomed and Leach [7] derived the real classification and showed that a second-order equation can admit 0, 1, 2, 3, or the maximum 8 dimension real point symmetry Lie algebra. The original Lie classification and the classification in the real domain [7] for second-order ODEs are, inter alia, compared in Ibragimov and Mahomed [8]. Algebraic linearizability criteria were initiated by Lie himself [1], who showed that such second-order equations possessing a Lie algebra of dimension 2 and of rank 1 are linearizable via point transformations. This falls under Types II and IV Lie canonical forms in Lie's classifi- cation. The Types I and III cases with focus on linearizability were achieved in Sarlet et al. [9] and Mahomed and Leach [10]. The reader is also referred to the survey [11]. In the study of scalar linear ODEs of order n, n ≥ 3, Mahomed and Leach [12] (see also earlier studies [8, 11] as well as the contribution by Krause and Michel [13]) demonstrated that the point symmetry algebra can be n + 1,n + 2 or n + 4. Thus, for n ≥ 3, scalar linear ODEs are not necessarily equivalent to each other via point transformation. Moreover, for n ≥ 3, there exist linear as well as nonlinearizable ODEs with n + 2 and n + 3 symmetry algebras [12]. It is important to remark that second-order ODEs are quite different to higher-order equations n ≥ 3 as per the point symmetry algebras they admit. Apart from the maximum dimension of the point symmetry algebra being 2+6 for second-order ODEs and that for higher-order, n ≥ 3, equations n + 4 (see Lie [1, 2]), there are two more notable differences. Secondly, that all linear second-order ODEs are equivalent to the free particle equation, whereas a linear higher-order n ≥ 3 ODE has three equivalence classes depending upon whether it has n + 1,n + 2 or the maximum number n + 4 of point symmetries [12]. Thirdly, the complete or full algebra of point symmetries of a second-order ODE is a subalgebra of its maximum algebra sl(3,R), whereas the full algebra of a higher-order n ≥ 3 ODE is not necessarily a subalgebra of its maximum Lie point symmetry algebra [12]. The point symmetry Lie algebra classification of third-order ODEs as well as linearization by point transformation have been investigated in a number of relevant publications (see earlier studies [12–22]). Furthermore, integrability and reductions for third-order ODEs were investigated in Ibragimov and Nucci [23]. Scalar fourth-order ODEs were considered in recent works from the point of view of Lie point symmetry classification in terms of four-dimensional algebras, canonical forms, as well as integrability (see previous research [24–26]). A complete Lie point symmetry classification of fourth-order ODEs and algebraic linearization were also attempted [27]. Linearization criteria, by point transformation, for such ODEs, were found as well [28]. The classification of scalar nth-order ODEs which possess nontrivial irreducible contact Lie symmetry algebras was completed in the work of Wafo Soh et al [29]. For scalar third-order ODEs, the contact symmetry algebra is a subalgebra of the 10-dimensional contact symmetry algebra of 𝑦′′′ = 0 except for linear equations that admit four- and five-dimensional point symmetry algebras [29]. It is shown in this work that fourth-order scalar ODEs do not admit irreducible contact symmetry algebras. Further, it was proved in this paper [29] that there are three types of contact symmetry algebras (of dimensions 6, 7, and 10) admissible for nth-order ODEs for n > 4, up to local contact transformations. The present work is a natural generalization of previous contributions on the classification of scalar second- and third-order ODEs. Here we present the complete classification of nth-order, n ≥ 4, ODEs according to the Lie point sym- metry algebras they admit. Previously, for order n > 3, only linearizability of nth-order ODEs was performed into three classes [12]. We explicitly present the new maximal n dimensional point symmetry Lie algebras admissible and their new representative or canonical equations as well as higher-dimensional point symmetry algebras and new canonical ODEs. We also have shown how one can obtain lower-dimensional point symmetry algebras and determine the corresponding ODEs. All these results are stated as main theorems in Section 7. Furthermore, scalar higher-order ODEs arise in various applications, including non-Newtonian fluids, when the governing high-order PDEs are reduced to ODEs by similarity transformations, as well as in Euler–Bernoulli beam theory. The algebraic analysis presented here enables researchers in these and allied areas to search for Lie symmetry reductions and solutions of the scalar higher-order ODEs. 8450 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Section 2 deals with notation which are well-known in books (see, e.g., earlier studies [30–34]) which we have utilized in Shah et al. [27] as well as provide an overview of the methods used herein. In Sections 3 and 4, we classify all equations of order n which admit n+1 and n+2 dimensional Lie symmetry algebras, respectively, including those that are linearizable by point transformation. Then in Section 5, we find all the fundamental invariants of order n and n−1 for five and higher dimensions and present the representative ODEs as well. In Section 6, we discuss admissibility for n + 3 dimensional algebras. Thereafter, in Section 7, we obtain a complete classification for n ≥ 4 for the maximal n, lower as well as higher symmetry algebra cases. We finally present concluding remarks. 2 NOTATION AND METHODOLOGY We utilize the vector fields in the plane presented in the important work [4] for the real Lie symmetry classification of scalar nth-order ODEs performed in this seminal study. By (m,n), we denote the real algebra realization, where m is the type of algebra in González-López et al. [4] and n is the dimension of the Lie algebra. One writes a general vector field or generator in the real plane as Xi = 𝜉i(x, 𝑦)𝜕x + 𝜂i(x, 𝑦)𝜕𝑦, i = 1, … ,n. Here (x, 𝑦) ∈ R 2 and 𝜕x denotes 𝜕∕𝜕x. Note also that n is the dimension of a Lie algebra of which Xis are the generators. Other notations will be introduced as they arise in the sequel. Now let L be an m dimensional Lie subalgebra of vector fields in the real plane. To find an invariant equation of order n, we as usual consider the normal form of an nth-order ODE 𝑦(n) = H ( x, 𝑦, 𝑦′, 𝑦′′, … , 𝑦(n−1)) . (2.1) For a generator X to be a symmetry of (2.1), the symmetry condition X [n] (𝑦(n) − H ) |𝑦(n)=H = 0 (2.2) must hold on the equation, where X [n] denotes the nth prolongation. If condition (2.2) is satisfied for every Xi, i = 1, 2, … ,m, then the resulting nth-order equation is said to be invariant under L and L is called the symmetry Lie algebra of the equation. Further, let L be an m dimensional Lie subalgebra of vector fields defined on a subspace D ⊂ R 2. Then the nth-order prolonged Lie algebra is defined on a subspace D(n) ⊂ R n+2. Suppose that r0 is the rank of L in D. Then rn will be the rank of the prolonged L in D (n). The rank here means the rank of the matrix whose rows are coefficients of the m generating vector fields of L. If we denote dn to be the number of differential invariants of order n, then we have dn = n + 2 − rn, n ≥ 0. Example 2.1. Consider simply X = 𝜕x. Here n = 0 and r0 = 1. Thus, d0 = 0 + 2 − 1 = 1, so we have one zeroth-order differential invariant which is u = 𝑦. Example 2.2. Now we take X1 = 𝜕x, X2 = 𝜕𝑦. Again, n = 0, ro = 2, and d0 = 0+2−2 = 0, and there is no zeroth-order invariant. Now for the first prolongations of X1 and X2, n = 1, r1 = 2, and d1 = 1 + 2 − 2 = 1. Thus, there is a first- order differential invariant u = 𝑦′. We now mention invariant differentiation and its operator. If u, v are invariants, then by Lie's theorem, Dxv∕Dxu is also an invariant. This process is also called invariant differentiation (see, e.g., Ibragimov [31]). Recall that Dx is the total differentiation operator. We can write this as Dxv Dxu = (Dxu)−1Dxv = 𝜆Dxv. We call 𝜆Dx =  the invariant differentiation operator once we know 𝜆. 8451 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Suppose we have an unknown 𝜆(x, 𝑦, 𝑦′, … , 𝑦(n)). We require 𝜆Dxv to be invariant, and therefore, we need X [n](𝜆Dxv) = 0. (2.3) If we refer to X [n] as X by ignoring the prolongation sign, then we can simply write X = X + 𝜉Dx, where X = w𝜕𝑦 + Dxw𝜕𝑦′ + D2 xw𝜕𝑦′′ + ... + Dn x w𝜕𝑦(n) is called the canonical operator and w = 𝜂 − 𝑦′𝜉. Equation (2.3) then becomes ( X + 𝜉Dx ) (𝜆Dxv) = 0 ⇒ 𝜉Dx(𝜆Dxv) + X(𝜆)Dxv + 𝜆X(Dxv) = 0 ⇒ 𝜉Dx𝜆Dxv + 𝜉𝜆D2 xv + (X𝜆 − 𝜉Dx𝜆)Dxv + 𝜆X(Dxv) = 0 ⇒ 𝜉Dx𝜆Dxv + 𝜆𝜉D2 xv + X(𝜆)Dxv − 𝜉Dx𝜆Dxv + 𝜆X(Dxv) = 0 ⇒ 𝜉𝜆D2 xv + X(𝜆)Dxv + 𝜆Dx(Xv − 𝜉Dxv) = 0 ⇒ (X(𝜆) − 𝜆Dx𝜉)Dxv = 0. This yields the known result (see Ibragimov [31]) X(𝜆) = 𝜆Dx𝜉. (2.4) Hence, 𝜆 satisfies the nonhomogenous linear PDE (2.4). One only requires one nontrivial solution for 𝜆 which can be a constant as well. Example 2.3. Let X1 = 𝜕x and X2 = 𝜕𝑦. We know that u = 𝑦′ is a first-order differential invariant of X1 and X2. Applying condition (2.4), we have X1𝜆 = 0,X2𝜆 = 0. This clearly shows that we can set 𝜆 = 1, and therefore, the invariant differentiation operator can be taken as  = (1)Dx = Dx. Thus, 𝑦′ = Dx𝑦 ′ = 𝑦′′ is a second-order differential invariant. We briefly consider Lie determinants. Definition 2.1. Consider an m-dimensional Lie subalgebra of vector fields whose generators are given as Xk = 𝜉k𝜕x + 𝜂k𝜕𝑦, k = 1, 2, ...,m. Then the determinant of the following matrix M ⎛⎜⎜⎜⎝ 𝜉1 𝜂1 𝜂 [1] 1 𝜂 [2] 1 … 𝜂 [m−2] 1 𝜉2 𝜂2 𝜂 [1] 2 𝜂 [2] 2 … 𝜂 [m−2] 2 ⋮ ⋮ ⋱ ⋮ 𝜉m 𝜂m 𝜂 [1] m 𝜂 [2] m … 𝜂 [m−2] m ⎞⎟⎟⎟⎠ is called the Lie determinant which corresponds to the m dimensional Lie algebra for m ≥ 2. We denote the Lie determinant by ΛL. Lie proved that for an m dimensional Lie subalgebra of vector fields L, the Lie determinant gives rise to all the invariant equations of order ≤ m − 2, [2]. Similarly, it can be noticed that the rank of the prolonged algebra is m, that is, maximal unless the Lie determinant vanishes in which case the rank of L diminishes. Here the algebra is prolonged up to order m− 2. These invariant equations are called the singular invariant equations of L. The fundamental differential invariants which are not singular must be of order m − 1 and m and the higher-order differential invariants can then be found from the fundamental invariants. 8452 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Note here that once we find the (m − 1)th-order differential invariant say 𝜙 and the invariant differentiation operator  = 𝜆Dx, we can determine the mth-order differential invariant as 𝜙. We commence our classification for n+1 dimensions below, in Section 3, and then n+2 dimensions in Section 4. These include linearization. Thereafter, in Section 5, we find all the fundamental invariants for five and higher dimensions and provide the canonical scalar ODEs. Then we proceed to n+3 dimensions, Section 6, which have a paucity of three classes. Finally, in Section 7, we obtain the complete Lie symmetry classification of scalar ODEs of any higher-order n > 3 and encapsulate the main results in the stated theorems. 3 n + 1 DIMENSIONAL ALGEBRAS AND ODEs 3.1 Nonlinear equations It is easy to observe that (20,n + 1), r = n − 1,n > 2 ∶ X1 = 𝜕∕𝜕𝑦,X2 = x𝜕∕𝜕𝑦,X3 = 𝜉1(x)𝜕∕𝜕𝑦, … ,Xn−1 = 𝜉n−1(x)𝜕∕𝜕𝑦, where 𝜉i(x)s are not linear in x and are linearly independent arbitrary functions, is not admitted by an equation as there can only be n solution symmetries. The linearizable case (21,n + 1) is considered in subsection 3.2. Also, (22,n + 1) and (23,n+1) are not possessed as maximal Lie algebras. We consider the other n+1 dimensional algebras. These are as below. (24,n + 1), r = n − 2,n ≥ 3 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x + 𝛼𝑦𝜕𝑦,X4 = x𝜕𝑦,X5 = x2𝜕𝑦, … ,Xn+1 = xn−2𝜕𝑦. The generators, except X3, imply that an nth-order equation of the form (2.1) admitting these generators must be of the form 𝑦(n) = H ( 𝑦(n−1)) .The nth prolongation of X3 is as follows: x𝜕x + 𝛼𝑦𝜕𝑦 + (𝛼−1)𝑦′𝜕𝑦′+, … ,+(𝛼−n)𝑦(n)𝜕𝑦(n) . By applying this to the resultant equation and solving, we find the general form of an nth-order equation admitting this algebra to be 𝑦(n) = K ( 𝑦(n−1)) 𝛼−n 𝛼−n+1 , K ≠ 0, (3.1) where K ≠ 0 is an arbitrary constant and 𝛼 ≠ n − 1. For K = 0 or 𝛼 = n − 1, it is easy to see that the general form of such an equation admitting this algebra must be 𝑦(n) = 0, and we know from Lie that this simplest equation admits the maximal n + 4 dimensional algebra of which such an algebra is a subalgebra. (25,n + 1), r = n − 1,n ≥ 2 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕𝑦,X4 = x2𝜕𝑦, ...,Xn = xn−2𝜕𝑦,Xn+1 = x𝜕x + (r𝑦 + xr)𝜕𝑦. The nth prolongation of Xn+1 is as follows: x𝜕x+(r𝑦+xr)𝜕𝑦+(rxr−1+(r−1)𝑦′)𝜕𝑦′+...+((r(r−1)...(r−n+1))xr−n+(r−n)𝑦(n))𝜕𝑦(n) . The invariant equation is 𝑦(n) = K exp ( −𝑦(n−1) (n − 1)! ) , K ≠ 0. (3.2) (26,n + 1), r = n − 3,n ≥ 4 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x,X4 = 𝑦𝜕𝑦,X5 = x𝜕𝑦, … ,Xn+1 = xn−3𝜕𝑦. Here the equation turns out to be (K is constant) 𝑦(n) = K ( 𝑦(n−1))2 𝑦(n−2) , K ≠ 0,n∕(n − 1). (3.3) If K = n∕(n−1), then there is one more symmetry X = x2𝜕x+(n−3)x𝑦𝜕𝑦 as discussed in Section 4 as the type (28, n+2). (27,n + 1), r = n − 3,n ≥ 4 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = 2x𝜕x + r𝑦𝜕𝑦,X4 = x2𝜕x + rx𝑦𝜕𝑦,X5 = x𝜕𝑦, … ,Xn+1 = xn−3𝜕𝑦. The nth prolongation of X3 and X4 is as follows: X3+ ∑n k=1(r−2k)𝑦(k)𝜕𝑦(k) and X4+ ∑n k=1(k(r−k+1)𝑦(k−1) +x(r−2k)𝑦(k))𝜕𝑦(k) , respectively. The equation is (K is constant) 𝑦(n) = n n − 1 ( 𝑦(n−1))2 𝑦(n−2) + K ( 𝑦(n−2)) n+3 n−1 , K ≠ 0. (3.4) 8453 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. If K = 0, then there is one more symmetry X = 𝑦𝜕𝑦 as pursued in Section 4. (28,n + 1), r = n − 4,n ≥ 5 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x,X4 = 𝑦𝜕𝑦,X5 = x2𝜕x + (n − 4)x𝑦𝜕𝑦,X6 = x𝜕𝑦, … ,Xn+1 = xn−4𝜕𝑦. The nth-order invariant equation is 𝑦(n) = ( 𝑦(n−2))3( 𝑦(n−3))−2 [ n(3(n − 2)K1 − 2n + 2) (n − 2)2 + K((n − 2)K1 − (n − 1)) 3 2 ] , (3.5) where K1 = 𝑦(n−3)𝑦(n−1)(𝑦(n−2))−2 and K constant. 3.2 Linearizable equations Here we consider linearization for higher-order n ≥ 3 equations. We demonstrate how one obtains the linear form. The reader is referred to Mahomed and Leach [12] for details. We have the algebra with realization (21,n + 1), r = n − 1: X1 = 𝜕𝑦, X2 = 𝑦𝜕𝑦, X3 = 𝜉1(x)𝜕𝑦, ..., Xn+1 = 𝜉r(x)𝜕𝑦 which can be simplified by introducing coordinates: x = 𝜉1(x), 𝑦 = 𝑦. Ignoring the bars, the generators of this algebra can be transformed to X1 = 𝜕𝑦, X2 = 𝑦𝜕𝑦, X3 = x𝜕𝑦, X4 = 𝜉2(x)𝜕𝑦,..., Xn+1 = 𝜉r(x)𝜕𝑦. Thus, we consider (21,n + 1), r = n − 2 ∶ X1 = 𝜕𝑦,X2 = 𝑦𝜕𝑦,X3 = x𝜕𝑦,X4 = 𝜉1(x)𝜕𝑦, … ,Xn+1 = 𝜉r(x)𝜕𝑦. We have the result from Mahomed and Leach [12] which we state as follows. Proposition 3.1 (see Mahomed & Leach [12]). (21,n + 1) is the symmetry algebra of the nth, n ≥ 3, order linear homogenous equation 𝑦(n) = n−2∑ i=2 Ai(x)𝑦(i+1), (3.6) such that each 𝜉i for i = 1, 2, … ,n − 2, form independent solutions of this equation, that is, the 𝜉is satisfy the system of homogenous equations 𝜉 (n) k = n−2∑ i=1 Ai(x)𝜉(i+1) k , k = 1, ...,n − 2. (3.7) Note that if Ais are constant, then a further symmetry 𝜕x arises and one gets the algebra (23, n + 2) as discussed in the next Section 4.2. Furthermore, if the Ais satisfy conditions (3.20) and (3.21) in Mahomed and Leach [12], then one has the maximal algebra (28, n + 4) admitted as also looked at in Section 4.2. 4 n + 2 DIMENSIONAL ALGEBRAS AND REPRESENTATIVE ODEs 4.1 Nonlinear equations For the n + 2 dimensional algebras, (20,n + 2), (21,n + 2), and (22,n + 2) are clearly not admissible algebras. The linearization case (23,n + 2) is looked at in Section 4.2. (24,n + 2), r = n − 1,n ≥ 2 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x + 𝛼𝑦𝜕𝑦,X4 = x𝜕𝑦,X5 = x2𝜕𝑦, … ,Xn+2 = xn−1𝜕𝑦. ΛL = 1 · 2! · 3!...(n − 1)!(𝛼 − n)𝑦(n), where ΛL here and in what follows is the Lie determinant of Definition 2.1. (25,n + 2), r = n,n ≥ 1 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕𝑦,X4 = x2𝜕𝑦, … ,Xn+1 = xn−1𝜕𝑦,Xn+2 = x𝜕x + (r𝑦 + xr)𝜕𝑦. ΛL = 1 · 2! · 3!....(n − 2)! · (n − 1)! · n! 8454 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. (26,n + 2), r = n − 2,n ≥ 3 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x,X4 = 𝑦𝜕𝑦,X5 = x𝜕𝑦, … ,Xn+2 = xn−2𝜕𝑦. ΛL = 1 · 2! · 3!...(n − 3)! · (n − 2)!𝑦(n−1)𝑦(n). (27,n + 2), r = n − 2,n ≥ 3 ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = 2x𝜕x + r𝑦𝜕𝑦,X4 = x2𝜕x + rx𝑦𝜕𝑦,X5 = x𝜕𝑦, … ,Xn+2 = xn−2𝜕𝑦. ΛL = 1 · 2! · 3!...(n − 2)! · n2(𝑦(n−1))2 . (28,n + 2), r = n − 3,n ≥ 4 ∶ X1 = 𝜕x,X2 = x𝜕x,X3 = 𝑦𝜕𝑦,X4 = x2𝜕x + rx𝑦𝜕𝑦,X5 = 𝜕𝑦,X6 = x𝜕𝑦, … ,Xn+2 = xn−3𝜕𝑦. The equation is 𝑦(n) = n n − 1 ( 𝑦(n−1))2 𝑦(n−2) . (4.1) Here we have only one case, the last, which constitute an invariant equation with maximal n + 2 dimensional real symmetry algebra. The rest do not form an equation or are singular equations with the maximal n+ 4 dimension algebra. Next, we need to again deal with linearization. We also mention (28, n + 4) as it results in linearization. This is stated at the end of Section 4.2. 4.2 Linearizable equations for n + 2 symmetries We obtain the form for the reduced linear equation that is a consequence of linearizability (see Mahomed & Leach [12]). (23,n + 2), r = n ∶ X1 = 𝜂1(x)𝜕𝑦,X2 = 𝜂2(x)𝜕𝑦, … ,Xn = 𝜂n(x)𝜕𝑦,Xn+1 = 𝑦𝜕𝑦,Xn+2 = 𝜕x. We sate the following proposition (see Mahomed and Leach [12] for details). Proposition 4.1 (see Mahomed & Leach [12]). The generators given in (23,n + 2) form a Lie symmetry algebra of the homogenous constant coefficient equation 𝑦(n) = n−1∑ i=0 Ai𝑦 (i), (4.2) if the 𝜂i, i = 1, … ,n, form a fundamental set of solutions of the constant coefficient equation itself. We conclude by considering the following algebra which is maximal. (28,n + 4), r = n − 1,n ≥ 3 ∶ X1 = 𝜕x,X2 = x𝜕x,X3 = 𝑦𝜕𝑦,X4 = x2𝜕x + rx𝑦𝜕𝑦,X5 = 𝜕𝑦,X6 = x𝜕𝑦, … ,Xn+4 = xn−1𝜕𝑦. This algebra has as representative, the simplest, nth, n ≥ 3, order ODE 𝑦(n) = 0, (4.3) which admits the maximal n + 4 dimensional algebra as is well-known from the landmark works of Lie. Further, if (4.2) has Ais satisfying the conditions (3.20) and (3.21) as in Mahomed and Leach [12], then the maximal algebra (28, n + 4) is admitted by the equation. 5 n DIMENSIONAL ALGEBRAS AND CORRESPONDING EQUATIONS In this section, we determine all the fundamental invariants (invariants of order n and n − 1) for five and higher dimen- sions as well as present the representative ODEs. Note that the algebras (20,n) and (21,n) are not admissible as maximal symmetry algebras. (5, 5): The generators of this algebra are X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x − 𝑦𝜕𝑦, X4 = 𝑦𝜕x, X5 = x𝜕𝑦. 8455 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. This is a five-dimensional algebra; hence, r = 5. We find the third prolongations of the generators of this algebra. Then the Lie determinant, which is the determinant of the matrix M = ⎛⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 0 1 0 0 0 0 x 1 0 0 x −𝑦 −2𝑦′ −3𝑦′′ −4𝑦′′′ 𝑦 0 −𝑦′2 −3𝑦′𝑦′′ − ( 3𝑦′′2 + 4𝑦′𝑦′′′ ) , ⎞⎟⎟⎟⎟⎟⎠ is ΛL = 9𝑦′′3. This shows that 𝑦′′ = 0 is the only equation of order ≤ 3 invariant under this algebra. The fundamental differential invariants can be found in the usual way. For the given algebra, we already know a fourth-order invariant 𝜙1 = 3𝑦(4)𝑦′′ −5 3 − 5𝑦′′′2𝑦′′−8∕3 . Thus, a fourth-order ODE 𝜙1 = 3K, K a constant, has (5,5). To find a fifth-order differential invariant, we can solve the determining system of linear partial differential equations X [5] i ( 𝑦(5) − H ) |eq = 0, for i = 1, 2, … , 5. The solution easily results in the fifth-order equation 𝑦(5) = −40 9 𝑦′′′ 3 𝑦′′ −2 + 5𝑦′′′𝑦(4)𝑦′′−1 + 𝑦′′ 2H(𝜙1) (5.1) invariant under this algebra, where 𝜙1 is the fourth-order invariant of the algebra and H is an arbitrary function of its argument. We can thus write the fifth-order differential invariant to be 𝜙2 = ( 𝑦′′ 2 𝑦(5) + 40 9 𝑦′′′ 3 − 5𝑦′′𝑦′′′𝑦(4) ) ∕𝑦′′4. Here 𝜙1 and 𝜙2 are the fundamental differential invariants of this algebra. All higher-order differential invariants can be deduced by Lie's invariant differentiation 𝜙n+1 = Dx(𝜙n)∕Dx(𝜙n−1) or by using the invariant derivative operator  = 𝑦′′ −1∕3Dx so that higher-order equations possessing this algebra are n−4𝜙1 = H ( 𝜙1, … ,n−5𝜙1 ) , n ≥ 5, (5.2) where Dx is the total derivative operator. (15, 5) ∶ X1 = 𝜕x,X2 = 𝜕𝑦,X3 = x𝜕x,X4 = 𝑦𝜕𝑦,X5 = x2𝜕x. The generators X1 to X3 and X5 give rise to the third-order invariant K1 = 𝑦′ −3 𝑦′′′ − 3 2 𝑦′ −4 𝑦′′ 2 , and the invariant differentiation operator is 𝑦′−1Dx. Now writing X4 in terms of K1, K2 = 𝑦′ −1DxK1 and K3 = 𝑦′ −1DxK2 results in X̃4 = −2K1𝜕∕𝜕K1 − 3K2𝜕∕𝜕K2 − 4K3𝜕∕𝜕K3. This provides the invariants 𝜙1 = K2K−3∕2 1 , 𝜙2 = 𝑦′ −1K−2 1 DxK2. We can evaluate K2 as K2 = 𝑦′ −4 𝑦(4) − 6𝑦′−5 𝑦′′𝑦′′′ + 6𝑦′−6 𝑦′′ 3 . 8456 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Also, similarly DxK2. Thus, one has the fundamental invariants 𝜙1 = 𝑦′ 2 𝑦(4) − 6 ( 𝑦′𝑦′′𝑦′′′ − 𝑦′′ 3 ) ( 𝑦′𝑦′′′ − 3 2 𝑦′′2 ) 3 2 and 𝜙2 = 𝑦′ 3 𝑦(5) − 10𝑦′2𝑦′′𝑦(4) − 6𝑦′2𝑦′′′2 + 48𝑦′𝑦′′2𝑦′′′ − 36𝑦′′4( 𝑦′𝑦′′′ − 3 2 𝑦′′2 )2 with the fifth-order equation as 𝑦′ 3 𝑦(5) − 10𝑦′2𝑦′′𝑦(4) − 6𝑦′2𝑦′′′2 + 48𝑦′𝑦′′2𝑦′′′ − 36𝑦′′4( 𝑦′𝑦′′′ − 3 2 𝑦′′2 )2 = H(𝜙1), (5.3) and with  = 𝑦′ −1K−1∕2 1 Dx, we can invoke (5.2) for higher-order invariant equations. Note that a fourth-order ODE admitting (15,5) is 𝜙1 = K, K constant. (6, 6): X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x, X4 = 𝑦𝜕x, X5 = x𝜕𝑦, X6 = 𝑦𝜕𝑦. The generators X1, X2, X5, and X6 result in the invariants K1 = 3𝑦′′−1 𝑦(4) − 5𝑦′′′2𝑦′′−2 , K2 = 9𝑦′′−1 𝑦(5) − 45𝑦′′−2 𝑦′′′𝑦(4) + 40𝑦′′−3 𝑦′′′ 3 . Writing X4 in terms of K1 and K2, we find (up to scaling) X̃4 = 2K1𝜕K1 + 3K2𝜕K2 which provides the fifth-order invariant of (6, 6) 𝜙1 = K−3∕2 1 K2, = 9𝑦′′2𝑦(5) − 45𝑦′′𝑦′′′𝑦(4) + 40𝑦′′′3( 3𝑦′′𝑦(4) − 5𝑦′′′2 ) 3 2 . The invariant differentiation operator is  = K−1∕2 1 Dx, and hence, one can derive the sixth-order invariant K−3 1 ( − 3 2 K2DxK1 + K1DxK2 ) . The fifth-order invariant equation is K−3∕2 1 K2 = 9𝑦′′2𝑦(5) − 45𝑦′′𝑦′′′𝑦(4) + 40𝑦′′′3 (3𝑦′′𝑦(4) − 5𝑦′′′2) 3 2 = K, (5.4) where K is constant, and with  = 𝑦′′(3𝑦′′𝑦(4) − 5𝑦′′′2)−1∕2Dx, we have n−5𝜙1 = H(𝜙1, … ,n−6𝜙1), n ≥ 6, (5.5) for the sixth- and higher-order equations. Thus, a sixth-order invariant ODE is K−3 1 ( −3 2 K2DxK1 + K1DxK2 ) = H ( K−3∕2 1 K2 ) . (5.6) Note that K1 = 0 is the fourth-order singular invariant equation having this algebra. 8457 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. (16, 6): X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x, X4 = 𝑦𝜕𝑦, X5 = x2𝜕x, X6 = 𝑦2𝜕𝑦. Here the invariants, using X1, X2, X4 and X6, are K1 = 𝑦′′′𝑦−1 − 3 2 𝑦′ −2 𝑦′′ 2 , as well as K2 = DxK1 and K3 = DxK2. Utilizing X5 in (K1,K2,K3) space, we end up with, up to scaling, X̃5 = 2xK1𝜕K1 + (3xK2 + 2K1)𝜕K2 + (4xK3 + 5K2)𝜕K3 . Hence, a fifth-order invariant transpires as𝜙1 = K−3 1 ( 5(DxK1)2 − 4K1D2 xK1 ) and invariant fifth-order ODE is (K constant) K−3 1 ( 5(DxK1)2 − 4K1D2 xK1 ) = K. (5.7) The invariant differentiation operator is  = K−1∕2 1 Dx, and one can deduce the sixth-order invariant. Higher-order invariant equations are obtained as in (5.2). The sixth-order invariant ODE is therefore K−1∕2 1 Dx ( K−3 1 ( 5(DxK1)2 − 4K1D2 xK1 )) = H ( K−3 1 ( 5(DxK1)2 − 4K1D2 xK1 ) . (5.8) We remark that K1 = 0 is the third-order singular invariant equation admitting this algebra. (7, 6): X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x + 𝑦𝜕𝑦, X4 = 𝑦𝜕x − x𝜕𝑦, X5 = (x2 − 𝑦2)𝜕x + 2x𝑦𝜕𝑦, X6 = 2x𝑦𝜕x + (𝑦2 − x2)𝜕𝑦. The generators X1 to X4 give rise to the invariant K1 = ( 1 + 𝑦′ 2 ) 𝑦′′ −2 𝑦′′′ − 3𝑦′, as well as the fourth- and fifth-order invariants K2 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK1, K3 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK2. Now X5 in (K1,K2,K3) space yields X̃5 = ( 1 + 𝑦′ 2 ) 𝑦′′ −1 [−4𝑦′K1𝜕K1 + ( 4𝑦′K2 1 − 6𝑦′K2 − 4K1 ) 𝜕K2 + ( 14𝑦′K1K2 + 4𝑦′K1 − 4𝑦′K3 1 − 8𝑦′K3 − 10K2 + 8K2 1 ) 𝜕K3 ] . The invariant deduced is of order 5 (which is admitted by X6 as well since [X4,X5] = X6) given by 𝜙1 = K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 ) . The fifth-order invariant ODE hence is K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 ) = K. (5.9) The operator of invariant differentiation is  = ( 1 + 𝑦′ 2 ) 𝑦′′ −1K−1∕2 1 Dx and the sixth- and higher-order equations are given by (5.5). One has the sixth-order invariant ODE( 1 + 𝑦′ 2 ) 𝑦′′ −1K−1∕2 1 Dx ( K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 )) = H(K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 ) . (5.10) Here, K1 = 0 is the third-order singular invariant equation. (8, 8): X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕𝑦, X4 = 𝑦𝜕𝑦, X5 = 𝑦𝜕x, X6 = x𝜕x, X7 = x2𝜕x + x𝑦𝜕𝑦, X8 = x𝑦𝜕x + 𝑦2𝜕𝑦. 8458 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. The operators X1 to X6 result in the fifth-order invariant of the algebra (6,6), namely, J1 = K−3∕2 1 K2, where K1 and K2 are as in (6,6). The operator X7 in terms of J1 and its invariant derivatives J2 = K−1∕2 1 DxJ1 and J3 = K−1∕2 1 DxJ2 turns out to be X̃7 = − 9K−1 1 𝑦′′ −1 𝑦′′′J1𝜕∕𝜕J1 + ( −12K−1 1 J2𝑦 ′′−1 𝑦′′′ + 3J2 1 K−1 1 𝑦′′ −1 𝑦′′′ − 3J1K−1∕2 1 ) 𝜕∕𝜕J2 + ( 10J1J2K−1 1 𝑦′′ −1 𝑦′′′ − J3 1 K−1 1 𝑦′′ −1 𝑦′′′ + J1K−1 1 𝑦′′ −1 𝑦′′′ − 15J3K−1 1 𝑦′′ −1 𝑦′′′ − 7J2K−1∕2 1 + 3 2 J2 1 K−1∕2 1 ) 𝜕∕𝜕J3 which yields the invariant of order 7 of this algebra (8,8) (note [X5,X7] = X8) as 𝜙1 = 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 , and thus, the seventh-order invariant ODE is (K is a constant) 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 = K. (5.11) The invariant differential operator of this algebra is  = K−1∕3 2 Dx. Therefore, eighth-order ODEs admitting this eight dimensional algebra is given by K−1∕3 2 Dx ( 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 ) = H(𝜙1) (5.12) and higher-order by invariant differentiation as n−7𝜙1 = H ( 𝜙1, … ,n−8𝜙1 ) , n ≥ 8. For this algebra, the singular invariant equation is K2 = 0 which is fifth-order. This is also given in Section 6. (22,n) ∶ r = n − 1,n ≥ 2 ∶ X1 = 𝜕x,X2 = 𝜂1(x)𝜕𝑦, … ,Xn = 𝜂n−1(x)𝜕𝑦. The nth-order equation is 𝑦(n) + a1𝑦 (n−1) + · · · + an−1𝑦 ′ = H ( 𝑦(n−1) + a1𝑦 (n−2) + · · · + an−1𝑦 ) , (5.13) where the ais are constants and the 𝜂is satisfy the linear (n − 1)th-order equation 𝜂 (n−1) i + a1𝜂 (n−2) i + · · · + an−1𝜂i = 0, i = 1, … ,n − 1. (23,n) ∶ r = n − 2, n ≥ 3: X1 = 𝜕x, X2 = 𝑦𝜕𝑦, X3 = 𝜂1(x)𝜕𝑦, … ,Xn = 𝜂n−2(x)𝜕𝑦. The nth-order equation that admits this algebra is D2 x(𝑦(n−2) + a1𝑦 (n−3) + · · · + an−2𝑦) 𝑦(n−2) + a1𝑦(n−3) + · · · + an−2𝑦 = H ( Dx ln |𝑦(n−2) + a1𝑦 (n−3) + · · · + an−2𝑦|) , (5.14) where the ais are constants and the 𝜂is satisfy the linear (n − 2)th-order equation 𝜂 (n−2) i + a1𝜂 (n−3) i + · · · + an−2𝜂i = 0, i = 1, … ,n − 2. (24,n) ∶ r = n − 3, n ≥ 4: X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x + 𝛼𝑦𝜕𝑦, X4 = x𝜕𝑦,..., Xn = xn−3𝜕𝑦. Fundamental invariants, if 𝛼 ≠ n − 1, are 𝜙1 = ( 𝑦(n−1))𝛼−n+2( 𝑦(n−2))−(𝛼−n+1) , 𝜙2 = 𝑦(n) ( 𝑦(n−1))−( 𝛼−n 𝛼−n+1 ) , 8459 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. and for 𝛼 = n − 1 𝜙1 = 𝑦(n−1), 𝜙2 = 𝑦(n)𝑦(n−2). Thus, the invariant equation is 𝑦(n) ( 𝑦(n−1))−( 𝛼−n 𝛼−n+1 ) = H (( 𝑦(n−1))𝛼−n+2( 𝑦(n−2))−(𝛼−n+1) ) , 𝛼 ≠ n − 1. (5.15) If 𝛼 = n − 1, the invariant equation simply is 𝑦(n) = ( 𝑦(n−2))−1H ( 𝑦(n−1)) . (5.16) (25,n) ∶ r = n − 2, n ≥ 3: X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕𝑦, X4 = x2𝜕𝑦, … ,Xn−1 = xn−3𝜕𝑦, Xn = x𝜕x + ( (n − 2)𝑦 + x(n−2)) 𝜕𝑦. The fundamental differential invariants are 𝜙1 = 𝑦(n−1) exp 𝑦(n−2) (n − 2)! , 𝜙2 = 𝑦(n) exp 2𝑦(n−2) (n − 2)! and invariant equation 𝑦(n) exp 2𝑦(n−2) (n − 2)! = H ( 𝑦(n−1) exp 𝑦(n−2) (n − 2)! ) . (5.17) (26,n): r = n − 4, n ≥ 5: X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x, X4 = 𝑦𝜕𝑦, X5 = x𝜕𝑦, · · · ,Xn = xn−4𝜕𝑦. The fundamental differential invariants: 𝜙1 = 𝑦(n−3)𝑦(n−1)( 𝑦(n−2) )2 , 𝜙2 = ( 𝑦(n−3))2 𝑦(n)( 𝑦(n−2) )3 and invariant ODE ( 𝑦(n−3))2 𝑦(n)( 𝑦(n−2) )3 = H ( 𝑦(n−3)𝑦(n−1)( 𝑦(n−2) )2 ) . (5.18) (27,n): r = n − 4, n ≥ 5: X1 = 𝜕x, X2 = 𝜕𝑦, X3 = 2x𝜕x + r𝑦𝜕𝑦, X4 = x2𝜕x + rx𝑦𝜕𝑦, X5 = x𝜕𝑦, · · · ,Xn = xn−4𝜕𝑦. The fundamental invariants are as follows: 𝜙1 = 𝑦(n−1)(𝑦(n−3)) n+2 2−n − n − 1 n − 2 ( 𝑦(n−2))2( 𝑦(n−3)) −2n n−2 𝜙2 = 𝑦(n) ( 𝑦(n−3)) n+4 2−n + 2n(n − 1) (n − 2)2 ( 𝑦(n−2))3( 𝑦(n−3)) 3n 2−n + 3n 2 − n 𝑦(n−1)𝑦(n−2)(𝑦(n−3)) 2(n+1) 2−n . The invariant equation here is 𝑦(n) ( 𝑦(n−3)) n+4 2−n + 2n(n − 1) (n − 2)2 ( 𝑦(n−2))3( 𝑦(n−3)) 3n 2−n + 3n 2 − n 𝑦(n−1)𝑦(n−2)(𝑦(n−3)) 2(n+1) 2−n = H(𝜙1). (5.19) (28,n): r = n − 5, n ≥ 6: X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕𝑦, … ,Xn−3 = xn−5𝜕𝑦, Xn−2 = x𝜕x, Xn−1 = 𝑦𝜕𝑦, Xn = x2𝜕x + rx𝑦𝜕𝑦. The fundamental invariants are as follows: 𝜙1 = (n − 3)2K2 − 3(n − 1)(n − 3)K1 + 2(n − 1)(n − 2) (n − 3)2((n − 3)K1 − (n − 2)) 3 2 , 𝜙2 = (n − 3)3K3 − 4n(n − 3)2K2 + 6n(n − 1)(n − 3)K1 − 3n(n − 1(n − 2) (n − 3)3((n − 3)K1 − (n − 2))2 , 8460 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. where Ki = 𝑦(n−3+i)(𝑦(n−4))i( 𝑦(n−3) )i+1 , i = 1, 2, 3. The invariant ODE is (n − 3)3K3 − 4n(n − 3)2K2 + 6n(n − 1)(n − 3)K1 − 3n(n − 1(n − 2) (n − 3)3((n − 3)K1 − (n − 2))2 = H(𝜙1). (5.20) 6 n + 3 DIMENSIONAL ALGEBRAS AND EQUATIONS Scalar first-order equations admit infinite number of point symmetries. A second-order equation does not admit a five-dimensional symmetry algebra, as is well-known. The only third-order equations that admit six-dimensional algebras are 𝑦′′′ = 3𝑦′′2 2𝑦′ and𝑦′′′ = 3𝑦′𝑦′′2 1 + 𝑦′2 , (6.1) where the algebras are 𝔰𝔩(2,R)⊕𝔰𝔩(2,R), and 𝔰𝔬(3, 1), respectively. These are the algebras (16,6) and (7,6) as stated in the previous Section 5. These are also known from the initial seminal works of Lie. In Shah et al. [27], it was shown that a fourth-order equation does not admit a maximal seven-dimensional algebra. For fifth-order equations, the only equation which admits an eighth-dimensional algebra is 𝑦(5) = 5𝑦′′′𝑦(4) 𝑦′′ − 40𝑦′′′3 9𝑦′′2 , (6.2) whose symmetry algebra is 𝔰𝔩(3,R) (also referred to as (8,8)) as stated in the previous section. We check all the possible algebras of dimension n + 3 for n ≥ 5. Since an nth-order linear equation cannot have more than n independent solutions, the algebras (m,n + 3) for m = 20, … , 25 are not admitted. We discuss the remaining possible algebras: (26,n + 3): X1 = 𝜕x, X2 = 𝜕𝑦, X3 = x𝜕x, X4 = 𝑦𝜕𝑦, X5 = x𝜕𝑦, · · · ,Xn+3 = xn−1𝜕𝑦. The generators X1, X2, and X5, · · · ,Xn+3 imply that the equation must be of the form 𝑦(n) = K. The X4 then implies that K must vanish. Then X3 is automatically admitted by this equation. However, the maximal symmetry algebra of this equation is n + 4. In the same way, it can be shown easily that the algebras (27,n + 3) and (28,n + 3) are admitted by an nth-order equation if and only if the equation is equivalent to 𝑦(n) = 0 and hence these algebras are admitted by the equation but are not maximal. We state the general result on admission of (n+3) dimensional algebra in the next section. In the following section, we review the classification for n = 4 and discuss n ≥ 5 in some detail. Then we present the results on how one can obtain a complete classification for n ≥ 4. We state relevant theorems of our main results. 7 CLASSIFICATION OF HIGHER, n ≥ 4, ORDER ODEs In this section, we review the main aspects on the classification of the fourth-order ODEs and then discuss n ≥ 5. We thus consider the Lie algebraic classification of ODEs of any high order. In general, we can classify scalar ODEs into three subclasses as follows: Subclass (1): nth-order equations admitting n + 1, n + 2, n + 3 and the maximal n + 4 dimensional algebras. These are higher symmetries admitted by a scalar ODE. Subclass (2): nth-order equations admitting n dimensional algebras. For this, we have found the fundamental invariants of n dimensional algebras which are of order n − 1 and n. Subclass (3): nth-order equations admitting algebras of dimension lower than n. All these subclasses (1), (2), and (3) are completed herein for n ≥ 4 and the algebra of dimension n + 4 is already a well-known algebra since the initial work of Lie with corresponding equation 𝑦(n) = 0 that possesses this maximal dimension Lie algebra. This is the algebra (28,n + 4) as stated in Section 4.2. Also, for all subclasses, we provide the 8461 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. TABLE 1 The nth-order equations admitting one- and two-dimensional algebras for n ≥ 3. Algebra Generators nth-order invariant ODE (9, 1) 𝜕x 𝑦(n) = H ( 𝑦, 𝑦′, 𝑦′′, … , 𝑦(n−1)) (10, 2) 𝜕x , x𝜕x 𝑦(n) = 𝑦′ nH ( 𝑦, 𝑦′′𝑦′ −2 , … , 𝑦(n−1)𝑦′ 1−n ) (20, 2) 𝜕𝑦, x𝜕𝑦 𝑦(n) = H ( x, 𝑦′′, ...𝑦(n−1)) (22, 2) 𝜕x , 𝜕𝑦 𝑦(n) = H ( 𝑦′, 𝑦′′, … , 𝑦(n−1)) A2,1 𝜕x , x𝜕x + 𝑦𝜕𝑦 𝑦(n) = 𝑦1−nH ( 𝑦′, 𝑦′′𝑦, 𝑦′′′𝑦2, … , 𝑦(n−1)𝑦n−2) Abbreviation: ODE, ordinary differential equation. TABLE 2 Scalar fourth-order equations admitting real three-dimensional algebras. Algebra 𝝀 Invariants 𝝓i Fourth-order ODE (1, 3) a−1∕2 exp(−𝛼 arctan 𝑦′) 𝜙2 = 𝑦′′a−3∕2 exp(−𝛼 arctan 𝑦′) 𝜙4 = H(𝜙2, 𝜙3) 𝛼 ≥ 0 a = 1 + 𝑦′ 2 𝜙3 = 𝑦′′ −2 𝑦′′′a − 3𝑦′ (2, 3) xa−1∕2 𝜙2 = a−3∕2(x𝑦′′ − a𝑦′) xa−1∕2Dx𝜙3 = H(𝜙2, 𝜙3) a = 1 + 𝑦′ 2 𝜙3 = a−3x2(a𝑦′′′ − 3𝑦′𝑦′′2) (3, 3) ra−1∕2, a = 1 + 𝑦′ 2 𝜙2 = a−3∕2(r𝑦′′ + 2a(𝑦 − x𝑦′)) ra−1∕2Dx𝜙3 r = 1 + x2 + 𝑦2 𝜙3 = r2a−3(a𝑦′′′ − 3𝑦′𝑦′′2) = H(𝜙2, 𝜙3) (11,3) 𝑦′ −1 𝜙0 = 𝑦 𝑦(4)𝑦′ −4 + 6𝑦′−6 𝑦′′ 3 𝜙3 = 𝑦′ −4(2𝑦′𝑦′′′ − 3𝑦′′2) −6𝑦′−5 𝑦′′𝑦′′′ = H(𝜙0, 𝜙3) (12,3) 𝑦′ 1∕(𝛼−1) 𝜙2 = 𝑦′′𝑦′ 2−𝛼 𝛼−1 𝑦(4) = 𝑦′′′ 𝛼−4 𝛼−3 H(𝜙2, 𝜙3)|𝛼| < 1 𝛼 ≠ 0 𝜙3 = 𝑦′′′𝑦′′ 3−𝛼 𝛼−2 𝛼 = 1 𝑦′′ −1 𝜙1 = 𝑦′, 𝜙3 = 𝑦′′ −2 𝑦′′′ 𝑦(4) = 𝑦′′′ 3∕2H(𝜙1, 𝜙3) (17,3) xb−1∕2 𝜙2 = (x𝑦′′ − b𝑦′)b−3∕2 xb−1∕2Dx𝜙3 = H(𝜙2, 𝜙3) b = 1 − 𝑦′ 2 𝜙3 = x2b−3(b𝑦′′′ + 3𝑦′𝑦′′2) (18,3) 𝑦2 𝜙2 = 𝑦3𝑦′′ 𝑦7𝑦(4) + 8𝑦′𝑦6𝑦′′′ 𝜙3 = 𝑦5𝑦′′′ + 3𝑦′𝑦′′𝑦4 +12𝑦5𝑦′ 2 𝑦′′ = H(𝜙2, 𝜙3) (20,3) 1 𝜙0 = x, 𝜙3 = 𝑦′′′ − 𝑦′′ 𝜉′′ ′ 𝜉′′ 𝑦(4) = 𝑦′′′ 𝜉(4) 𝜉′′ ′ + H(𝜙0, 𝜙3) (21,3) 1 𝜙0 = x, 𝜙3 = 𝑦′′′∕𝑦′′ 𝑦(4) = 𝑦′′′H(𝜙0, 𝜙3) (22,3) 1 𝜙2 = E(𝑦) = 𝑦′′ + a1𝑦 ′ + a2𝑦 Dx𝜙3 = H(𝜙2, 𝜙3) 𝜙3 = Dx𝜙2, 𝜂i satisfy E(𝜂i) = 0, i = 1, 2, ai const (23,3) 1 𝜙2 = 𝑦′′∕𝑦′, 𝜙3 = 𝑦′′′∕𝑦′′ 𝑦(4) = 𝑦′′′H(𝜙2, 𝜙3) (25,3) exp 𝑦′ 𝜙2 = 𝑦′′ exp 𝑦′, 𝜙3 = 𝑦′′ −2 𝑦′′′ 𝑦(4) = 𝑦′′′ 3∕2H(𝜙2, 𝜙3) Abbreviation: ODE, ordinary differential equation. procedure to obtain the representative ODEs. These are presented in tabular forms and the main results in theorems at the end of this section. For algebras of dimensions 1 and 2, the algebras and canonical equation are given in Table 1. The nth-order equations admitting one- and two-dimensional algebras for n ≥ 3 are easy to determine as in Table 1. For any three-dimensional algebra, we already know the invariants of orders 2, 3, and 4 from previous works [7, 8, 15, 16] as well as in Shah et al. [27]. Then by invariant differentiation, we can find higher-order invariants up to the required order. Lie's recursive formula may also be used. We provide the types of 3D algebras that are possessed by fourth-order ODEs. They are as follows: (1, 3) ∶ 𝛼 ≥ 0, 𝜕x, 𝜕𝑦, (𝛼x + 𝑦)𝜕x + (𝛼𝑦 − x)𝜕𝑦, (2, 3) ∶ 𝜕𝑦, x𝜕x + 𝑦𝜕𝑦, 2x𝑦px + (𝑦2 − x2)𝜕𝑦, (3, 3) ∶ 𝑦𝜕x − x𝜕𝑦, (1 + x2 − 𝑦2)𝜕x + 2x𝑦𝜕𝑦, 2x𝑦𝜕x + (1 + 𝑦2 − x2)𝜕𝑦, (11, 3) ∶ 𝜕x, x𝜕x, x2𝜕x, (12, 3) ∶ 0 < 𝛼 ≤ 1, 𝜕x, 𝜕𝑦, x𝜕x + 𝛼𝑦𝜕𝑦, (17, 3) ∶ 𝜕𝑦, x𝜕x + 𝑦𝜕𝑦, 2x𝑦𝜕x + (x2 + 𝑦2)𝜕𝑦, (18, 3) ∶ 𝜕𝑦, x𝜕x + 𝑦𝜕𝑦, 2x𝑦px + 𝑦2𝜕𝑦, (20, 3) ∶ 𝜕𝑦, x𝜕𝑦, 𝜉(x)𝜕𝑦, 𝜉 is not linear in x and arbitrary, (21, 3) ∶ 𝜕𝑦, 𝑦𝜕𝑦, x𝜕𝑦, (22, 3) ∶ 𝜕x, 𝜂1(x)𝜕𝑦, 𝜂2(x)𝜕𝑦, 𝜂is are linearly independent arbitrary functions, (23, 3) ∶ 𝜕x, 𝑦𝜕𝑦, 𝜕𝑦, (25, 3) ∶ 𝜕x, 𝜕𝑦, x𝜕x + (x + 𝑦)𝜕𝑦, 8462 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Algebra 𝝀 Invariant/s Fourth-order ODE (4, 4) ( 1 + 𝑦′ 2 ) 𝑦′′ −1 𝑦′′ −2 𝑦′′′ ( 1 + 𝑦′ 2 ) − 3𝑦′ 𝑦(4) = 𝑦′′ 3 ( 1 + 𝑦′ 2 )−2 (15𝑦′2 +10𝜙𝑦′ + H(𝜙)) (13, 4) 𝑦′𝑦′′ −1 𝑦′𝑦′′ −2 𝑦′′′ 𝑦(4) = 𝑦′ −2 𝑦′′ 3H(𝜙), H ≠ 0 (14, 4) 𝑦′ −1 𝑦′′′𝑦′ −3 − 3 2 𝑦′′ 2 𝑦′ −4 𝑦(4) = 𝑦′ 4H(𝜙) + 6𝑦′2 𝑦′′𝜙 +3𝑦′−2 𝑦′′ 3 (19, 4) 𝑦1∕2𝑦′′ −1∕2 𝑦1∕2𝑦′′ −3∕2 𝑦′′′ + 3𝑦′𝑦−1∕2𝑦′′ −1∕2 𝑦(4) = 4 3 𝑦′′ −1 𝑦′′′ 2 +𝑦−1𝑦′′ 2H(𝜙), H ≠ 0 (22,4) 1 E(𝑦) ≡ 𝑦′′′ + a1𝑦 ′′ + a2𝑦 ′ + a3𝑦 DxE(𝑦) = H(E(𝑦)), H ≠ 0 𝜂i solves E(𝜂i) = 0 for ais const. (23,4) 1 Dx ln |E(𝑦)| ≡ Dx ln |𝑦′′ + a1𝑦 ′ + a2𝑦| D2 xE(𝑦) = E(𝑦)H(𝜙), 𝜂i satisfy E(𝜂i) = 0 for ai const. H ≠ 0 (24,4) 𝑦′′ 1∕(𝛼−2), 𝑦′′′ 𝛼−2 𝑦′′3−𝛼 , 𝛼 ≠ 3 𝑦(4) = 𝑦′′′ 𝛼−4 𝛼−3 H(𝜙),H ≠ 0 𝑦′′ 𝑦′′′, 𝛼 = 3 𝑦(4) = 𝑦′′ −1H(𝜙), H ≠ 0 𝑦′′′ −1 𝜙 = 𝑦′′, 𝑦(4)∕𝑦′′′2, 𝛼 = 2 𝑦(4) = 𝑦′′′ 2H(𝜙), H ≠ 0 (25,4) exp(𝑦′′∕2) 𝑦′′′ exp(𝑦′′∕2) 𝑦(4) = exp(−𝑦′′)H(𝜙), H ≠ 0 Abbreviation: ODE, ordinary differential equation. TABLE 3 The fourth-order equations admitting four-dimensional algebras. Higher algebra Fourth-order ODE (5, 5) 𝑦(4) = K𝑦′′ 5 3 + 5 3 𝑦′′′ 2 𝑦′′ −1, K ≠ 0, (15, 5) 𝑦′2 𝑦(4)−6(𝑦′𝑦′′𝑦′′′−𝑦′′3)( 𝑦′𝑦′′′− 3 2 𝑦′′2 ) 3 2 = K, (21,5) 𝑦(4) = ∑2 i=1 Ai(x)𝑦(i+1), 𝜉k satisfy 𝜉 (4) k = ∑2 i=1 Ai(x)𝜉(i+1) k , k = 1, 2, (24, 5) 𝑦(4) = K𝑦′′′ 𝛼−4 𝛼−3 , 𝛼 ≠ 3 K ≠ 0, (25, 5) 𝑦(4) = K exp(−𝑦′′′∕6), K ≠ 0, (26, 5) 𝑦(4) = K𝑦′′′ 2 𝑦′′ −1, K ≠ 0, 4∕3, 5∕3, (27, 5) 𝑦(4) = K𝑦′′ 7∕3 + 4 3 𝑦′′′ 2 𝑦′′ −1, K ≠ 0 (6, 6) 𝑦(4) = 5 3 𝑦′′′ 2 𝑦′′ −1 (23, 6) 𝑦(4) = ∑3 i=0 Ai𝑦 (i), Ai const., 𝜂i satisfy same equation (28, 6) 𝑦(4) = 4 3 𝑦′′′ 2 𝑦′′ −1 (28, 8) 𝑦(4) = 0 Note: Note that K is a constant and for (21,5) as well as (23, 6); the Ais do not satisfy the maximal symmetry conditions of Mahomed and Leach [12] as mentioned in Sections 3.2 and 4.2. Abbreviation: ODE, ordinary differential equation. TABLE 4 The fourth-order invariant ODEs that correspond to their five- and higher-dimensional symmetry algebras. The algebras (2,3), (17,3), and (18,3) above are equivalent to the realizations as given in González-López et al. [4] via the transformations x̄ = 𝑦, �̄� = x, x̄ = x + 𝑦, �̄� = 𝑦 − x, and x̄ = 𝑦, �̄� = |x|1∕2, respectively. We compactly review scalar fourth-order ODEs which admit real three-dimensional Lie algebras as alluded to above. The reader is also referred to Shah et al. [27] for further details. This is important for higher than fourth-order symmetry classification of scalar ODEs. We provide the main results in Table 2. Remark. In Table 2, the ODEs for four types of algebras are as follows. (1, 3) : 𝜙4 = 𝑦(4)a2𝑦′′ −3 − 2a2𝑦′′ −4 𝑦′′′ 2 + 2a𝑦′𝑦′′−2 𝑦′′′ − 3a, (2, 3) : xa−1∕2Dx𝜙3 = x2a−5∕2 ( 2𝑦′′′ + x𝑦(4) − 10x𝑦′𝑦′′𝑦′′′a−1 − 6𝑦′𝑦′′2a−1 − 3x𝑦′′3a−1 + 18x𝑦′2𝑦′′3a−2 ) , (3, 3) : ra−1∕2Dx𝜙3 = r2a−5∕2 ( 4(x + 𝑦𝑦′)𝑦′′′ − 10ra−1𝑦′𝑦′′𝑦′′′ + r𝑦(4) − 3ra−1𝑦′′ 3 − 12a−1𝑦′𝑦′′ 2(x + 𝑦𝑦′) + 18ra−2𝑦′ 2 𝑦′′ 3 ) , (17, 3) : xb−1∕2Dx𝜙3 = x2b−5∕2 ( 2𝑦′′′ + x𝑦(4) + 10xb−1𝑦′𝑦′′𝑦′′′ + 6𝑦′𝑦′′2b−1 + 3x𝑦′′3b−1 + 18xb−2𝑦′ 2 𝑦′′ 3 ) The a, b, and r are as in Table 2. There are 12 types of real three-dimensional Lie algebras admitted by scalar fourth-order ODEs as listed in Table 2. One can utilize the operator of invariant differentiation  = 𝜆Dx for fourth- and higher-order ODEs possessing three-dimensional algebras. Therefore, nth-order, n ≥ 4, scalar ODEs with three-dimensional symmetry algebras have the form n−3𝜙3 = H(𝜙, 𝜙3, … ,n−4𝜙3), n ≥ 4, where 𝜙 is 𝜙0, 𝜙1, or 𝜙2 as given in Table 2. Hence, for 8463 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. n = 4, we have the fourth-order ODE given by 𝜙3 = H(𝜙, 𝜙3) as in Table 2 with two arguments in H. In general, for higher-order ODEs, there are n − 2 arguments. We present the list of four-dimensional algebra types as used in Shah et al. [27]. These are (4, 4): 𝜕x, 𝜕𝑦, x𝜕x + 𝑦𝜕𝑦, 𝑦𝜕x − x𝜕𝑦, (13, 4): 𝜕x, 𝜕𝑦, x𝜕x, 𝑦𝜕𝑦, (14, 4): 𝜕x, 𝜕𝑦, x𝜕x, x2𝜕x, (19, 4): 𝜕x, x𝜕x, 𝑦𝜕𝑦, x2𝜕x + x𝑦𝜕𝑦, Also, the algebras (22,4), (23,4), (24,4), and (25,4) are easily found by setting n = 4 in the general n dimensional cases discussed in Section 5. We now briefly review fourth-order equations that possess real four-dimensional algebras in a compact way than discussed in Shah et al. [27]. This is presented in Table 3. Remark. There are multiple cases of (22,4) in Shah et al. [27]. These are considered as a single case in Table 3. We observe that there are eight types of four-dimensional algebras admissible by fourth-order ODEs, and also, there are seven types of five-dimensional algebras as listed in Table 4, namely, (5,5), (15,5) (Section 5), (21,5) (linear homogeneous ODE, Proposition 3.1), (24,5), (25,5), (26,5), and (27,5) (these four are given in Section 3.1 for n = 4). There are as well three types of six-dimensional algebras (as in Table 4) which are (6,6) (singular invariant ODE stated in Section 5), (23,6) (linear constant coefficient ODE, Proposition 4.1), and (28,6) (Section 4.1). Further, there is the maximum eighth-dimensional case (28,8) (simplest linear ODE, Section 4.2 end). Thus, altogether, there are 11 types of higher than four-dimensional algebra types. All these 11 types are mentioned in the previous Sections 3–5 as indicated and summarized in Table 4. Note that invariant differentiation of the invariants in Table 3 will give fifth- and higher-order invariants. It is also important to include the algebras (20,m) and (21,m) for higher than mth-order ODEs. We now focus on these two algebra types. The algebras (20,3) and (21,3) are stated in Table 2 and included here. For (20,m): 𝜕∕𝜕𝑦, x𝜕∕𝜕𝑦, 𝜉1𝜕∕𝜕𝑦, … , 𝜉m−2𝜕∕𝜕𝑦, 3 ≤ m < n, one has 𝜆 = 1 and the invariants 𝜙 = x and 𝜙m = 𝑦(m) − a1𝑦 (m−1) − … − am−2𝑦 ′′, where 𝜉i, i = 1, … ,m − 2 satisfy 𝜉 (m) i − a1𝜉 (m−1) i − … − am−2𝜉 ′′ i = 0. The nth-order ODE with (20,m) is (n−m)𝜙m = H ( 𝜙, 𝜙m, … ,(n−m−1)𝜙m ) (7.1) with  = Dx. In the case (21,m): 𝜕∕𝜕𝑦, 𝑦𝜕∕𝜕𝑦, x𝜕∕𝜕𝑦, 𝜉1𝜕∕𝜕𝑦, … , 𝜉m−3𝜕∕𝜕𝑦, 3 ≤ m < n, we have 𝜆 = 1 and invariants 𝜙 = x, 𝜙m = Dx ln |𝜙m−1|, where𝜙m−1 = 𝑦(m−1)−a1𝑦 (m−2)−…−am−3𝑦 ′′ and 𝜉i, i = 1, … ,m−3 satisfy 𝜉(m−1) i −a1𝜉 (m−2) i −…−am−3𝜉 ′′ i = 0. The nth-order ODE with (21,m) is of the form (7.1) with appropriate 𝜙, 𝜙m and  = Dx. In the case of higher than fourth-order ODEs admitting four-dimensional symmetry algebras, we resort to Table 3 and also have (20, 4) and (21, 4). For (20,4), the invariants as above are 𝜙 = x, 𝜙4 ≡ E(𝑦) = 𝑦(4) −a1𝑦 ′′′ −a2𝑦 ′′, where 𝜉i, i = 1, 2 satisfy E(𝜉i) = 0 for ai constants. One thus obtains the fifth-order ODE Dx𝜙4 = H(𝜙, 𝜙4)which has (20,4). In the case (21,4), the invariants are 𝜙 = x and 𝜙4 = Dx ln |𝑦′′′ − 𝑦′′𝜉′′′∕𝜉′′| and one has again the form Dx𝜙4 = H(𝜙, 𝜙4). For higher than 4, nth-order ODEs with four-dimensional algebra, we have the form (7.1) with m = 4. Therefore, for higher nth, n ≥ 5, order ODEs with four-dimensional algebras, there are 10 types. These follow from Table 3 and inclusion of (20,4) and (21,4). One deduces these by invariant differentiation as (H has n− 3 arguments) n−4𝜙4 = H(𝜙, 𝜙4, … ,n−5𝜙4), n ≥ 5. We now consider fifth-order invariant equations admitting one or more dimensional algebras. The one- (single type) and two-symmetry algebras (four types) are known as in Table 1. In the case of three-dimensional algebras, there are 12 types as a consequence of Table 2 and the ensuing discussions. Thus, a fifth-order ODE possessing 3D algebras has the form 2𝜙3 = H(𝜙, 𝜙3,𝜙3), where  = 𝜆Dx for each of the 𝜆s as stated in Table 2. For four-dimensional algebras, there are 10 types as in Table 3 and immediate deliberations which included (20,4) and (21,4). Hence, a fifth-order ODE which admits a 4D algebra is of the form 𝜙4 = H(𝜙, 𝜙4) for each 𝜆 in Table 3. One can use each of the 𝜆s in Table 3 to find 𝜙4 for all the types. Now we focus on algebras of dimension 5 and greater. These are discussed and follow from our deliberations in the previous sections as well. The fifth-order ODEs admitting a five-dimensional algebra are presented in Table 5. Higher symmetries admitted by such ODEs are listed in Table 6. The fifth-order ODEs for algebras (5,5) and (15,5) are Equations (5.1) and (5.3). Those for the types (22,5) to (27,5) are the seven Equations (5.13) to (5.19). 8464 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Algebra Fifth-order ODE (5, 5) 𝑦(5) = −40 9 𝑦′′′ 3 𝑦′′ −2 + 5𝑦′′′𝑦(4)𝑦′′−1 + 𝑦′′ 2H(𝜙), H ≠ 0, 𝜙 = 3𝑦(4)𝑦′′ −5 3 − 5𝑦′′′2 𝑦′′ −8∕3, (15, 5) 𝑦′3 𝑦(5)−10𝑦′2 𝑦′′𝑦(4)−6𝑦′2 𝑦′′ ′2+48𝑦′𝑦′′2 𝑦′′′−36𝑦′′4( 𝑦′𝑦′′′− 3 2 𝑦′′2 )2 = H(𝜙) 𝜙 = 𝑦′2 𝑦(4)−6(𝑦′𝑦′′𝑦′′′−𝑦′′3)( 𝑦′𝑦′′′− 3 2 𝑦′′2 ) 3 2 , (22,5) Dx(E4(𝑦)) = H(E4(𝑦)), H ≠ 0, 𝜂i is solution of E4(𝜂i) = 0, i = 1, … , 4 for ai consts. (23,5) D2 x(E3(𝑦)) = E3(𝑦)H(Dx ln |E3(𝑦)|), H ≠ 0, 𝜂i is solution of E3(𝜂i) = 0, i = 1, … , 3 for ai consts. (24, 5) 𝑦(5) = ( 𝑦(4) ) 𝛼−5 𝛼−4 H( ( 𝑦(4) )𝛼−3 , ( 𝑦(3) )4−𝛼), 𝛼 ≠ 4, H ≠ 0, 𝑦(5) = (𝑦′′′)−1H ( 𝑦(4) ) , 𝛼 = 4, H ≠ 0, (25, 5) 𝑦(5) = exp ( − 2𝑦′′′ 3! ) H(𝑦(4) exp(𝑦′′′∕3!)), H ≠ 0, (26, 5) 𝑦(5)𝑦′′ 2 𝑦′′′ −3 = H(𝑦′′𝑦′′′−2 𝑦(4)), H ≠ 0, (27, 5) 𝑦(5)𝑦′′ −3 + 40 9 𝑦′′′ 3 𝑦′′ −5 − 5𝑦(4)𝑦′′′𝑦′′−4 = H(𝜙), H ≠ 0, 𝜙 = 𝑦(4)𝑦′′ −7∕3 − 4 3 𝑦′′′ 2 𝑦′′ −10∕3, H ≠ C𝜙3∕2, C const. Note: Note that En(z) ≡ z(n) + a1z(n−1) + … + anz and in (27,5) if H = C𝜙3∕2 for C constant, then the algebra (28, 6) occurs. Abbreviation: ODE, ordinary differential equation. TABLE 5 The fifth-order invariant ODEs that correspond to their five-dimensional symmetry algebras. Higher algebra Fifth-order equation (6, 6) 9𝑦′′2 𝑦(5)−45𝑦′′𝑦′′′𝑦(4)+40𝑦′′ ′3 (3𝑦′′𝑦(4)−5𝑦′′ ′2) 3 2 = K,K ≠ 0, (16,6) K−3 1 (5(DxK1)2 − 4K1D2 xK1) = K, K1 = 𝑦−1𝑦′′′ − 3 2 𝑦′ −2 𝑦′′ 2, (7,6) K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 ) = K, K1 = ( 1 + 𝑦′ 2 ) 𝑦′′ −2 𝑦′′′ − 3𝑦′, K2 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK1, K3 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK2, (21,6) 𝑦(5) = ∑3 i=1 Ai(x)𝑦(i+1), 𝜉k satisfy 𝜉 (5) k = ∑3 i=1 Ai(x)𝜉(i+1) k , k = 1, 2, 3, (24,6) 𝑦(5) = K ( 𝑦(4) ) 𝛼−5 𝛼−4 , 𝛼 ≠ 4,K ≠ 0, (25, 6) 𝑦(5) = K exp(−𝑦(4)∕4!), K ≠ 0, (26, 6) 𝑦(5) = K𝑦′′′ −1( 𝑦(4) )2, K ≠ 0, 5∕4, (27, 6) 𝑦(5) = K ( 𝑦(3) )2 + 5 4 ( 𝑦(4) )2 𝑦′′′ −1, K ≠ 0, (28, 6) ( 𝑦(5)𝑦′′2 𝑦′′ ′3 − 5(9K1−8) 9 ) ∕(3K1 − 4)3∕2 = K, K ≠ 0, K1 = 𝑦(4)𝑦′′𝑦′′′ −2 (23,7) 𝑦(5) = ∑4 i=0 Ai𝑦 (i), Ai const., 𝜂i satisfy the same equation. (28, 7) 𝑦(5) = 5 4 ( 𝑦(4) )2 𝑦′′′ −1 (8,8) 9𝑦′′2 𝑦(5) − 45𝑦′′𝑦′′′𝑦(4) + 40𝑦′′′3 = 0 (28, 9) 𝑦(5) = 0 Note: Note that K is a constant and in (21,6) (Ais not constant) and (23,7), the Ais do not satisfy the maximal symmetry condition as in Mahomed and Leach [12]; see Sections 3.2 and 4.2. Abbreviation: ODE, ordinary differential equation. TABLE 6 The fifth-order invariant ODEs that correspond to their higher symmetry algebras. The fifth-order equations for (6,6), (16,6), and (7,6) are (5.4), (5.7), and (5.9). Also for (24,6) to (28,6), the ODEs are (3.1) to (3.5) for n = 5. The linear case for (21,6) follows from Proposition 3.1 by setting n = 5. Moreover, the ODE for (8,8) is the singular invariant equation K2 = 0; (23,7) is the linear ODE (4.2); for (28,7), the ODE is (4.1) and for (28,9), one has the simplest Equation (4.3). One can proceed to classify sixth-order ODEs as follows. For one and two dimensions, these are in Table 1. As for three-dimensional algebras, there are 12 types and equations are of the form 3𝜙3 = H(𝜙, 𝜙3,𝜙3, 2𝜙3). In the case of four dimensions, we have 10 types with ODEs 2𝜙4 = H(𝜙, 𝜙4,𝜙4). Five dimensions result in 10 types with equations 𝜙5 = H(𝜙, 𝜙5) (Table 5 and (20,5), (21,5)). Note here that 𝜆 = 1∕Dx𝜙 in  = 𝜆Dx. The algebras for dimension 6 and ODEs are as follows. One has the algebra (6,6) with equation given by (5.6), (16,6) with Equation (5.8), and (7,6) with representative ODE (5.10). Each of these three types is presented in Section 5. 8465 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Also, one can easily deduce the algebras and forms for the equations for the six-dimensional algebras (22,6) to (28,6 ) from Section 5 by setting n = 6 for each type. The equations are (5.13) to (5.20) (there are two equations for (24.6)). Alto- gether, there are hence 10 maximal algebra types for the sixth-order ODEs admitting six-dimensional algebras. Moreover, for higher symmetries n + 1,n + 2,n + 4 for n = 6, there are nine types given by (21,7) (linear ODE in Proposition 3.1 with n = 6), (24,7) to (28,7) (Section 3.1 for n = 6 with corresponding Equations (3.1) to (3.5)) as well as (23,8) (linear Equation (4.2)), (28,8) ((4.1) for n = 6), and (28,10) simplest linear ODE with maximal n+ 4 = 10 symmetries, as in (4.3). Likewise, if we investigate the seventh-order ODEs, one has the same number of types of lower-dimensional algebras as for the sixth-order ODEs, namely, one-, two-, three-, four-, and five-dimensional algebras. For six dimensions, we have 12 types. The number of types of maximal algebras of dimension 7 are seven given by (22,7) to (28,7) in Section 5 with ODEs (5.13) to (5.20) with n = 7. Moreover, for higher symmetries, we end up with 10 types, namely, (21,8) (Proposition 3.1), (24,8) to (28,8) with Equations (3.1) to (3.5) with n = 7, (23,9) (linear, Proposition 4.1), (28,9) having (4.1), (28,11) with maximal symmetry equation, and (8,8) given by the ODE (5.11), namely, 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 = K, where K is constant. Now proceeding to the eighth-order ODEs, we have similar one to six lower dimension types as for the seventh-order ODEs. For dimension 7, we have nine types. One also has the eight maximal algebra types (22,8) to (28,8) with corresponding ODEs (5.13) to (5.20) for n = 8 as well as (8,8) given by Equation (5.12). There are also nine types of higher symmetries, namely, (21,9) (linear), (24,9) to (28,9), (23,10) (linear), (28,10), and (28,12) (linear maximal case). We have the generalization to order 9 and greater as follows: For n ≥ 9, there are the lower-dimension types similar to the eighth-order ODEs with dimension 8 as lower-dimension algebra having 10 types. Moreover, seven maximal types for nth-order ODEs admit n symmetries. These are (22,n) to (28,n) with representative ODEs (5.13) to (5.20) with two cases for (24,n). Also, for higher symmetries for nth-order equations, n ≥ 9, there are nine types, namely, (21,n + 1) (Proposition 3.1, linear), (24,n + 1) to (28,n + 1) with ODEs (3.1) to (3.5), (23,n + 2) (Proposition 4.1, linear), (28,n + 2) with ODE (4.1) and (28,n + 4) which is the maximal algebra with simplest ODE 𝑦(n) = 0. We now state the following important results on maximal n-dimensional and higher-dimensional symmetry algebras admitted by a scalar higher-order ODEs for n ≥ 4. These are consequences of our prior discussions. Theorem 7.1. The number of types of maximal n dimensional Lie symmetry algebras admitted by scalar nth-order ODEs for n ≥ 4 are as follows: 8 for n = 4, 5 with algebras and canonical ODEs given in Tables 3 and 5, 10 for n = 6 with algebras and representative equations Algebra sixth-order equation (6,6) K−3 1 ( − 3 2 K2DxK1 + K1DxK2 ) = H(K−3∕2 1 K2) K1 = 3𝑦′′−1 𝑦(4) − 5𝑦′′′2 𝑦′′ −2 ,K2 = 9𝑦′′−1 𝑦(5) − 45𝑦′′−2 𝑦′′′𝑦(4) + 40𝑦′′−3 𝑦′′′ 3 (16,6) K−1∕2 1 Dx𝜙 = H(𝜙), 𝜙 = K−3 1 (5(DxK1)2 − 4K1D2 xK1) K1 = 𝑦′′′𝑦−1 − 3 2 𝑦′ −2 𝑦′′ 2 (7,6) ( 1 + 𝑦′ 2 ) 𝑦′′ −1K−1∕2 1 Dx𝜙 = H(𝜙), 𝜙 = K−3 1 ( 2K1K3 + 4K2 1 K2 − 5 2 K2 2 + 2K4 1 − 2K2 1 ) , K1 = ( 1 + 𝑦′ 2 ) 𝑦′′ −2 𝑦′′′ − 3𝑦′, K2 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK1, K3 = 𝑦′′ −1 ( 1 + 𝑦′ 2 ) DxK2 (22,6) DxE5(𝑦) = H (E5(𝑦)), E5(𝜂i) = 0, i = 1, … , 5, H ≠ 0, (23,6) D2 xE4(𝑦) = E4(𝑦)H (Dx ln |E4(𝑦)|), E4(𝜂i) = 0, i = 1, … , 4, H ≠ 0 (24,6) 𝑦(6) = (𝑦(5)) 𝛼−6 𝛼−4 H ( (𝑦(5))𝛼−4(𝑦(4))5−𝛼 ) , 𝛼 ≠ 5, H ≠ 0, 𝑦(6) = ( 𝑦(4) )−1H(𝑦(5)), 𝛼 = 5 , H ≠ 0, (25,6) 𝑦(6) = exp(−𝑦(4)∕12)H ( 𝑦(5) exp(𝑦(4)∕24) ) , H ≠ 0, (26,6) 𝑦(6) = ( 𝑦(4) )3( 𝑦(3) )−2H ( 𝑦(5)𝑦(3) ( 𝑦(4) )−2 ) , H ≠ 0, (27,6) 𝑦(6) + 15 4 ( 𝑦(4) )3( 𝑦(3) )−2 − 9 2 (𝑦(5)𝑦(4) ( 𝑦(3) )−1 = ( 𝑦(3) )5∕2H(𝜙) 𝜙 = 𝑦(5) ( 𝑦(3) )−2 − 5 4 ( 𝑦(4) )2( 𝑦(3) )−3, H ≠ C𝜙3∕2, (28,6) K3 − 8K2 + 20K1 − 40 3 = (3K1 − 4)2H ( 9K2−45K1+40 9(3K1−4)3∕2 ) K1 = 𝑦(4)𝑦(2) ( 𝑦(3) )−2, K2 = 𝑦(5) ( 𝑦(2) )2( 𝑦(3) )−3, K3 = 𝑦(6) ( 𝑦(2) )3( 𝑦(3) )−4 8466 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. 7 for n = 7 with algebras and canonical ODEs given below in (7.2) by letting n = 7 8 for n = 8 with algebras and equations K−1∕3 2 Dx ( 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 ) = H(𝜙), where 𝜙 = 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 as well as by letting n = 8 in the seven cases (7.2). 7 for n ≥ 9 with algebras and representative equations Algebra nth-order Equation (22,n) DxEn−1(𝑦) = H (En−1(𝑦)), En−1(𝜂i) = 0, i = 1, … ,n − 1, H ≠ 0, (23,n) D2 xEn−2(𝑦) = En−2(𝑦)H (Dx ln |En−2(𝑦)|), En−2(𝜂i) = 0, i = 1, … ,n − 2, H ≠ 0 (24,n) 𝑦(n) = ( 𝑦(n−1)) 𝛼−n 𝛼−n+1 H (( 𝑦(n−1))𝛼−n+2( 𝑦(n−2))−(𝛼−n+1) ) , 𝛼 ≠ n − 1, H ≠ 0, 𝑦(n) = ( 𝑦(n−2))−1H ( 𝑦(n−1)), 𝛼 = n − 1, H ≠ 0, (25,n) 𝑦(n) exp 2𝑦(n−2) (n−2)! = H ( 𝑦(n−1) exp 𝑦(n−2) (n−2)! ) , H ≠ 0 (26,n) (𝑦(n−3))2 𝑦(n) (𝑦(n−2))3 = H ( 𝑦(n−3)𝑦(n−1) (𝑦(n−2))2 ) , H ≠ 0, (27,n) 𝑦(n) + 2n(n−1) (n−2)2 ( 𝑦(n−2))3( 𝑦(n−3))−2 + 3n 2−n 𝑦(n−1)𝑦(n−2)(𝑦(n−3))−1 = ( 𝑦(n−3)) n+4 n−2 H(𝜙), H ≠ C𝜙3∕2, 𝜙 = 𝑦(n−1)(𝑦(n−3)) n+2 2−n − n−1 n−2 ( 𝑦(n−2))2( 𝑦(n−3)) −2n n−2 (28,n) K3 − 4n n−3 K2 + 6n(n−1) (n−3)2 K1 − 3n(n−1)(n−2) (n−3)3 = ((n − 3)K1 − (n − 2))2)H(𝜙) 𝜙 = (n−3)2K2−3(n−1)(n−3)K1+2(n−1)(n−2) (n−3)2((n−3)K1−(n−2)) 3 2 Ki = 𝑦(n−3+i)(𝑦(n−4))i (𝑦(n−3))i+1 , i = 1, 2, 3, (7.2) Remark: Note that in the above, En(z) ≡ z(n) + a1z(n−1) + … + anz. If H = C𝜙3∕2, then (28,7) arises for the sixth-order ODEs or (28,n + 1) for (27,n). Theorem 7.2. The number of types of maximal n + 1 dimensional Lie symmetry algebras admitted by scalar nth-order ODEs for n ≥ 4 are as follows: 7 for n = 4, with algebras and canonical ODEs given in Table 4 9 for n = 5, with algebras and representative equations in Table 6 6 for n = 6, with algebras and equations given in (7.3) by setting n = 6 7 for n = 7, with algebras and canonical ODEs (8,8), 12J−2∕3 1 J2 − 28J−8∕3 1 J2 2 + 24J−5∕3 1 J3 + J4∕3 1 − 4J−2∕3 1 = K, where J1 = K−3∕2 1 K2, with K1 and K2 as in (6,6), J2 = K−1∕2 1 DxJ1 and J3 = K−1∕2 1 DxJ2 and the 6 algebras in (7.3) by setting n = 7. 6 for n ≥ 8, with algebras and representative ODEs Higher algebra nth-order equation (24, n + 1) 𝑦(n) = K ( 𝑦(n−1)) 𝛼−n 𝛼−n+1 , K ≠ 0 (25, n + 1) 𝑦(n) = K exp ( −𝑦(n−1) (n−1)! ) , K ≠ 0 (26, n + 1) 𝑦(n) = K (𝑦(n−1))2 𝑦(n−2) , K ≠ 0,n∕(n − 1) (27, n + 1) 𝑦(n) = n n−1 (𝑦(n−1))2 𝑦(n−2) + K ( 𝑦(n−2)) n+3 n−1 , K ≠ 0 (28, n + 1) 𝑦(n) = ( 𝑦(n−2))3( 𝑦(n−3))−2 [ n(3(n−2)K1−2n+2) (n−2)2 + K((n − 2)K1 − (n − 1)) 3 2 ] (21, n + 1) 𝑦(n) = ∑n−2 i=2 Ai(x)𝑦(i+1), 𝜉k, satisfy 𝜉(n)k = ∑n−2 i=1 Ai(x)𝜉(i+1) k , k = 1, ...,n − 2. (7.3) Remark. For (21,n + 1), the Ais are not constant or satisfy the maximal conditions as stated in section 3.2. Theorem 7.3. The number of types of n + 2 dimensional maximal symmetry algebras admitted by an nth-order ODEs for n ≥ 5 are two, one for linear class of equations, namely, (23,n + 2), with Equation (4.2) and one for nonlinear 8467 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. Equations (4.1) with algebra (28,n+2). For n = 4, there are three types, (23,6) results in a linear equation as well as (6,6) and (28,6) which give rise to nonlinear classes of equations. Theorem 7.4. The only higher-order n ≥ 3, ODEs which admit real n + 3 dimensional algebras are the two ODEs for n = 3, namely, (6.1) possessing 𝔰𝔩(2,R) ⊕ 𝔰𝔩(2,R) and 𝔰𝔬(3, 1), respectively, and the fifth-order equation which is (6.2) having 𝔰𝔩(3,R) as maximal algebra. Note that the maximum real Lie algebra admissible is (28,n + 4) as is well-known from Lie for 𝑦(n) = 0, n ≥ 3 and all scalar linear higher-order ODEs (3.20) and (3.21) as shown in Mahomed and Leach [12]. In order to recall, for lower-dimensional algebras admitted by higher-order ODEs n ≥ 4, one can refer to the one- and two-dimensional algebras listed in Table 1 as well as the 12 types of real 3D algebra realizations given in Table 2. Moreover, one should note that for lower dimensions, there also appears the two nonmaximal Lie algebras (20,m) and (21,m), when one classifies scalar ODEs n ≥ 4 as discussed. Therefore, we can state the following theorem for lower dimensions. Theorem 7.5. The number of types of lower m < n dimensional Lie algebras possessed by scalar nth-order ODEs for n ≥ 4 are given by the following: 1,4 for m = 1, 2 dimensional algebras with the algebras and ODEs as in Table 1 and 12 for m = 3 dimension algebras with algebras and ODEs given in Table 2 for the fourth-order equations and for higher n > 4 order equations obtained by (with algebras as in Table 2) invariant differentiation from (7.1) with m = 3, 10 for dimensions m = 4, 5 with 8 algebras in Tables 3 and 5 as well as the two algebras (20,m) and (21,m) with ODEs deduced from (7.1) via invariant derivatives. The equations are 𝜙4 = H(𝜙) and 𝜙5 = H(𝜙) in Tables 3 and 5. Twelve for dimension m = 6 with 10 algebras as in Theorem 7.1 together with (20,m) and (21,m) and equations determined by (7.1). The sixth-order ODEs 𝜙6 = H(𝜙) are stated in Theorem 7.1 Nine for dimension m = 7 with seven algebras in Theorem 7.1 as well as the two algebras (20,m) and (21,m) and ODEs derived by (7.1). The equations of the form 𝜙7 = H(𝜙) are known. Ten for dimension m = 8 with eight algebras in Theorem 7.1 and two algebras (20,m) and (21,m) and ODEs deduced by (7.1). ODEs of the form 𝜙8 = H(𝜙) are stated. Nine for dimension m ≥ 9 with seven algebras in Theorem 7.1 as well as the algebras (20,m) and (21,m) and ODEs deduced by (7.1); nth-order equations are given. Thus, we have a full understanding of the real symmetry Lie algebras admissible by scalar nth-order ODEs for n ≥ 2 keeping in mind that the second- and third-order ODEs are well-known from the literature as referenced. 8 CONCLUDING REMARKS We have completely classified scalar nth, n ≥ 4 order ODEs according to the real Lie algebras they possess by using the classification of Lie algebras in the plane as in the seminal contribution [4]. The classification for the second- and third-order equations are well-known in a number of influential works [2, 7, 8, 15, 16, 23]. In the case of fourth-order equations, there have been progress in a number of papers, namely, in previous research [24–27]. Here we have relooked at the maximal four-dimensional algebras in a compact form to enhance further classification of higher-order ODEs which admit four-dimensional algebras as subalgebras. We have shown that a higher nth-order equation cannot admit maximally n+3 point symmetries with the exception of third- and fifth-order ODEs. For third-order equations, six symmetries occur for two classes having algebras so(3, 1) and so(2, 2) which are famous from the pioneering contribution of Lie [2]. In the case of the fifth-order ODEs, one has sl(3,R) admitted for one class of equations which is quite interesting as usually this algebra is often alluded to as the maximal Lie algebra of scalar second-order ODEs. The results on the number of n + 1 dimensional symmetry algebras for scalar higher-order ODEs are provided with theorems on higher symmetries. Further, it is important to mention that n+2 dimensional algebras occur for the third-order equations only in the linear class having 3+2=5 symmetries (see, e.g., Ibragimov & Mahomed [8]); for the fourth-order, there are two further classes apart from the linear class as discussed (see also Shah et al. [27]) and for the fifth- and higher-order ODEs, there is one more class besides the linear class. These can be deduced from the discussions and results stated relating to the fourth- and higher-order ODEs. 8468 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WAQAS SHAH ET AL. We have derived all the maximal n dimensional symmetry algebras and their representative equations for n ≥ 4. Fur- thermore, one can easily extract the higher symmetry classification as well as pointed out. These are presented as main results. For lower-dimensional algebras, one can continue by considering further the two types (20,m) and (21,m) for m < n, where n is the order of the ODE, as well as the familiar two- and three-dimensional algebra types which are recalled in tabular form. This is stated as a theorem as well on when higher-order ODEs admit lower-dimensional Lie symmetry algebras and the ODEs that result. The theorems on linearization appear in two cases which are mentioned here as propositions with reference to the initial works [11, 12]. We have therefore achieved a complete classification of scalar nth, n ≥ 4, order ODEs in terms of the real Lie algebras of point symmetries they admit. AUTHOR CONTRIBUTIONS Said Waqas Shah: Conceptualization; investigation; writing—original draft; methodology; validation; software; for- mal analysis; writing—review and editing. F. M. Mahomed: Conceptualization; investigation; funding acquisition; writing—original draft; methodology; validation; writing—review and editing; formal analysis; supervision; resources. H. Azad: Conceptualization; investigation; funding acquisition; methodology; validation; writing—review and editing; formal analysis; supervision; resources. ACKNOWLEDGEMENTS FM thanks the NRF-DSI and Wits for research support. CONFLICT OF INTEREST STATEMENT This work does not have any conflicts of interest. ORCID Said Waqas Shah https://orcid.org/0000-0002-3015-7231 F. M. Mahomed https://orcid.org/0000-0002-6995-5820 REFERENCES 1. S. Lie, Differential equations, Chelsea, New York, 1967. [in German]. 2. S. 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Ibragimov, Elementary Lie group analysis and ordinary differential equations, Vol. 197, Wiley, New York, 1999. 32. P. J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer, New York, 1993. 33. P. J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. 34. G. W. Bluman and S. Anco, Symmetry and integration methods for differential equations, Springer, New York, 2002. How to cite this article: S. W. Shah, F. M. Mahomed, and H. Azad, Symmetry algebra classification of scalar nth-order ordinary differential equations, Math. Meth. Appl. Sci. 47 (2024), 8449–8470, DOI 10.1002/mma.10025. 8470 10991476, 2024, 11, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1002/m m a.10025 by Fazal M ahom ed - N ational H ealth A nd M edical R esearch C ouncil , W iley O nline L ibrary on [14/09/2024]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense info:doi/10.1002/mma.4923 info:doi/10.1002/mma.5208 info:doi/10.1155/2021/6619325 info:doi/10.1088/1751-8113/41/23/235206 info:doi/10.1002/mma.10025 Symmetry algebra classification of scalar nth-order ordinary differential equations Abstract 1 INTRODUCTION 2 NOTATION AND METHODOLOGY 3 n+1DIMENSIONAL ALGEBRAS AND ODEs 3.1. Nonlinear equations 3.2. Linearizable equations 4 n+2DIMENSIONAL ALGEBRAS AND REPRESENTATIVE ODEs 4.1. Nonlinear equations 4.2. Linearizable equations for n+2symmetries 5 nDIMENSIONAL ALGEBRAS AND CORRESPONDING EQUATIONS 6 n+3DIMENSIONAL ALGEBRAS AND EQUATIONS 7 CLASSIFICATION OF HIGHER, n4, ORDER ODEs 8 CONCLUDING REMARKS REFERENCES << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.3 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends false /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize false /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage false /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Remove /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages false /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth 8 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /FlateEncode /AutoFilterColorImages false /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages false /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth 8 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /FlateEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages false /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /PDFX1a:2001 ] /PDFX1aCheck true /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError false /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (Euroscale Coated v2) /PDFXOutputConditionIdentifier (FOGRA1) /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /False /CreateJDFFile false /Description << /CHS /CHT /DAN /DEU /ESP /FRA /ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF che devono essere conformi o verificati in base a PDF/X-1a:2001, uno standard ISO per lo scambio di contenuto grafico. 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