Acta Polytechnica https://doi.org/10.14311/AP.2023.63.0019 Acta Polytechnica 63(1):19–22, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague LINEARISATION OF A SECOND-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATION Adhir Maharaj∗, Peter G. L. Leach, Megan Govender, David P. Day Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of South Africa ∗ corresponding author: adhirm@dut.ac.za Abstract. We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f (v)∂v . The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3, R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties. Keywords: Lie symmetries, integrability, linearisation. 1. Introduction Nonlinear differential equations are ubiquitous in mathematically orientated scientific fields, such as physics, engineering, epidemiology etc. Therefore, the analysis and closed-form solutions of differential equa- tions are important to understand natural phenomena. In the search for solutions of differential equations, one discovers the beauty of the algebraic properties that the equations possess. Even though closed-form solutions are the primary objective, one cannot ignore the interesting properties of the equations [1–6]. In recent years, one such area in relativistic astrophysics involves the embedding of a four-dimensional differen- tiable manifold into a higher dimensional Euclidean space which gives rise to the so-called Karmarkar condition for Class I spacetimes [7]. The Karmarkar condition leads to a quadrature, which reduces the problem of determinig the gravitational behaviour of a gravitating system to a single generating function. This is then used to close the system of field equations in order to get a full description of the thermodynam- ical and gravitational evolution of the model. In a re- cent approach, Nikolaev and Maharaj [8] investigated the embedding properties of the Vaidya metric [9]. The Vaidya solution is the unique solution of the Ein- stein field equations describing the exterior spacetime filled with null radition of a spherical mass distribution undergoing dissipative gravitational collapse. In their work, Nikolaev and Maharaj showed that the Vaidya solution is not Class I embeddable but the generalised Vaidya metric describing an anisotropic and inhomo- geneous atmosphere comprising of a mixture of strings and null radiation gives rise to interesting embedding properties. Here, we consider the nonlinear partial dif- ferential equation arising from the generalised Vaidya metric be of Class I. The governing equation is 2r2mm′′ − r2m′2 − 2rmm′ + 3m2 = 0, (1) where the prime denotes differentiation of the depen- dent variable, m(v, r), with respect to the independent variable, r. Equation (1) is not v-dependent explicitly and possesses the point symmetry f (v)∂v where f (v) is an arbitrary function of v only. Using this symmetry, we obtain the invariants r = x and m = y(x), which re- duces (1) to a nonlinear nonautonomous second-order ordinary differential equation 2x2yy′′ − x2y′2 − 2xyy′ + 3y2 = 0, (2) where y is a function of x only. We use the Lie sym- metry approach to obtain the solution of (2). Using the solution of (2), we obtain the solution of (1). 2. Preliminaries Let (x, y) denote the variables of a two-dimensional space. Suppose that x is the independent variable and y is the dependent variable. An infinitesimal transformation in this space has the form x̄ = x + ϵξ(x, y) (3) ȳ = y + ϵη(x, y) (4) which can be regarded as generated by the differential operator Γ = ξ(x, y) ∂ ∂x + η(x, y) ∂ ∂y . (5) Since we are concerned with point symmetries in this paper, ξ and η depend upon x and y only. Under the infinitesimal transformation (3) and (4), the nth derivative transform is given by ζn = η(n) − n∑ j=1 ( n j ) y(n+1−j)ξ(j) (6) and Γn = ζn ∂ ∂y(n) , (7) 19 https://doi.org/10.14311/AP.2023.63.0019 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en A. Maharaj, P. G. L. Leach, M. Govender, D. P. Day Acta Polytechnica where the notation η(n), ξ(j) and y(n) denote the nth, jth and nth derivative of the dependent vari- able with respect to x. In the case of a function, f (x, y, y′, ..., y(n)), the infinitesimal transformation is generated by Γ + Γ1 + Γ2 + ... + Γn which we write as Γ[n], where [10] Γ[n] = Γ + n∑ i=1 [ η (i) − i∑ j=1 ( i j ) y (i+1−j) ξ (j) ] ∂ ∂y(i) , (8) is called the nth extension of Γ. In the case of an equation E(x, y, y′, ..., y(n)) = 0 (9) the equation is a constraint and the condition [11, 12] Γ a symmetry of the equation Γ[n]E|E=0 = 0, (10) i.e. the action of th nth extension of Γ on the function E is zero when the Equation (9) is taken into account. We note that E = 0 may be a scalar equation or a system of equations1. 3. Symmetry analysis The Lie point symmetries2 of (2) are Γ1 = x ∂ ∂x Γ2 = y ∂ ∂y Γ3 = x3/2 √ y ∂ ∂y Γ4 = √ xy ∂ ∂y Γ5 = 2x ∂ ∂x + 3y ∂ ∂y Γ6 = x2 ∂ ∂x + 3xy ∂ ∂y Γ7 = √ y x ∂ ∂x + ( y x )3/2 ∂y Γ8 = √ xy ∂ ∂x + 3y3/2 √ x ∂ ∂y which is a maximal number for a second-order ordinary differential equation and must be a representation of the sl(3, R) algebra in the Mubarakzyanov Classifi- cation Scheme [21–24]. Equation (2) is linearisable to d2Y dX 2 = 0, (11) 1An interested reader is referred to [13–16]. 2The Mathematica add-on package SYM [17–20] was used to obtain the symmetries. by means of a point transformation. The solution of (11) is Y = AX + B, (12) while the solution of (2) is not exactly obvious. How- ever, one can transform (2) to (11). We seek the trans- formation from (2) to (11) which casts Γ4 = √ xy∂y into canonical form. Γ4 assumes canonical form pro- vided ξ(x, y) ∂X ∂x + η(x, y) ∂X ∂y = 0 (13) ξ(x, y) ∂Y ∂x + η(x, y) ∂Y ∂y = 1, (14) where ξ = 0 and η = √ xy because (2) possesses a symmetry of the general form Γ = ξ∂x + η∂y . When we apply the method of characteristics for first-order partial differential equations to (13) and (14), we obtain dx 0 = dy √ xy = dX 0 (15) dx 0 = dy √ xy = dY 1 (16) for which the solutions are X = x, Y 2 = 4y x . (17) Under the transformation (17), Equation (2) takes the form in (11). Hence we may apply (17) to (12) to obtain the solution to (2), which is y(x) = 1 4 x(Ax + B)2, (18) where A and B are two constants of integration. By using the invariants r = x and m = y(x), the solution of (1) follows from (18) and is m(v, r) = 1 4 r(A(v)r + B(v))2, (19) where A(v) and B(v) are functions of integration. 4. The general case We consider a general case by setting y(x) = un, where u is a function of x in Equation (2), we obtain a more general second-order equation 2nx2uu′′ + n(n − 2)x2u′2 − 2nxuu′ + 3u2 = 0. (20) 20 vol. 63 no. 1/2023 Linearisation of a second-order nonlinear ordinary differential equation The Lie point symmetries of (20) are Λ1 = x ∂ ∂x Λ2 = ∂ ∂x + u nx ∂ ∂u Λ3 = 1 n x3/2u1−n/2 ∂ ∂u Λ4 = 1 n √ xu1−n/2 ∂ ∂u Λ5 = 2x ∂ ∂x + 3 n u ∂ ∂u Λ6 = x2 ∂ ∂x + 3 n xu ∂ ∂u Λ7 = √ un x ∂ ∂x + u1+n/2 nx3/2 ∂ ∂u Λ8 = √ xun ∂ ∂x + 3 n √ un+2 x ∂ ∂u . As (20) is a second-order ordinary differential equation and possesses eight Lie point symmetries, it is related to the generic second-order equation [25] d2Y dX 2 = 0. (21) When we apply the method of characteristics for first- order partial differential equations to (13) and (14), and using symmetry Λ4, we obtain dx 0 = du 1 n √ xu1−n/2 = dX 0 (22) dx 0 = du 1 n √ xu1−n/2 = dY 1 (23) for which the solutions are X = x, Y 2 = 4un x . (24) From the solution of (21), by means of the transfor- mation (24), we obtain the solution of (20) as u(x) = ( x 4 ) 1 n (C1x + C2) 2 n , (25) where C1 and C2 are constants of integration. 5. Conclusion Most studies of the algebraic properties of ordinary differential equations are focused on the first, second and third order equations, which is most natural since these are the equations which arise in the modelling of natural phenomena. In this paper, we performed the symmetry analysis of Equation (2) and showed that the equation possesses the sl(3, R) algebra. In turn, we reported the solution of (2) and thus obtained the solution of (1). A natural generalisation of (2) followed. By setting m(v, r) = zn, where z is a function of v and r in Equation (1), we obtain a more general partial differential differential 2nr2zz′′ + n(n − 2)r2z′2 − 2nrzz′ + 3z2 = 0, (26) where the prime denotes differentiation of the depen- dent variable, z(v, r), with respect to the independent variable, r. We note that, as in Equation (1), (26) is not explicitly dependent on v, and therefore pos- sesses the point symmetry g(v)∂v , where g(v) is an arbitrary function of v only. We use this symmetry to obtain the invariants r = x and z = u(x) which re- duce (26) to the second-order nonlinear Equation (20) with the solution given by (25). Using (25) and the invariants mentioned above, we obtain the solution for Equation (26) to be z(v, r) = ( r 4 ) 1 n (C1(v)r + C2(v)) 2 n , (27) where C1(v) and C2(v) are functions of integration. This paper demonstrates that the Equations (1), hence (26), which, at first glance, looks complicated, has some very interesting properties from the viewpoint of Symmetry analysis. Using the symmetry approach we were able to show that these equations are integrable and have closed-form solutions. Acknowledgements MG expresses grateful thanks to the National Research Foundation of South Africa and the Durban University of Technology for their continuing support. AM acknowl- edges support of the Durban University of Technology. PGLL appreciates the support of the National Research Foundation of South Africa, the University of KwaZulu- Natal and the Durban University of Technology. DPD acknowledges the support provided by Durban University of Technology. References [1] B. Abraham-Shrauner, P. G. L. Leach, K. S. Govinder, G. Ratcliff. Hidden and contact symmetries of ordinary differential equations. Journal of Physics A: Mathematical and General 28(23):6707, 1995. https://doi.org/10.1088/0305-4470/28/23/020 [2] K. Andriopoulos, P. G. L. Leach, A. Maharaj. On differential sequences. Applied Mathematics & Information Sciences 5(3):525–546, 2011. [3] C. Géronimi, M. R. Feix, P. G. L. Leach. Third order differential equation possessing three symmetries the two homogeneous ones plus the time translation. Tech. rep., SCAN-9905040, 1999. [4] A. K. Halder, A. Paliathanasis, P. G. L. Leach. Singularity analysis of a variant of the Painlevé–Ince equation. Applied Mathematics Letters 98:70–73, 2019. https://doi.org/10.1016/j.aml.2019.05.042 [5] A. Maharaj, P. G. L. Leach. Properties of the dominant behaviour of quadratic systems. Journal of Nonlinear Mathematical Physics 13(1):129–144, 2006. https://doi.org/10.2991/jnmp.2006.13.1.11 21 https://doi.org/10.1088/0305-4470/28/23/020 https://doi.org/10.1016/j.aml.2019.05.042 https://doi.org/10.2991/jnmp.2006.13.1.11 A. Maharaj, P. G. L. Leach, M. Govender, D. P. Day Acta Polytechnica [6] A. Maharaj, K. Andriopoulos, P. G. L. Leach. Properties of a differential sequence based upon the Kummer-Schwarz equation. Acta Polytechnica 60(5):428–434, 2020. https://doi.org/10.14311/AP.2020.60.0428 [7] K. Karmarkar. Gravitational metrics of spherical symmetry and class one. Proceedings of the Indian Academy of Sciences – Section A 27:56, 1948. https://doi.org/10.1007/BF03173443 [8] A. V. Nikolaev, S. D. Maharaj. Embedding with Vaidya geometry. The European Physical Journal C 80(7):1–9, 2020. https://doi.org/10.1140/epjc/s10052-020-8231-0 [9] P. Chunilal Vaidya. The external field of a radiating star in general relativity. Current Science 12:183, 1943. [10] F. M. Mahomed, P. G. L. Leach. Symmetry Lie algebras of nth order ordinary differential equations. Journal of Mathematical Analysis and Applications 151(1):80–107, 1990. https://doi.org/10.1016/0022-247X(90)90244-A [11] H. Stephani. Differential equations: Their solution using symmetries. Cambridge University Press, 1989. [12] G. W. Bluman, S. Kumei. Symmetries and differential equations, vol. 81. Springer Science & Business Media, 2013. [13] A. Maharaj, P. G. L. Leach. The method of reduction of order and linearization of the two-dimensional Ermakov system. Mathematical methods in the applied sciences 30(16):2125–2145, 2007. https://doi.org/10.1002/mma.919 [14] A. Maharaj, P. G. L. Leach. Application of symmetry and singularity analyses to mathematical models of biological systems. Mathematics and Computers in Simulation 96:104–123, 2014. https://doi.org/10.1016/j.matcom.2013.06.005 [15] P. G. L. Leach. Symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations. Journal of mathematical analysis and applications 336(2):987–994, 2007. https://doi.org/10.1016/j.jmaa.2007.03.045 [16] P. G. L. Leach, J. Miritzis. Analytic behaviour of competition among three species. Journal of Nonlinear Mathematical Physics 13(4):535–548, 2006. https://doi.org/10.2991/jnmp.2006.13.4.8 [17] K. Andriopoulos, S. Dimas, P. G. L. Leach, D. Tsoubelis. On the systematic approach to the classification of differential equations by group theoretical methods. Journal of computational and applied mathematics 230(1):224–232, 2009. https://doi.org/10.1016/j.cam.2008.11.002 [18] S. Dimas, D. Tsoubelis. SYM: A new symmetry- finding package for Mathematica. In Proceedings of the 10th international conference in modern group analysis, pp. 64–70. University of Cyprus Press, 2004. [19] S. Dimas, D. Tsoubelis. A new Mathematica-based program for solving overdetermined systems of PDEs. In 8th International Mathematica Symposium, pp. 1–5. Avignon, 2006. [20] S. Dimas. Partial differential equations, algebraic computing and nonlinear systems. Ph.D. thesis, University of Patras, Greece, 2008. [21] V. V. Morozov. Classification of nilpotent Lie algebras of sixth order. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika 4:161–171, 1958. [22] G. M. Mubarakzyanov. On solvable Lie algebras. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (1):114–123, 1963. [23] G. M. Mubarakzyanov. Classification of real structures of Lie algebras of fifth order. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (3):99–106, 1963. [24] G. M. Mubarakzyanov. Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (4):104–116, 1963. [25] F. M. Mahomed, P. G. L. Leach. The linear symmetries of a nonlinear differential equation. Quaestiones Mathematicae 8(3):241–274, 1985. https://doi.org/10.1080/16073606.1985.9631915 22 https://doi.org/10.14311/AP.2020.60.0428 https://doi.org/10.1007/BF03173443 https://doi.org/10.1140/epjc/s10052-020-8231-0 https://doi.org/10.1016/0022-247X(90)90244-A https://doi.org/10.1002/mma.919 https://doi.org/10.1016/j.matcom.2013.06.005 https://doi.org/10.1016/j.jmaa.2007.03.045 https://doi.org/10.2991/jnmp.2006.13.4.8 https://doi.org/10.1016/j.cam.2008.11.002 https://doi.org/10.1080/16073606.1985.9631915 Acta Polytechnica 63(1):19–22, 2023 1 Introduction 2 Preliminaries 3 Symmetry analysis 4 The general case 5 Conclusion Acknowledgements References