Winter Sinkala

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Research ArticleSome Remarks on the Solution of Linearisable Second-OrderOrdinary Differential Equations via Point TransformationsWinter SinkalaDepartment of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, Private Bag X1,Mthatha 5117, South AfricaCorrespondence should be addressed to Winter Sinkala; sinkalaw@gmail.comReceived 29 March 2020; Accepted 18 May 2020; Published 1 July 2020Academic Editor: Ji GaoCopyright © 2020 Winter Sinkala. )is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricateequations. A class of differential equations that are particularly amenable to solution techniques based on such transformations isthe class of linearisable second-order ordinary differential equations (ODEs). )ere are various characterisations of such ODEs.We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisablesecond-order ODEs. )e general solution of any given equation from this class is then easily obtainable from the generic solutionthrough a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate theapproach with three examples.1. IntroductionNonlinear ODEs arise in many different contexts, includingas mathematical models of real-world phenomena. Ana-lytical solutions of many such equations are often hard tofind, which is why a whole range of methods have beenproposed for investigating different types of nonlinearODEs. )ese methods include Painlevé singularity analysis,Lie symmetry analysis, Darboux method, and the Jacobi lastmultiplier method (see [1] and the references therein). )eLie symmetry method, which is based on the invariance of adifferential equation under a continuous group of pointtransformations, is widely used. Given a differential equa-tion, Lie point symmetries of the equation can be used toperform many things on the equation including con-structing explicit transformations that reduce the equationto a simpler one, when this is possible. When the simpler“target” equation is a linear equation, the problem is calledthe linearisation problem.)e pioneering work on this, withregard to second-order ODEs, is attributed to Sophus Lie(see [2] and the references there in). Lie proved that, to belinearisable, a second-order ODE must be at most cubicallysemilinear, and the coefficients in it must satisfy an over-determined system of conditions [3, 4]. A considerableamount of research has since been conducted on the line-arisation problem [2, 3, 5–7] leading, in particular, to avariety of ways of characterising linearisable second-orderODEs (see, e.g., )eorem 8 of [3]). A scaled-down version of)eorem 8 of [3] is presented here.Theorem 1. Let us consider a scalar second-order ODE in theformy′′ � f x, y, y′( 􏼁. (1))e following statements are equivalent [3]:(1) A scalar second-order equation (1) is linearisable viaa point transformation.(2) Equation (1) has the maximum eight-dimensionalsymmetry Lie algebra.(3) Equation (1) has the cubic in derivative formHindawiJournal of MathematicsVolume 2020, Article ID 2406961, 5 pageshttps://doi.org/10.1155/2020/2406961y′′ � A(x, y)y′3+ B(x, y)y′2+ C(x, y)y′ + D(x, y),(2)with the coefficients A to D satisfying the two in-variant conditions3Axx + 3AxC − 3AyD + 3ACx + Cyy− 6ADy + BCy − 2BBx − 2Bxy � 0,6AxD − 3ByD + 3ADx + Bxx − 2Cxy− 3BDy + 3Dyy + 2CCy − CBx � 0.(3)(4) Equation (1) has two noncommuting symmetries X1and X2 in a suitable basis withX1, X2􏼂 􏼃 � X1, X1 ≠ ρ(x, y)X2, (4)for any nonconstant function ρ such that a pointchange of variables X � X(x, y), Y � Y(x, y) whichbrings X1 and X2 to their canonical formX1 �zzY,X2 � XzzX+ YzzY,(5)reduces equation (1) toXY′′ �b3a+b327a2+ 1 +b23a􏼠 􏼡Y′ + bY′2+ aY′3, (6)where a(≠ 0) and b are constants.In this paper, we take advantage of representation (6)to construct a proxy solution for all linearisable second-order ODEs. We start by using the expanded Lie groupmethod to simplify equation (6) significantly. We thenconstruct two point transformations and use them torecover the general solution of equation (6) from thesolution of the simplest second-order ODE, the freeparticle equation Y′′ � 0. )e solution of any given lin-earisable second-order ODE can now be recovered fromthe solution of equation (6) in a routine fashion via apoint transformation constructed from only two sym-metries of the equation. We illustrate the solution routinethrough three examples.)e rest of the paper is organised as follows. In Section2, we use the expanded Lie group method to construct apoint transformation that reduces equation (6) to asimpler equation. In Section 3, we perform further re-duction and map equation (6) to the free particleequation. We subsequently deduce a generic solution ofall linearisable second-order ODEs. In Section 4, we il-lustrate through three examples how the general solutionof any linearisable second-order ODE may be deducedfrom the constructed generic solution. We give con-cluding remarks in Section 5.2. Reduction of Equation (6) via the ExpandedLie Group Method)e Lie symmetry method for studying differential equa-tions, initiated by Sophus Lie in the latter part of thenineteenth century, is based on continuous groups oftransformations that map solutions of a given differentialequation into other solutions of the same equation. )emethod extends and harmonises various specialisedmethods for solving ODEs. )ere is extensive literature onthe Lie symmetry method, to which we refer the interestedreader to see, for example, [4, 8–13].When we consider a continuous group of transforma-tions acting on the expanded space of variables, which in-cludes the equation parameters in addition to independentand dependent variables, we obtain an expanded Lie grouptransformation of the equation [14]. Such a group oftransformations represents a particular case of the equiva-lence group that preserves the class of equations under study.Let us take the parameter b in equation (6) as a secondindependent variable. Now, consider a one-parameter (ε) Liegroup of point transformations in (X, Y, b),􏽥X � f(X, Y, b, ε),􏽥Y � g(X, Y, b, ε),􏽥b � h(b, ε),(7)for some functions f, g, and h, with an infinitesimal gen-erator of the formX � ξ(X, Y, b)zX + τ(X, Y, b)zY + β(b)zb, (8)which leaves equation (6) invariant. Equation (6) admits (7)if X satisfies the invariance requirement:X[2] b3a+b327a2+ 1 +b23a􏼠 􏼡Y′ + bY′2+ aY′3− XY′′􏼨 􏼩􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌(6)� 0,(9)where X[2] is the second extension of X. We obtain, as aparticular solution of (9), thatξ �1 + X22X,τ � Y1 + X24X+121 −1 + X22X2􏼠 􏼡􏼢 􏼣 −b6aX−X3a,β � 1.(10))e corresponding one-parameter (ε) Lie group oftransformations is􏽥X �������������eε 1 + X2( ) − 1􏽰, (11)􏽥Y �e(ε/2)(3aY + bX) − (b + ε)������������eε 1 + X2( ) − 1􏽰3a, (12)􏽥b � b + ε, (13)2 Journal of Mathematicsso that, under this transformation, equation (6) necessarilybecomes􏽥X􏽥Y′′ �􏽥b3a+􏽥b327a2+ 1 +􏽥b23a⎛⎝ ⎞⎠􏽥Y′ + 􏽥b􏽥Y′2+ a􏽥Y′3. (14)If we now set ε � − b in transformations (11)–(13), 􏽥bequals zero, and equation (14) reduces to􏽥X􏽥Y′′ � 􏽥Y′ + a􏽥Y′3. (15)Clearly, equation (15) is mapped back to equation (6) viapoint transformations (11)–(13) with ε � − b, i.e.,􏽥X ��������������e− b 1 + X2( ) − 1􏽱,􏽥Y � e(− b/2)Y +b3aX􏼠 􏼡.(16))is means that a solution of equation (15) is mapped toa solution of equation (6) via transformation (16).3. Reduction of (15) to the FreeParticle EquationWe seek an invertible point transformation betweenequation (15), which we restate here in the original vari-ables X and Y,XY′′ � Y′ + aY′3, (17)and the free particle equationW′′ � 0, W � W(Z). (18)We exploit the equivalence of the symmetry Lie algebrasof the two equations to construct the point transformation.Lie point symmetries admitted by equations (17) and(18) areΦ1 �1X􏼒 􏼓zX,Φ2 �− 2a2Y3X􏼠 􏼡zX + 3aY2+ X2􏼐 􏼑zY,Φ3 �aY2X+ X􏼠 􏼡zX,Φ4 �aY2X􏼠 􏼡zX − YzY,Φ5 �X4 − a2Y42X􏼠 􏼡zX + Y aY2+ X2􏼐 􏼑zY,Φ6 �YX􏼒 􏼓zX,Φ7 �2aYX􏼒 􏼓zX − zY,Φ8 � YX −aY3X􏼠 􏼡zX + 2Y2zY,(19)Ω1 � zZ,Ω2 � zW,Ω3 � ZzZ,Ω4 � ZzW,Ω5 � WzWΩ6 � WzZ,Ω7 � ZWzZ + W2zW,Ω8 � Z2zZ − ZWzW,(20)respectively. )e Lie algebras arising from (19) and (20) areequivalent, which means an isomorphism leading to thesame commutator table for the two Lie algebras can befound. We may rearrange operators (19) to form a new basisΓ1, . . . , Γ8􏼈 􏼉 such thatΓi, Γj􏽨 􏽩 � CkijΓk,Ωi,Ωj􏽨 􏽩 � CkijΩk,(21)with the same structure constants Ckij􏽮 􏽯. )e followingrearrangements provide the desired new basis for thesymmetry Lie algebra arising from (19):Γ1 � cΦ6,Γ2 � cλΦ8,Γ3 �12Φ3,Γ4 � λΦ4,Γ5 � −Φ32+Φ4 + cδΦ8,Γ6 � −Φ12λ+δcΦ6λ,Γ7 � −δΦ32λ+2δΦ4λ+aΦ62cλ−Φ72cλ+cδ2Φ8λ,Γ8 � −Φ22c+ δΦ5 +aΦ82c,(22)where c, λ, and δ are arbitrary constants with λc≠ 0.We nowseek a point transformation that maps equation (17) toequation (18) of the formZ � α(X, Y),W � β(X, Y),(23)for functions α and β. According to Chapter 6 of [9], thefunctions α and β must be such that the conditionsΩiZ � Γiα(X, Y),ΩiW � Γiβ(X, Y), i � 1, . . . , 8,(24)Journal of Mathematics 3are satisfied. )e equations in (24) translate into an over-determined system of sixteen elementary linear PDEs thatdefine the functions α and β. )e equations are easily solved,and we obtainZ � α(X, Y) �aY2 + X22cY,W � β(X, Y) �δλ−12cλY.(25))e solution of (17) is now recovered from the solutionof (18),W � mZ + c, (26)through point transformation (25). We obtainδλ−12cλY� maY2 + X22cY􏼠 􏼡 + c, (27)or, equivalently,aY2+ AY + X2+ B � 0, (28)whereA �2cmc −δλ􏼠 􏼡,B �1λm.(29))e solution of generic equation (6) now follows frompoint transformation (16) and equation (28), stated in termsof the variables 􏽥X and 􏽥Y of equation (15). It isaY2+2b3XY + 1 +b29a􏼠 􏼡X2+ J2 Y +b3aX􏼠 􏼡 + J1 � 0,(30)whereJ1 � eb(B − 1) + 1,J2 � Ae(b/2).(31)Equation (30) represents a convenient solution of everylinearisable second-order ordinary differential equation; inthat, the general solution of every such equation is realisablefrom equation (30) via a point transformation constructedusing only two suitably chosen symmetries of the equationaccording to )eorem 1.4. Illustrative ExamplesExample 1. )e ODEy′′ + 3yy′ + y3 � 0, (32)called the modified Emdem equation, which arises in avariety of contexts [1], admits the maximal 8 symmetries.Among the admitted symmetries are two noncommutingsymmetry generatorsX1 � zx,X2 � xzx − yzy,(33)which satisfy condition (4) of )eorem 1. Symmetries (33)are reduced to their canonical form (5) via the pointtransformationX � k1/y( 􏼁,Y � x + k2/y( 􏼁,(34)where k1(≠ 0) and k2 are arbitrary constants (also, see [3]).Under this, point transformation equation (32) is reduced toequation (6) with a � − k21 and b � 3k1(k2 + 1), and thesolution in (30) is transformed intoy �2k21x − J2k21x2 − J2x − J1, (35)which is the desired general solution of (32).Example 2. Consider ODE No. 6.180 of Kamke [15]:y′′(y − 1)2x2 − 2y′2x2 − y′(y − 1)x − 2y(y − 1)2 � 0.(36))is equation satisfies the conditions in Item 3 of)eorem 1 and therefore admits the maximal 8 Lie pointsymmetries, two of which areX1 �y − 2y − 1􏼠 􏼡zx +2yxzy.X2 � xzx.(37))ese infinitesimal symmetries satisfy condition (4)from)eorem 1 and are reduceable to their canonical forms(5) via the point transformationX �k2x����1 − y􏽰y,Y �k1����1 − y􏽰y+ 1 −1y􏼠 􏼡x,(38)where k1 and k2(≠ 0) are arbitrary constants. Under this,transformation equation (36) is reduced to equation (6),with a � − k22 and b � 3k1k2.)e general solution of (36) nowfollows easily from equation (30) written in terms of x and yvia point transformation (38), with a and b set to − k22 and3k1k2, respectively. We obtainy �x k22x − J2( 􏼁k22x2 − J2x − J1. (39)Example 3. Consider the linearisable equation in p.100 of[11]:4 Journal of Mathematicsxy′′ � 2 y′( 􏼁2 + y′ + y2. (40))e symmetriesX1 � 1 −1xy􏼠 􏼡zx +yxzy,X2 � x2−xy􏼠 􏼡zx +(2 − 3xy)zy,(41)are admitted by (40) and satisfy condition (14). )e pointtransformation that maps symmetries (41) to their canonicalform (5) isX ����������2xy − 1y2− 1􏽳,Y �xy − 1y.(42))is transformation also reduces equation (40) toequation (6), with a � 1 and b � 0. Writing solution (31) interms of x and y via point transformation (42) and settinga � 1 and b � 0, we obtain the solution to (40):y �J2x2 + J2x + J1 − 1. (43)5. Concluding Remarks)ere are many characterisations of linearisable second-order ODEs. One important characterisation is that everysuch equation is reduceable to a generic second-orderODE, equation (6), via a point transformation constructedfrom two suitably chosen symmetries. We have used theexpanded Lie group approach to simplify equation (6)significantly via a point transformation constructed to setthe parameter b in equation (6) to zero. )e reducedequation is subsequently mapped to the free particleequation, y′′ � 0, via another invertible point transfor-mation constructed by “aligning” the respective symme-tries of the two equations. )e constructed pointtransformations are used in succession to obtain thegeneral solution to equation (6), which is the desired proxysolution for all linearisable second-order differentialequations. )is allows construction of solutions of allequations in this class algorithmically using only twosuitably chosen symmetries of the equation. We have il-lustrated the solution routine with three examples.Data AvailabilityNo data were used to support this study.Conflicts of Interest)e author declares that there are no conflicts of interestregarding the publication of this paper.Acknowledgments)e author would like to thank the Directorate of ResearchDevelopment and Innovation ofWalter Sisulu University forcontinued financial support.References[1] M. Senthilvelan, V. K. Chandrasekar, and R. Mohanasubha,“Symmetries of nonlinear ordinary differential equations: themodified emden equation as a case study,” Pramana-Journalof Physics, vol. 85, no. 5, pp. 755–787, 2015.[2] M. Safdar, S. Ali, and F. M. Mahomed, “Linearization ofsystems of four second-order ordinary differential equations,”Pramana-Journal of Physics, vol. 73, no. 3, pp. 581–594, 2011.[3] F. M. 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Baumann, Symmetry Analysis of Differential Equations withMathematica, Springer, Berlin, Germany, 2000.[9] G. W. Bluman and S. Kumei, Symmetries and DifferentialEquations, Springer-Verlag, Berlin, Germany, 1989.[10] B. J. Cantwell, Introduction to Symmetry Analysis, CambridgeUniversity Press, Cambridge, UK, 2002.[11] P. E. Hydon, Symmetry Methods for Differential Equations: ABeginner’s Guide, Cambridge University Press, Cambridge,UK, 2000.[12] L. V. Ovsiannikov, Group Analysis of Differential Equations,Academic Press, Cambridge, MA, USA, 1982.[13] P. J. Olver, Applications of Lie Groups to Differential Equa-tions, Springer-Verlag, Berlin, Germany, 1993.[14] G. I. Burde, “Expanded Lie group transformations and sim-ilarity reductions of differential equations,” Proceedings ofInstitute of Mathematics of NAS of Ukraine, vol. 43, no. 1,pp. 93–101, 2002.[15] E. Kamke, Differentialgleichungen: Lösungsmethoden undLösungen, I, Gewöhnliche Differential-Gleichungen, B. G.Teubner, Leipzig, Germany, 1977.Journal of Mathematics 5

Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

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Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

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