Some remarks on the solution of linearisable second-order ordinary differential equations via point transformations

Transformation of ordinary differential equations (ODEs) to other equivalence equations offers a fruitful route towards solution of many intricate equations. Linearisable second-order ODEs in particular is a class of ODEs that is amenable to solution techniques based on such transformations. There are various characterisations of linearisable second-order ODEs. We exploit a particular characterisation of this class of ODEs and the expanded Lie group method to construct a generic solution for all Linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach by finding general solutions of three linearisable second-order ODEs.


Introduction
Nonlinear ODEs arise in many different contexts as mathematical models of real-world phenomena. Analytical solutions of such equations are often hard to find, which is why a whole range of methods have been proposed for investigating different types of nonlnear ODEs. These methods include (i) Painlevé singularity analysis, (ii) Lie symmetry analysis, (iii) Darboux method, and (iv) the Jacobi last multiplier method (see [13] and the references therein). The Lie symmetry method is particularly widely used. It is based on the invariance of a differential equation under a continuous group of point transformations. Given a nonlinear ODE, Lie symmetries of the equation can be used to derive new solutions of the equation from old ones, to reduce the order of the equation, to discover whether or not the equation can be linearised and to construct an explicit transformation that reduces the equation to a simpler one, typically a linear equation. This last application is called the linearisation problem. The pioneering work on this is attributed to Sophus Lie (see [7] and the references there in). Sophus proved that to be linearisable, a second-order ODE must be at most cubically semilinear and the coefficients in it must be satisfy an overdetermined system of conditions [5,2]. A considerable amount of research has since been conducted on the linearisation problem [5,6,14,7,12,14], leading in particular to a variety of ways of characterising linearisable second-order ODEs (see, e,g. [ with the coefficients A to D satisfying the two invariant conditions 4. Eq. (1.1) has two non-commuting symmetries X 1 , X 2 , in a suitable basis with [X 1 , X 2 ] = X 1 , X 1 = ρ(x, y)X 2 (1.4) for any non-constant function ρ such that a point change of variables X = X(x, y), Y = Y (x, y) which brings X 1 and X 2 to their canonical form reduces Eq. (1.1) to where a( = 0) and b are constants.
In this paper we take advantage of the representation (1.6) of all linearisable second-order ODEs to propose a plan for finding solutions of these equations algorithmically through point transformations. We start by using the expanded Lie group method to simplify Eq. (1.6) significantly. We then construct two point transformations and use them to recover the general solution of (1.6) from the solution of the simplest second-order ODE, the free particle equation Y ′′ = 0. The solution of a given linearisable secondorder ordinary differential equation can now be recovered from the solution of Eq. (1.6) in a routine fashion using a point transformation constructed from only two symmetries of the equation. We illustrate the general solution plan through three examples.
The rest of the paper is organised as follows. In Section 2 we introduce the expanded Lie group method and use it to reduce Eq. (1.6). Further reduction of Eq. (1.6) to the particle equation is done in Section 3. In Section 4 we provide three illustrative examples. We give concluding remarks in Section 5.

Reduction of Eq. (1.6) via the expanded Lie group method
The Lie symmetry method for studying differential equations, initiated by Sophus Lie in the latter part of the nineteenth century, is based on continuous groups of transformations that map solutions of a given differential equation into other solutions of the same equation. The method extends and harmonises various specialised methods for solving ODEs. There is extensive literature on the Lie symmetry method, to which we refer the interested reader (see, for example [2,19,8,16,17,9,18]).
When we consider a continuous group of transformations acting on the expanded space of variables which includes the equation parameters in addition to independent and dependent variables, we obtain an expanded Lie group transformation of the equation [1]. Such a group of transformations represents a particular case of the equivalence group that preserves the class of equations under study.
Let us take the parameter b in Eq. (1.6) as a second independent variable. Now consider a one-parameter (ε) Lie group of point transformations in (X, Y, b): for some functions f , g and h, with an infinitesimal generator of the form where X [2] is the second extension of X. We obtain, as a particular solution of (2.3), that The corresponding one-parameter (ε) Lie group of transformations is: so that under this transformation Eq. (1.6) necessarily becomes (2.10) If we now set ε = −b in the transformation (2.7)-(2.9), b equals zero and Eq. (2.10) reduces to  3 Reduction of (2.11) to the free particle equation We seek an invertible point transformation between Eq. (2.11), which we restate here in the original variables X and Y , and the free particle equation We exploit the equivalence of the symmetry Lie algebras of the two equations to construct the point transformation.

Point symmetries admitted by equations (3.1) and (3.2) have infinitesimal symmetry generators
and with the same structure constants {C k ij }. The following rearrangements provide the desired new basis for the symmetry Lie algebra arising from (3.3): where γ, λ and δ are arbtrary constants with λγ = 0. We now seek a point transformation that maps Eq. (3.1) to Eq. (3.2) of the form for functions α and β. According to [8,Chapter 6] the functions α and β must be such that the conditions are satisfied. The equations in (3.6) translate into an overdetermined system of sixteen elementary PDEs that define the functions α and β. The equations are easily solved and we obtain The solution of (3.1) is now recovered from the solution of (3.2), through the point transformation (3.7). We obtain 9) or, equivalently, The solution of the generic equation (1.6) now follows from the point transformation (2.12) and Eq. (3.10), stated in terms of the variables X and Y of Eq. (2.11). It is called the modified Emdem equation, which arises in a vriety of contexts [13], admits the maximal 8 symmetries. Among the admitted symmetries are two noncomuting symmetry generators that satisfy condition (1.4) of Theorem 1.1. The symmetries (4.2) are reduced to their canonical form (1.5) via the point transformation where k 1 = 0 and k 2 are arbitrary constants (see also [5]). Under this point transformation Eq. (4.1) is reduced to (1.6) with a = −k 2 1 and b = 3k 1 (k 2 + 1), and the solution in (3.11) is transformed into which is the desired general solution of (4.1).
Example 4.2 Consider ODE No. 6.180 of Kamke [4]: This equation satisfies the conditions in Item 3 of Theorem 1.1 and therefore admits the maximal 8 Lie point symmetries, two of which are: These infinitesimal symmetries satisfy condition (1.4) from Theorem 1.1 and are reduceable to their canonical forms (1.5) via the point transformation where k 1 and k 2 = 0 are arbitrary constants. Under this transformation Eq. (4.5) is reduced to (1.6), with a = −k 2 2 and b = 3k 1 k 2 . The general solution of (4.5) now follows easily from Eq. (3.11) written in terms of x and y via the point transformation (4.7), with a and b set to −k 2 2 and 3k 1 k 2 , respectively. We obtain The symmetries are admitted by (4.9) and satisfy the condition (1.4). The point transformation that maps the symmetries (4.10) to their canonical form (1.5) is This transformation also reduces Eq. (4.9) to (1.6), with a = 1 and b = 0. Writing solution (3.11) in terms of x and y via the point transformation (4.11) and setting a = 1 and b = 0, we obtain the solution to (4.9), (4.12)

Concluding remarks
There are many characterisations of linearisable second-order differential equations. One important characterisation is that every such equation is reduceable to a generic second-order ODE, (1.6), via a point transformation constructed from two suitably chosen symmetries of the ODE. We have shown through the expanded Lie group approach that equation (1.6) can be reduced significantly with the innocuous specification of parameter b in the equation to be zero. The reduced equation (1.6) (with b set to zero) is now mapped to the free particle equation y ′′ = 0 via an invertible point transformation constructed by "aligning" the respective symmetries of the two equations. The constructed point transformations are used in succession to obtain the general solution to (1.6), a proxy solution for all linearisable second-order differential equations. This allows construction of solutions to all equations in this class algorithmically using only two suitably chosen symmetries of the equation. We have illustrated the solution procedure with three examples.