Proceedings of 10th International Conference in MOdern GRoup ANalysis 2005, 143–151 On Some Aspects of Ordinary Differential Equations Invariant under Translation in the Independent Variable and Rescaling Sibusiso MOYO † and Peter G.L. LEACH ‡ † Department of Mathematics, Durban Institute of Technology, P O Box 953, Steve Biko Campus, Durban 4000, South Africa E-mail: moyos@dit.ac.za ‡ School of Mathematical Sciences, Howard College, University of KwaZulu-Natal, Durban 4041, South Africa E-mail: leachp@nu.ac.za We study the class of general second-order ordinary differential equations in- variant under translation in the independent variable and rescaling from the La- grangian perspective and show that the differential equation, y 00 +yy 0 +ky 3 = 0, is a unique member of this class. Other aspects of equations arising from this analysis are also discussed. 1 Introduction The study of second-order ordinary differential equations invariant under transla- tion in the independent variable and rescaling has received an incredible amount of attention [1–3]. For example the differential equation, y 00 + yy 0 + ky 3 =0, (1) arises in the study of univalent functions [4], the study of the stability of gaseous spheres [5], the Riccati equation [6] and in the modeling of the fusion of pellets [7]. Furthermore, for rational values of k ∈ (1/9, 1/8) the solution can be expressed in parametric form [8] and (1) passes the weak Painlev´ e test. For k =1/9 the equa- tion possesses eight Lie-point symmetries [3] with the algebra sl(3,R) which im- plies that equation (1) is equivalent to Y 00 = 0 under a point transformation given by X = x - 1/y, Y = x 2 /2 - x/y. For k 6=1/9 the equation has only the two point symmetries G 1 = ∂ x , G 2 = -x∂ x + y∂ y (2) with the algebra A 2 . Leach et al [1] pointed out that the value of k =1/8 was critical in that the solution of (1) passes from nonoscillatory to oscillatory. The main aim of this paper is to give a method to determine Lagrangians of second-order ordinary differential equations invariant under (2) and to analyse the equations in terms of the Painlev´ e analysis. Once a Lagrangian is known, Noether’s approach can be used to determine the corresponding integrals [9].