On solutions of third and fourth-order time dependent Riccati equations and the generalized Chazy system Partha Guha a,⇑ , A. Ghose Choudhury b , Barun Khanra c a S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700098, India b Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Kolkata 700009, India c Sailendra Sircar Vidyalaya, 62A Shyampukur Street, Kolkata 700004, India article info Article history: Received 1 August 2011 Received in revised form 6 January 2012 Accepted 14 January 2012 Available online 23 March 2012 Keywords: Nonlocal transformation First integrals Third and fourth-order Riccati equation Generalized Chazy equation F-XVI Bureau symbol PI abstract We introduce a new transformation (nonlocal) to find the general solutions of some equa- tions belonging to the third and fourth-order time dependent Riccati class of equations. These are in turn related to the Chazy polynomial class and the time dependent F-XVI Bureau symbol PI equations respectively. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction The problem of constructing a solution of a given ordinary differential equation is a non trivial one and constitutes a vital aspect of their analysis. Although there are a number of well defined methods for the solution of linear ordinary differential equations (ODEs), the same however, cannot be said in the case of nonlinear ODEs. A popular method for solving a differen- tial equation consists in using a suitable transformation to reduce it to a simpler differential equation whose solution is known. In fact since the class of linear equations is acknowledged to be the simplest class of equations, most of our efforts are directed towards finding a transformation which reduces the ODE to a linear equation. This is referred to as the linear- ization problem. Point transformation, contact transformation, reduction of order, differential substitution, Generalized Sundman transformation etc. are some of the tools commonly used for solving the linearization problem. The problem of linearization of second-order ODEs was solved, as far back as 1883, by Lie [22] using point transformation. A point transformation is a mapping (t, x) ´ (T, X) of the form T = G(t, x), X = F(t, x) with the transformation often involving one or more continuous real parameters. The corresponding problem in case of third-order ordinary differential equation was studied by Grebot [17], using a re- stricted class of point transformation, namely t = /(x), u = w(x, y). Such transformations are also known as fiber-preserving transformation. Ibragimov and Meleshko solved the problem of linearization of third-order ordinary differential equation by means of point transformation in [19]. On the other hand Euler et al. [11] have deduced the necessary and sufficient conditions for a third order ODE to be equivalent to the equation X 000 = 0, here X 0 ¼ dX dT , applying a generalized Sundman 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.01.022 ⇑ Corresponding author. E-mail addresses: partha@bose.res.in (P. Guha), a_ghosechoudhury@rediffmail.com (A. Ghose Choudhury), barunkhanra@rediffmail.com (B. Khanra). Commun Nonlinear Sci Numer Simulat 17 (2012) 4053–4063 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns