Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

Abstract

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

“…Theorem 6.1. The complete set of reciprocal transformations of the two-dimensional stationary gas dynamics equations, considered up to the equivalence transformations corresponding to (11) and the involution E 2 , consists of the transformations ( 5) and (7):…”

“…In particular, the Sundman type transformation is effective for solving nonlinear ODEs. For example, it has been applied for linearization problems of ordinary differential equations 2–5 and for finding families of analytical solutions 6,7 …”