Chazy Classes IX--XII of third-order differential equations


Christopher M. Cosgrove


Research Report 98-23
Date: 24 June 1998


In this paper, we study Classes IX--XII of the thirteen classes introduced by Chazy (1911) in his classification of third-order differential equations in the polynomial class having the Painleve property. Classes IX and X are the only Chazy classes that have remained unsolved to this day. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.)

The method involves transforming to a more tractable class of fourth- and fifth-order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher-order Painleve transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents.

Also, for completeness, we include a review of the well-known Classes III and XII, which have received much attention in the literature because of their movable natural barriers and beautiful symmetries. In an appendix, we present the full list of Chazy equations (in the third-order polynomial class) and the solutions of those that are not dealt with in the body of this paper.

Key phrases

Painleve analysis. Painleve transcendents. ordinary differential equations. hyperelliptic functions. nonlinear equations

AMS Subject Classification (1991)

Primary: 34A34
Secondary: 34A05, 34A20, 33E30


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