Some Fourth-Order Ordinary Differential Equations which Pass the Painleve Test

Abstract

An approach to the Painleve analysis of fourth-order ordinary differential equations is presented. Some fourth-order ordinary differential equations which pass the Painleve test are found. As is well known, at the turn of the century Painleve and his school discovered six ordinary differential equations (ODEs) that define new functions. This was achieved by classifying second order ODEs of a certain form having what is today referred to as the Painleve property (the general solution should be free of movable critical points). The aim of this paper is to find some fourth-order differential equations having the Painleve property. To do this we consider fourth order differential equations of two forms