A Systematic Analysis of the Properties of the Generalised Painlev\'e--Ince Equation
Abstract
We consider the generalized Painlev\'e--Ince equation, equation* x+α xx+β x3=0 equation* and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as β =α 2/9~the given differential equation is maximally symmetric and well-known that it pass the Painlev\'e test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev\'e--Ince equation fails at the Painlev\'e test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable x=1/y. We conclude that the Painlev\'e--Ince equation is integrable is terms of Lie symmetries and of the Painlev\'e test.