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تسجيل مستخدم جديدA Systematic Analysis of the Properties of the Generalised Painleve--Ince Equation
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We consider the generalized Painleve--Ince equation, begin{equation*} ddot{x}+alpha xdot{x}+beta x^{3}=0 end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $beta =alpha ^{2}/9~$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.
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