Phase Portraits of the Equation $$\ddot{x}+ax\dot{x}+bx^{3}=0$$

Abstract

The second-order differential equation \(\ddot{x}+ax\dot{x}+bx^{3}=0\) with \(a,b\in\mathbb{R}\) has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits according to its parameters \(a\) and \(b\). This classification is done in the Poincaré disc in order to control the orbits that escape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincaré disc of the first-order differential system associated to the second-order differential equation. Additionally, we show that this system is always integrable, providing explicitly its first integrals.