Analysis and solution of a nonlinear second-order differential equation through rescaling and through a dynamical point of view

The solutions of the equation ÿ+yẏ+βy3=0, where β is a free parameter, are investigated. For β= (1)/(9) the equation is linearizable through an eight-parameter symmetry group and is completely integrable. For β≠ (1)/(9) only two symmetries subsist, but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of β and according to the initial conditions.