Instability of invariant boundary conditions of a generalized Lane-Emden equation of the second-kind

A generator of Lie point symmetries admitted by a Lane-Emden equation of the second-kind for arbitrary shape factor k is used to determine invariant boundary conditions admitted by the equation. The generator of Lie point symmetries is then used to reduce the order of the Lane-Emden equation. A phase plane analysis of the reduced equation indicates that the stability of the invariant boundary condition y^′ = 0 on the line x = 0 changes with changing shape factor k. We show that for values of the shape factor k > 1 the boundary condition y^′ = 0 is stable on the line x = 0 while it is unstable for k ≤ 1.