The behavior of solutions to a special Abel equation of the second kind near a nodal singular point

The propagation of a diffusion-reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in the Thomas–Fermi model, and some other models in natural sciences lead to bounded solutions of a certain autonomous nonlinear second-order ordinary differential equation reducible to an Abel equation of the second kind. In this study, a sufficient condition is obtained under which all solutions to a special second-kind Abel equation that pass through a nodal singular point of the equation can be represented by a convergent power series (in terms of fractional powers of the variable) in a neighborhood of this point. Under this condition, new parametric representations of bounded solutions to the corresponding autonomous nonlinear equation are derived. These representations are efficient for numerical implementation.