On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations

We investigate the properties of a general class of differential equations described by with a positive integer and , with , real functions of . For , these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values of connects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values of , we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology. 1. Introduction In [1], a differential equation was derived, in the context of the theoretical analysis of a security protocol, that can be written in the following form: In [2], the differential equations (1) are written as partial fractions in such a way that the form of their solution matches the recursive pattern of the definition of the Hyper-Lambert functions that were proposed in [3] as a generalization of the well-known Lambert function (see [4] for an introduction to this function and its properties). The differential equations defined by (1) can be seen as belonging to the general differential equation form with , real functions. The class defined by (2) appears to generalize, naturally, the Abel class of differential equations (see [5]). Unfortunately, no general solution strategy exists (to the best of our knowledge) for the solution of this general class for , in analogy with the methodology that exists for the Abel class. Thus, we could not reduce the problem of finding the solution of (1) into the problem of solving (2) for various values of . In this paper, we propose a general methodology which for odd values of provides a closed-form expression of the solution of (1) based on closed-form solutions of the roots of the equation where (the polynomial on the right-hand side of (1)) and . Moreover, this closed-form solutions are linked with the generalized Hyper-Lambert functions. However, for certain even values of (at least for values of where is