Abstract | In this paper we solve a particular instance of a class of differential equations that appears not to be
widely documented in the literature and results from the theoretical analysis of a probabilistic key agreement protocol parameterized by an integer k > 1. This class consists of differential equations of the form dy(t) dt = Pk+1 i=0 fi(t)y(t)i, with k > 1 and fi(t); 0 · i · k + 1 specific real functions of t. This form appears to generalize the class of Abel differential equations of the first kind in that it involves the sought function to powers greater than
3, for k > 2. For k > 2 the class of differential equations is not amenable to the general solution methodology of the Abel class of differential equations. To the best of our knowledge, there are no previous efforts to tackle, analytically, differential equations of this form. In this paper we focus on the case k = 3. We show that the solution to the resulting differential equation may be written through the use of a class of Generalized Hyper-
Lambert functions. These functions generalize the well known LambertW function, which frequently arises in the analysis of engineering applications. The solution to the differential equation is achieved, interestingly, through a reduction to a particular real root of the polynomial of degree four that appears in the right hand-side of the differential equation. This technique can be of independent interest and it can be, more generally, applicable to other differential equations of a similar form not amenable to known analytic techniques. |