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 dy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t), with k>1 a positive integer and fi(t), 0≤i≤k+1, with fi(t), real functions of t. For k=2, these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for k>2 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 k 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 k, 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