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

Abstract

We investigate the properties of a general class of differential equations described by $dy(t)/dt=f_{k+\mathrm{1}}(t)y(t{)}^{k+\mathrm{1}}+f_k(t)y(t{)}^k+\cdots +f_{\mathrm{2}}(t)y(t{)}^{\mathrm{2}}+f_{\mathrm{1}}(t)y(t)+f_{\mathrm{0}}(t),$ with $k>\mathrm{1}$ a positive integer and $f_i(t),\mathrm{\hspace{0.17em}0}\le i\le k+\mathrm{1}$, with $f_i(t)$, real functions of $t$. For $k=\mathrm{2}$, these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for $k>\mathrm{2}$ 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.