Analysis to the solutions of Abel's differential equation of the first kind under the transformation y=u(x)z(x)+v(x)

Abstract

In this work we study different analytical solutions which can be obtained from a new Abel equation of first kind, under the transformation 2076 E. Salinas-Hernández et al. y = u(x)z(x) + v(x), changing the variable to z(x), where the coefficients of this equation allow the construction of a system of auxiliary equation with φ 1 (x), φ 2 (x) and φ 3 (x) as free functions to the system. From the form of the system, different cases are obtained, whose details are described in this work.

Quasi-Lie families, schemes, invariants and their applications to Abel equations

“…For instance, Lie symmetries of Abel equations were studied in [32]. Their solutions via transformations y = u(x)z(x) + v(x) were investigated in [42] and through the referred to as Julia's condition in [6]; equivalence and integrable cases are presented in [21]; etc. Meanwhile, some solutions in closed form appeared in [39] and periodic solutions have been studied in [2].…”

A Mathematical Model of Epidemics—A Tutorial for Students

“…Next, we present the main stream of the derivation of the analytical solution. Some detailed calculations are given as questions, and the solutions can be found in Appendix B. Equations (21)- (23) are simultaneous differential equations; we can rewrite them as the differential equation of a single function. The procedure is the same as that used to solve a linear simultaneous equation.…”

“…If we set µ = 0, this model becomes the standard SIR model; Equation (63) is reduced to Equation (A4) in the case of µ = 0. Furthermore, some recent developments regarding the solution of the Abel equation of the first kind have been reported [22,23]. Next, we briefly analyze the properties of the SIR model with vital dynamics.…”