Summation formula for solutions of Riccati-Abel equation

TL;DR

This paper investigates the Riccati-Abel equation, revealing why it lacks a general addition formula and establishing a solution method that applies specifically to pairs of solutions, linking it to linear differential equations and complex algebra.

Contribution

It explains the absence of a general addition formula for Riccati-Abel solutions and develops a method applicable to solution pairs, connecting it to linear differential equations and third-order complex algebra.

Findings

Addition formula exists only for solution pairs

Solutions relate to linear differential equations

Connections to third-order complex algebra

Abstract

The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not admit an addition formula. In the present paper we explain a nature of this fault and elaborate a method of solution of this problem. We show that the addition formula for Riccati-Abel equation can be established only for pair of solutions. Furthermore, it is shown that analogously with ordinary Riccati equation, the relationships with linear differential equations and the general complex algebra of third order can be established only for the pair of solutions of Riccati-Abel equation.