Solution of the scalar Riccati equation

TL;DR

This paper provides a direct explicit solution to the scalar Riccati equation and related second-order linear ODEs, establishing a novel correspondence with automorphism groups and deriving applications in quantum mechanics and special functions.

Contribution

It introduces a general explicit solution method for the scalar Riccati equation and second-order linear ODEs, linking solutions to automorphism groups and deriving new formulas for special functions.

Findings

Explicit solution for the scalar Riccati equation.

Bijective correspondence with automorphism group paths.

New formulas for Airy functions and applications to Schrödinger and Miura transforms.

Abstract

The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has emerged according to which the general solution may be determined if a particular solution is known; but no general method exists to determine a particular solution explicitly, except in sporadic special cases. The purpose of the present article is to solve the scalar Riccati equation in general form, as well as the general linear ODE of second order, directly by explicit construction. In the case of the Riccati equation, the solution sets up a bijective correspondence between triples of locally integrable functions on the real line, and locally absolutely continuous paths through the identity in the automorphism group of the Riemann sphere. As applications of the results, we obtain an explicit solution to the one-dimensional Schr\"odinger equation, an inversion formula for the Miura transform, and a new formula for Airy functions.