Analytic General Solution of the Riccati equation

TL;DR

This paper introduces a new integrability condition for the Riccati equation, enabling the derivation of its general solution and extending to second-order linear ODEs, with implications for physics and dynamical systems.

Contribution

It presents a novel integrability condition for the Riccati equation and derives its general solution, extending the approach to second-order linear differential equations.

Findings

Derived a new integrability condition for the Riccati equation.

Presented the analytic general solution of the Riccati equation.

Extended the solution method to second-order linear ODEs.

Abstract

A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytic general solution is presented, which can be extended to second-order linear ordinary differential equation. This result may provide valuable mathematical criteria for in-depth research on quantum mechanics, relativity and dynamical systems.