Solving the Riccati Equation

TL;DR

This paper introduces a generalized recursive integrating factor method combined with a non-linear transformation to solve the Riccati equation, providing a systematic approach and illustrative examples.

Contribution

It presents a novel method that transforms the Riccati equation into a second-order linear differential equation for easier solution.

Findings

Successfully derives the general solution for the Riccati equation

Demonstrates the method with illustrative application examples

Provides a systematic approach for solving Riccati equations

Abstract

In this study, the Riccati equation is resolved using the generalized recursive integrating factor method. By applying a non-linear transformation to the dependent variable $y(x)$ of the Riccati equation, a second-order linear differential equation is derived for a variable $u(x)$ that is related to $y(x)$ through the aforementioned transformation. The second-order differential equation is then addressed using the aforementioned integrating factors method to derive the general solution for $u(x)$, which is subsequently transformed back to obtain the general solution for $y(x)$, thereby resolving the Riccati equation. The general solution to the Riccati equation is presented, followed by solving a few illustrative application examples.