Solving the Riccati Equation

TL;DR

The paper addresses finding a general solution to the nonlinear Riccati equation by transforming it into a second-order linear differential equation using $y(x)=-\frac{u'(x)}{q_2(x)u(x)}$. It then solves the resulting equation for $u(x)$ with the generalized recursive integrating factors method and reconstructs $y(x)$ via the inverse transformation, yielding a closed-form solution structure $y(x)=y_p(x)+y_q(x)$ with explicit forms for $y_p$ and $y_q$ in terms of auxiliary functions $\alpha(x)$ and $\beta(x)$. The approach is demonstrated through two examples that produce explicit closed-form Riccati solutions, illustrating the method's analytic power and potential applicability to other nonlinear ODEs. While the results provide a general solution framework, the paper notes that convergence properties of the obtained expressions remain an open question requiring further investigation.

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.