New further integrability cases for the Riccati equation

TL;DR

This paper introduces new conditions under which the Riccati equation can be integrated explicitly, expanding the set of solvable cases and applying these results to equations in physics like Schrödinger and Navier-Stokes.

Contribution

It presents novel integrability conditions for the Riccati equation with fixed coefficient forms and for the reduced case, enabling solutions via quadratures.

Findings

New integrability conditions for Riccati equations with specific coefficient forms.

Explicit solutions for the reduced Riccati equation under certain differential conditions.

Applications to integrating Schrödinger and Navier-Stokes equations.

Abstract

New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x)+[f(x)-b^{2}(x)]/4c(x)$, where $f(x)$ is an arbitrary function. The second integrability case is obtained for the "reduced" Riccati equation with $b(x)\equiv 0$. If the coefficients $a(x)$ and $c(x)$ satisfy the condition $\pm d\sqrt{f(x)/c(x)}/dx=a(x)+f(x)$, where $f(x)$ is an arbitrary function, then the general solution of the "reduced" Riccati equation can be obtained by quadratures. The applications of the integrability condition of the "reduced" Riccati equation for the integration of the Schr\"odinger and Navier-Stokes equations are briefly discussed.