New integrability case for the Riccati equation

TL;DR

This paper introduces a new integrability condition for the Riccati equation, enabling the derivation of general solutions through an auxiliary equation involving a generating function, with applications to physics problems.

Contribution

It presents a novel integrability condition for the Riccati equation using an auxiliary equation and generating function, expanding solution methods for nonlinear differential equations.

Findings

Derived general solutions for Riccati equations under new conditions

Validated the method with specific Riccati-type differential equations

Applied the integrability condition to physical models like oscillators and solitons

Abstract

A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if the coefficients $a(x)$, $b(x)$, $c(x)$, and the function $f(x)$ satisfy a particular constraint. The validity and reliability of the method are tested by obtaining the general solutions of some Riccati type differential equations. Some applications of the integrability conditions for the case of the damped harmonic oscillator with time dependent frequency, and for solitonic wave, are briefly discussed.