New further integrability cases for the Riccati equation

Applied Mathematics and Computation, Volume 219, Issue 14,  2013, Pages 7465-7471
M.K. Mak, T. Harko.

Department of Physics, Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong, PR China and

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom.

 

Abstract

New further integrability conditions of the Riccati equation dy/dx=a(x)+b(x)y+c(x)y² 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)-b²(x)]/4c(x), where f(x) is an arbitrary function. The second integrability case is obtained for the “reduced” Riccati equation with b(x)≡0. If the coefficientsa(x) and c(x) satisfy the condition , 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 Schrödinger and Navier–Stokes equations are briefly discussed.

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Additional Information:

The Riccati equation $y'(x)=a(x)+b(x)y(x)+c(x)y^2(x)$, first considered in a particular form in 1723, is one of the basic, and most studied non-linear ordinary differential equations. If a particular solution $y_1(x)$ of the Riccati equation is known, then a more general solution containing a single arbitrary constant can be obtained by means of the substitution $y=y_1(x)+1/v(x)$, which reduces the Riccati equation to a first order linear differential equation. In our paper, by extending the work initiated in [1], we have obtained two new integrability conditions for the Riccati equation, one for the ”full” equation, and one for its ”reduced” form, ($b(x)\equiv 0$, respectively. Both integrability cases are based on the correspondence between the initial Riccati equation and a more general equation containing a solution generating function $f(x)$. If the coefficients of the Riccati equations and the function $f(x)$ satisfy some differential integrability conditions, the general solutions of the Riccati equation is explicitly obtained. Every second-order linear differential equation can be transformed into a Riccati equation, and therefore large classes of physical models can be analyzed by using their reduction to a first order Riccati type equation. One of such second order equation reducible to a Riccati equation is the Schr\”odinger equation, which plays a fundamental role in quantum physics. With the use of the obtained integrability conditions the general solution of the one-dimensional Schr\”odinger equation for potentials of the form $V(x)=f_0x^2+E\pm\sqrt{f_0}$, and $V(x)=f_0x^n-E$, $n\neq 0$, respectively, can be obtained in an exact analytical form. The Navier-Stokes equation for a steady viscous flow, is one of the most complex equations of mathematical physics. Therefore reducing it to a simpler form, or establishing, by using some physically reasonable assumptions, a connection between the Navier-Stokes equation and other equations of the mathematical physics, is of fundamental importance in obtaining exact analytical solutions of the Navier-Stokes equation. Along a streamline the two-dimensional Navier-Stokes equation can be written as a Riccati equation. By using the new integrability conditions general solutions of the two-dimensional Navier-Stokes equation can be constructed. The time-dependent strain field of the velocity of a viscous fluid is related to the second partial $z$-derivative of the pressure by a Riccati equation. Therefore the new integrability conditions of the Riccati equation allow to obtain the strain rate for a viscous fluid flow for specific forms of the fluid pressure.

[1] M. K. Mak and T. Harko, New integrability case for the Riccati equation, Applied Mathematics and Computation 218 (2012), 10974-10981