New further integrability cases for the Riccati equation

New further integrability conditions of the Riccati equation dy/dx=a(x)+b(x)y+c(x)y2dy/dx=a(x)+b(x)y+c(x)y2 are presented. The first case corresponds to fixed functional forms of the coefficients a(x)a(x) and c(x)c(x) of the Riccati equation, and of the function F(x)=a(x)+[f(x)-b2(x)]/4c(x)F(x)=a(x)+f(x)-b2(x)/4c(x), where f(x)f(x) is an arbitrary function. The second integrability case is obtained for the “reduced” Riccati equation with b(x)≡0b(x)≡0. If the coefficients a(x)a(x) and c(x)c(x) satisfy the condition ±df(x)/c(x)/dx=a(x)+f(x), where f(x)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.