Question

The Riccati equation is a nonlinear differential equation that is of considerable importance in particle dynamics. By use of a change of the dependent variable y = -[1/P(x)]dz/dx, show that the Riccati differential equation becomes the one below on the ending point. Notice that the resulting transformation is a second-order linear differential equation.

Starting point:

(dy/dx) =P(x) y^2 +Q(x) y +R(x)

Ending point:

(d^2z/dx^2)-(Q+(1/p)(dp/dx) ) dz/dx +PRz=0

Answer 1

$$y^2=\frac{1}{P^2}\left(\frac{dz}{dx}\right)^2\\\frac{dy}{dx}=-\frac{1}{P}\frac{d^2z}{dx^2}+\frac{1}{P^2}\frac{dz}{dx}\frac{dP}{dx}$$

$$-\frac{1}{P}\frac{d^2z}{dx^2}+\frac{1}{P^2}\frac{dz}{dx}\frac{dP}{dx}=\frac{1}{P}\left(\frac{dz}{dx}\right)^2-Q\frac{1}{P}\frac{dz}{dx}+R$$

$$\frac{d^2z}{dx^2}-\frac{1}{P}\frac{dz}{dx}\frac{dP}{dx}+\left(\frac{dz}{dx}\right)^2-Q\frac{dz}{dx}+RP=0$$