projective coordinates and Ricatti's equation

Another useful conversion of a second order differential equation into a nonlinear first order equation, which works just as well for a pair of first order equations, is to introduce a projective transformation:

$$\begin{array}{cccc} \begin{array}{ccc} u & = & xy \\ v ... ...c} y^2 & = & uv \\ x^2 & = & u/v \end{array} \end{array}$$

whose variables satisfy

$$\frac{du}{ds} \frac{dx}{ds} y + x\frac{dy}{ds}$$ = (302)

= (cy + dx) y + x (ay + bx) (303)

= c uv + (a + d) u + b uv^-1 (304)

$$u^{-1}\frac{du}{ds}$$ = (a + d) + c v + b v^-1 (305)

$$\frac{dv}{ds} y\frac{dx^{-1}}{ds} + \frac{dy}{ds}x^{-1}$$ = (306)

= -yx^-2(cy+dx) + (ay+bx)x^-1 (307)

= - c v² + (a - d) v + b (308)

which gives the interesting result that both the logarithmic derivative of u and the derivative of v itself, depend only on v. The equation for v is nonlinear, but quadratic, and of the first order. An equation with this format is called a Ricatti equation.

Again summarizing,

$$u^{-1}\frac{du}{ds}$$ = (a + d) + c v + b v^-1 (309)

$$\frac{dv}{ds}$$ = - c v² + (a - d) v + b. (310)

Of course, the objective is to obtain x and y through the intermediary of u and v.

If the same example used for the Prüfer transformation, namely the zero potential Schrödinger equation, is used as a test case for the Ricatti equation, the result is:

$$u^{-1}\frac{du}{ds}$$ = v - E v^-1 (311)

$$\frac{dv}{ds}$$ = - v² - E (312)

whereupon

$$v(0) - \surd E\ \tan s,$$ v(s) = (313)

which is an agreeable result. Just as the self-adjoint form of the Prüfer transformation inproved the appearance of its results, similar adjustments can be made in the Ricatti equation. Incorporating a $\surd E$ in place of E would improve the format of the equation for u, but both Ricatti and Prüfer transformations would benefit from a change of independent variable that would absorb $\surd E$.