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Next: solving a Schwartz derivative Up: Second order differential equations Previous: polar coordinates and Prüfer's

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

whose variables satisfy
$\displaystyle \frac{du}{ds}$ = $\displaystyle \frac{dx}{ds} y + x\frac{dy}{ds}$ (302)
  = (cy + dx) y + x (ay + bx) (303)
  = c uv + (a + d) u + b uv-1 (304)
$\displaystyle u^{-1}\frac{du}{ds}$ = (a + d) + c v + b v-1 (305)


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

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

$\displaystyle u^{-1}\frac{du}{ds}$ = v - E v-1 (311)
$\displaystyle \frac{dv}{ds}$ = - v2 - E (312)

whereupon
v(s) = $\displaystyle v(0) - \surd E\ \tan s,$ (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 $\surd E$.


next up previous contents
Next: solving a Schwartz derivative Up: Second order differential equations Previous: polar coordinates and Prüfer's
Microcomputadoras
2001-04-05