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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:

whose variables satisfy
 = (302) = (cy + dx) y + x (ay + bx) (303) = c uv + (a + d) u + b uv-1 (304) = (a + d) + c v + b v-1 (305)

 = (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,

 = (a + d) + c v + b v-1 (309) = - 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:

 = v - E v-1 (311) = - v2 - E (312)

whereupon
 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 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 .

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