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Previous: polar coordinates and Prüfer's
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
Microcomputadoras
2001-04-05