From Osher Doctorow
Not only the Golden Mean, but generalizations of it can be obtained by
setting:
1) y = t in the Riccati Differential Equation dy/dt = A + By + Cy^2
(A, B, C constant)
For the Golden Mean, set A = 0, B = C = 1. We get:
2) dy/dt = = 1 = t + t^2
which is to say:
3) t^2 + t - 1 = 0
which has solution:
4) t = [-1 +/- sqrt(1 + 4) ]/2
which for the plus sign is the Golden Mean. The case with the minus
( - ) sign is a well-known type of mean (its name escapes me at the
moment).
Readers can try various values of A, B, C to generalize the above.
Osher Doctorow
.
|
