Re: dy/dt = ky^2 (Riccati) and its Solution

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/25/04 

Date: Wed, 25 Aug 2004 02:22:35 +0000 (UTC)

On 24 Aug 04 14:34:32 -0400 (EDT), Osher Doctorow wrote:
>Exponentials are also of enormous importance in Lie groups and Lie
>algebras and in complex analysis besides their importance in generat-
>ing trigonometric functions. See "Exponentials form a basis of
>discrete holomorphic functions," Christian Mercat (Technische U.
>Berlin) arXiv:math-ph/0210016 8 Oct 2002. Mercat also proves that
>on a convex set, the discrete polynomials also form a basis.

It is interesting to note that the logistic differential equation,
which is a special type of Riccati differential equation, is just
at the tip of the transition from dy/dt = ky^2 to dy/dt = A(t)y^2
where A(t) is not constant, although more generally it satisfies
dy/dt = ay(b - y). In fact, we have:

1) d[exp(ax)/[bexp(ax) + c] = ac[exp(ax)/(bexp(ax) + c)[1/(bexp(ax
+ c)] = acy.y/exp(ax) = acy^2A(x)

where y = exp(ax)/[bexp(ax + c)] and A(x) = 1/[exp(ax) + c].

Osher Doctorow