Riccati Type 1 Solutions

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/26/04 

Date: Thu, 26 Aug 2004 13:44:48 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Riccati Type 1 Solutions
Copyright by Owner Osher Doctorow Ph.D.
First Published 2004

Define a Riccati Type 1 Solution as a rational solution:

1) y = P(t)/Q(t)

of the Riccati differential equation:

2) dy/dt = A(t) + B(t)y + C(t)y^2

such that y is a solution of (2) iff P(t) and Q(t) are solutions of
equation (2). We can specialize this to dy/dt = ky^2 and dy/dt =
ky(c - y) subtypes of Riccati and also dy/dt = ky, etc.

We know that y = exp(t), y = exp(-t), y = t, y = 1/t are solutions
of (2), and so are y = t + k, 1/(t + k), y = kexp(t), y = kexp(-t),
y = k1exp(t) + k2, etc. So ratios of these may be Type 1 Solutions,
and computation verifies that.

What happens if we differentiate (1)? We get:

3) dy/dt = [QP' - PQ']/Q^2

where the arguments are suppressed for brevity. I will try to
discuss this further shortly.

Osher Doctorow