Quantum Gravity 295.2: Bi-Logistic q(x, y) Differential Equation using Probable Causation/Influence (PI)

We return now to the equation:

1) q(x, y) = ap + bp^2 + cF + dF, a and c > 0, b and d < 0 constants,
p = 1 + y - x with y = P(B) < = x = P(A), F = F(x, y) the joint
cumulative distribution function (cdf) of continuous random variables
X, Y at X = x, Y = y. This equation also extends to the cases where
both a and b are 0 but not c and d, or both c and d are 0 but not a
and b.

A special case of this equation is:

2) q(x, y) = ap(1 - p) + cF(1 - F)

which Readers will recognize as a generalized "Bi-Logistic"
Differential equation in the sense that it is Logistic in p if c is 0
and Logistic in F if a is 0, assuming that q(x, y) is a time
derivative of p and/or F.

A solution of (2) for q(x, y) a time derivative of p and/or (or +/-) F
or a linear combination of both is:

3) p = exp(k1t)/[1 + exp(k1t)], F = exp(k2t)/[1 + exp(k2t)], k1 and k2
real constants.

As an example if k1 = 1 in (3) and c= 0 in (2), consider:

4) p = exp(t)/[1 + exp(t)]

Then:

5) p(1 - p) = exp(t)/(1 + exp(t))^2 = dp/dt

which Readers can easily verify with a little algebra.

I did research on a linked system of differential equations of type
(2) (among other things) for my Ph.D. about 3 decades ago.

Osher Doctorow

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