From Osher Doctorow

Let's look at the Riccati Differential equation:

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

Since we can generalize this to Dt(y) = A(t) + B(t)y + C(t)y^2 where
Dt(y) is the partial derivative of y with respect to t, so that y =
y(x, t) for example, we could also formally take the partial derivative
of both sides of the last equation with respect to y:

2) Dyt(y) = B(t) + 2C(t)y

Since in my recent threads I've pointed out that the second mixed
partial derivative for probability density functions and related
functions relates to the acceleration of the physical Universe, let's
see what's required for Dyt(y) to change from negative to positive with
time. We have the transition at:

3) B(t) + 2C(t)y = 0

and therefo

4) y = -B(t)/(2C(t))

Since C(t) is usually taken negative and B(t) positive in the Logistic
and Simple Exponential subcases of the Riccati Differential equation,
those give a nonnegative y as for example a "radius" of the universe or
a related probability. For y to make the transition from less than
-B(t/(2C(t)) to greater than this is equivalent to Dyt(y) changing from
negative to positive with time. So for example if y is the radius of a
spherical Universe, there is a radius -B(t)/(2C(t)) at which the
transition from deceleration to acceleration occurs in time.

Osher Doctorow

From Osher Doctorow

The expression:

1) Dyt(y)

seems awfully strange. For one thing, the order of differentiation
doesn't seem to be interchangeable.

Still, it's not much worse than some of Feynman's expressions (although
Feynman had nothing to do with this).

It could be argued that for Dty or Dyt(y) to make sense, the quantity
to which the operator Dty or Dyt is applied must be a function of t and
y but not equal to t or to y. But this seems like an over-strong
restriction.

It could also be argued that in the original Riccati Differential
equation:

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

this only makes sense if y = y(t) is a function of time and that if we
generalize this equation to Dt(y) from dy/dt where Dt(y) is the partial
derivative of y with respect to t, then y = y(x, t) for some x other
than y is required in order for x and t to be "independent" or else no
such x different from t can be found.

Nevertheless, there is a "formal" partial derivative of the right hand
side of (2) with respect to y, and since the operator involved is Dy
(the partial derivative with respect to y), the expression Dy(dy/dt)
also has to make sense formally.

Osher Doctorow