A rather surprising equation comes from the Volterra-Like Convolution:

1) (y *L y)(x, t) = xy(t)y(x - t)

and in particular:

2) (y *L y)(x, 0) = y(0)xy(x) = (for short) y(0)xy or yo xy

First, square the left hand side of (2):

3) (y *L y)(x, 0)^2 = yo^2 x^2 y^2

Then differentiate the left hand side of (2) with respect to x:

4) d(y *L y)(x, 0)/dx = yo(y + xdy/dx)   (left hand side for short
written d(y *L y)/dx )

Now set up a Riccati Differential Equation of form:

5) d(y *L y)(x, 0)/dx = A + B (y *L y) + C(y *L y)^2  where A = A(t),
B = B(t), C = C(t).

We get:

6) yo(y + xdy/dx) = A + By(0)xy + Cy(0)^2x^2 y^2

and dividing through by yo assumed not to be 0 in (6):

7) y + xdy/dx = (A/yo) + Bxy + Cy(0) x^2 y^2

which, subtracting y from both sides of (7), yields:

8) xdy/dx = (A/yo) + (Bx - 1)y + Cy(0)x^2 y^2

or dividing both sides by nonzero x:

9) dy/dx = A/(xyo) + [(Bx - 1)/x]y + Cy(0)xy^2

With x = t, equation (9) has the unusual properties of selecting some
rather simple coefficients of the constant y, and y^2 terms of the
Riccati Differential Equation (9) involving 1/t, (t - 1)/t, and t in
the respective first, second, and third right hand side terms of (9),
which for example for simple choices of A, B, C, such as A = 1, B = 1,
C = -1 would have an especially simple form rather difficult to derive
from either "intuition" or other theories.

Hopefully, I will continue this shortly.

Osher Doctorow