Quantum Gravity 201.1: Golden Mean Function From dy/dt = t^2 + 2ty + y^2

From Osher Doctorow

I just explored:

1) dy/dt = t^2 + 2ty + y^2 = (t + y)^2

whose solution is:

2) y = tan(t) - t + y(0)

Lets examine the "Golden Mean "generating" function u:

3) u = y^2 - y - 1

which, when u = 0, yields y = golden mean = (1 + sqrt(5))/2.

Let's set u = dy/dt:

4) dy/dt = y^2 - y - 1

Now equate (1) and (4):

5) dy/dt = t^2 + 2ty + y^2 = y^2 - y - 1

which reduces in terms of t and y to:

6) t^2 + 2ty + y + 1 = 0

or, factoring:

7) y(1 + 2t) = -(1 + t^2)

and hence:

8) y = -(1 + t^2)/(1 + 2t) if t is not -1/2

In fact, y can be made positive if 1 + 2t < 0, that is to say t <
-1/2.

By long division, (8) yields:

9) y = -{(1/2)t + [1 - (1/2)t]/[1 + 2t] } if t is not -1/2

Osher Doctorow

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