The first assumption is that developer productivity declines with a constant rate $r$:

(1) $\Large \frac{dP(t)}{dt}$ = $\large -r \cdot P(t)$

Divide by $P(t)$:

(2) $\Large \frac{1}{P(t)}$ $\large \cdot$ $\Large \frac{dP(t)}{dt}$ = $\large -r$

(3) $\large \frac{dP(t)}{P(t)}$ = $\large -r dt$

(4) $\large \int \frac{dP(t)}{P(t)}$ = $-r \int t dt$

(5) $\large ln(P(t))= -rt +C$

Exponantiate both sides:

(6) $\large P(t) = e^{-rt}e^C$

(7) $\large P(t) = e^C \cdot e^{-rt} =e^C \cdot 1$

As P(0) = P_0 we get:

(8) $\large P(0) = e^C \cdot e^{-r \cdot 0} = e^C \cdot 1$

So $\large P(0) = e^C$ giving:
(9) $\large P(t) = P_0 \cdot e^{-rt}$

See: http://en.wikipedia.org/wiki/Exponential_decay#Solution_of_the_differential_equation
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math_public
created
Sun, 11 Dec 2011 22:31:01 GMT
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dirkjan
modified
Sun, 11 Dec 2011 22:31:01 GMT
modifier
dirkjan
creator
dirkjan