consider an animal population P(t) with constant death rate 0.02 (deaths per animal per month) and with birth rate proportional to P. suppose that P(0)=200 and P'(0)=2. when is P=1000?
$\displaystyle D = \frac{1}{5}$constant death rate 0.02 (deaths per animal per month)
$\displaystyle B = kP$and with birth rate proportional to P.
And with...
$\displaystyle \frac{dP}{dt} = B - D$
... we have...
$\displaystyle \frac{dP}{dt} = kP - \frac{1}{5}$
... which is suitably arranged as either...
$\displaystyle \frac{1}{kP - \frac{1}{5}}\ k\ \frac{dP}{dt} = k$
... or else...
$\displaystyle \frac{dP}{dt} - kP = - \frac{1}{5}$
... according to taste.
Just in case a picture helps...
... where (key in spoiler) ...
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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