1. ## population problem

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?

2. ## Re: population problem

constant death rate 0.02 (deaths per animal per month)
$\displaystyle D = \frac{1}{5}$

and with birth rate proportional to P.
$\displaystyle B = kP$

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:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case t), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

But for the integrating factor this is wrapped inside the legs-uncrossed version of...

... the product rule, where, again, straight continuous lines are differentiating downwards with respect to t.

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