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The number, $N$, of animals of a certain species at time $t$ years increases at a rate of $aN$ per year by births, but decreases at a rate of $\mu t$ per year by deaths, where $a$ and $\mu$ are positive constants.

Modelled as continuous variables, $N$ and $t$ are related by the differential equation: $dN/dt=aN-\mu t$

Given that $N=N(0)$ when $t=0$, find $N$ in terms of $t$, $a$, $\mu$ and $N(0)$.

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    I know how to use integrating factor to solve this type of questions, but I'm confused by the "Given that $N=N(0)$ when t=0 and express N in terms of $N(0)$ and ...".2012-07-15

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You can use an integrating factor.

$e^{-at}N'(t) - ae^{-at}N(t) = -\mu t e^{-at}$

Now undo the product rule.

$\left(e^{-at} N(t)\right)' = -\mu te^{-at}$

Now integrate to see that $ e^{-at}N(t) - N(0) = -\mu \int_0^t se^{-as}\,ds.$

To finish, integrate by parts and solve for $N$.

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    @anon fixed the omission Thanks.2012-07-14