Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $N(0)$.

share|improve this question
1  
integrating factor method hint: $e^{at}(e^{-at}N)'=N'-aN$. –  anon Jul 14 '12 at 17:29
    
This is just a very standard first-order linear ODE... What have you tried? –  KennyTM Jul 14 '12 at 17:29
    
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 ...". –  Vic. Jul 15 '12 at 2:55

1 Answer 1

up vote 2 down vote accepted

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 $N$.

share|improve this answer
    
@anon fixed the omission Thanks. –  ncmathsadist Jul 14 '12 at 17:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.