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Expert: - 2/17/2009


Question
i know how to do (i). But i dont know how to continue the rest. Please guide me. Here are the full question (i) in a certain pond, the rate of increase of the number of fish is proportional to the number of fish,n,present time t.Assuming that n can be regarded as a continuous variable,write down a differential equation relating n & t,& hence show that n=Ae^kt where A & k are constants. (ii)In a revised model, it is assumed also that fish are removed from the pond,by anglers & by natural wastage,at the constant rate of p per unit time,so that dn/dt = kn-p.Given that k=2,p=100 & that initially there were 500 fish in the pond,solve this differential equation,expressing n in terms of t.(iii)give a reason why this revised model is not satisfactory for large values of t. Cheers

Answer
Let n be the fish population.  That's really n(t) where t is time.

What this question says is that the increase in n(t)
is proporional to n and t.  This says that dn/dt = knt.
Divide both sides by n and integrate that,
then you have the equation.

Take dn/dt = kn - p and solve this differential equation
{ not quite a simple ordinary differential equation }.

Put in k = 2 and p = 100.

Note that as t increases, the function gets to far out of control
to describe a fish population.  

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