Population Modeling using DE's

[prace]

Hi,

I was wondering if anyone out here on a Friday night could help me understand population modeling. Here is what I have as a problem (this is pretty simple because my goal here is to understand the thinking behind the madness )

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The population of a certain community is known to increase at a rate proportional to the number of people present at any time. If the population has doubled in 5 years, how long will it take to triple, to quadruple?
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So I understand that the rate = dy/dt and proportional translates to "something" = "some constant" times "something", or (dy/dt)=Ky, or in this case, dP/dt = kP.

Solving this DE I get the equation P(t)= P(initial)e^(kt). So all I am told is that the population doubles in 5 years. So what can I do? I can't assume an arbitrary number as the initial population can I? So I have one equation with three unknowns, P(initial), P(t), and k.

Any insight would be great. Thanks!

 

[courtrigrad]

you know that \frac{dP}{dt} = kP. The solution is:

P(t) = P_{0}e^{kt}

You also know that 2P_{0}= P_{0}e^{5k}. What can you do from here? Solve for k:

e^{5k} = 2

k = \frac{\ln 2}{5}.
Then solve the equations for t:

3P_{0} = P_{0}e^{(\frac{\ln 2}{5})t}
4P_{0} = P_{0}e^{(\frac{\ln 2}{5})t}

 

[prace]

Oh man... So here the P_{0}'s will cancel and all I have to do is take the natural logs of both sides to get t? man... I think I was making it a lot harder than it really was. Thank you for taking the time to show me this.