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 )

-
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?
-

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.