 ## Applications of Differential Equations

### Population Dynamics

(continued from last page...)

### The Malthusian Model

This model made some rather wild assumptions about how populations grow, so we don't expect it to be the greatest predictor of what really happens in a population over an extended time period. Nevertheless, let's investigate it. The initial value problem is as follows:

dp/dt = k p
p(0) = po

where p is the population at time t, k is the growth constant, and po is the initial population size.

Now it's actually pretty easy to solve this initial value problem by hand, but let's have Mathematica do it, since it's readily available to us. Go ahead and tell Mathematica to solve this differential equation, and use the initial datum from our US census data as the initial condition. (Don't forget, use p[t], not just p, and use == in both the differential equation and the initial condition. Also, since we don't yet know the value of k, there'll be a k in your answer.) Read the result into the variable malthus.

You've probably noticed that I'm not typing out every command for you anymore. That's because you need to get better at entering these problems for yourself. Go on over to Mathematica and get on with the job.

Let's go see what you should have gotten... If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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