Homework 1

Instructions: You can use either LATEX or MS Word to typeset this homework. In the latter case, you must use Equation Editor to typeset all formulas. The inished homework needs to be uploaded as PDF and should ideally look like a handout.

1. Use separation of variables to solve

dx

= a + b x, dt

x(0) = x0 .

Validate your answer symbolically and in Matlab.   For numerical validation use a = 2, b = — 1, x0   = 10.  Experiment with the time interval when you use ode45 to make the plot look attractive.

2. Show that the solution of

dPdt = a P2 ,    P (0) = P0

becomes ininite in inite time. Attempt to solve the IVP using ode45 on the time interval that extends beyond the singularity—the point at which the solution becomes ininite. What does ode45 do?

3. Use separation of variables to solve

dPdt = P (1 —   P2 ),    P (0) = P0 .

Validate your symbolic solution in Matlab for P0  = .1, .5, 1, 1.5; you can put all plots on the same axes and indicate the values of the initial condition in the legend. Is this a realistic model of population growth?

4. Solve the logistic equation

dPdt = a P —   b P2 ,    P (t0 ) = P0 ,

without looking into your notes or the handout (you do not need to typeset the solution in your homework). Time yourself—you will need to do this on the irst exam and you do not want to spend more than 6– 7 minutes.  Next download world  census.txt from Canvas: this is mid-year world population data from 1950 until 2008. Fit the logistic curve to the data and ind the limiting world population—the carrying

capacity of Earth.