Background. The population model is .

To find a and b, consider the "linearization" .

Computer Lab Work.

Exercise 1. Use the following data for 1790 to 1930.
Note. In order to get the year, add 1800 to the first coordinate.

One way to enter the data as a matrix is to use the Input menu at the top
of the screen, then select the Create Table/Matrix/Pallet and ask for
15 rows and 2 columns. The computer will display the following:

Then move the cursor and to the appropriate location and enter the desired number.
When you are finished it should look like.

Or, the data could be entered in the the list form.

Select the second coordinates which are the population values and store them in q.

Select the second coordinates which are the population values for 1800-1920 and store them in P.

Form the difference quotients which are an approximation to dP/dt and store them in dP.

Put together the desired point {P,1/P dP} for the years 1800-1920 and store them in pts and plot the points.

Find the "least squares line" fitting these points.

Plot the least squares line and the points.

Use Mathematica to find the general solution to the differential
equation , and store the solution as the function f[t].

Dig out the constants a and b for the differential equation from the least
squares fit,and notice that Mathematica has substituted these values into f[t].

Form the equation f[0] = P[[0]] which means, set the value of the function
corresponding to 1800 equal to the value of the population for that year.
Then solve the equation for the constant C[1] and substitute this value
back into the formula for f[t] and plot f[t].

In order to make the list of points for the years 1800-1920 it will be necessary to type
them all in again, or make a copy of "data" and learn how to edit out the first and last
rows using the cut command. When you are done, call the new set "points."
Then plot the points, and then show the graph and points together.

Isn't that a nice fit !

Use f[t] to predict the values of the population for the years 1930, 1960, 1990.

The model does not reflect the global populations changes.
Do you know some historical facts that might have caused these changes ?
What are they ?

Exercise 2. Use the data in your text for the years 1890-1990.
Put together the desired point {P, 1/P dP} for the years 1900-1980.
Find a new function f[t] that can be used to better predict the population.
Use this f[t] to predict the values of the population for the years 2000, 2010, 2020.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998