Homework #2 Math 3200, Differential equations Spring 2000 - Kerley

The following projects are to be completed by Feb 11, 2000. A presentation by each group will be made beginning the week of Feb 14^th in Gilbreath 205. HTML files with supporting gif files will be implemented in Netscape and shown to the class. If you prefer, you can produce a file in Word with graphs inserted in the word documents(how?. Read on). Such graphs can be generated using a windows program such as PCMatlab, Maple, or MathCad and saved as a bmp file or windows metafile or some appropriate file. These bmp files can then be inserted in the word document. You will need to save your word file as a word document. In addition, you need to save it as a second file in HTML format. In saving your word document as an HTML, gif files are generated which will be needed for the demonstration in Netscape. I will also put each presentation on the ETSU server with the required HTML files and gif files at Each group needs to provide a hard copy of your word document which will include 2 graphs for distribution to the class. Some may want to use WordPerfect rather than Word. Either will work.

Everybody must submit the two following problems.

p 267, Exercise 7

Find the integral curve of the given equation, transform the given data set and use a least squares line to estimate k and C.

p 305, Exercise 17

Find the equilibria of each of the following. Then use a vector field to determine which are stable and which are unstable. Assume N > 0 and k >0.

Underwood and Vukosavijevic

Exercise 13 A glass of water is heated to 180 F and then left to cool in a room with a constant temperature of 80 F. Temperature measurements are made each minute resulting in the following data:

(a) Let y(t) be the temperature of the glass of water at time t. Explain why the mathematical model of the glass of water’s cooling is

(b) Find the integral curve of the equation in part (a).

(c) Transform the data using the integral curve in part (b), and then use least squares to estimate k and C.

(d) What will the temperature of the water be after 15 minutes?

Christian, McAllister, and Touchstone

Exercise 17 The same soup can with the same hole is filled to a different height and again observed, resulting in the data

How long until the soup can is empty?

Harvey, Street, and Zimmer (Work 2 problems below)

The following data set lists the 30 year average normal temperatures by month for Juneau, Alaska, where t is in months and y is in F. 

The average yearly temperature is M = 40.1 F. Transform the data using (3.76), and then estimate a and b by applying least squares to the transformed data. Graph the data and the seasonal curve (3.73) which best fits the data.

Here are several sunset times in Eastern Standard time for Johnson City, TN in the year 1999.

The average sunset time is M = 6.5417 (i.e., about 6:32 p.m.). Transform the data using (3.76), and then estimate a and b by applying least squares to the transformed data. On June 15, which is t = 166 days after the beginning of the year, the sun set at 7:49 eastern standard time. Use the model to predict the sunset on June 15.

Howard, Phillips, and Keeling (Work 2 problems below)

Exercise 26 Often the carrying capacity N is determined by the way the data is collected. In particular, the table below lists the number of people infected per 10,000 population during a 1996 flu outbreak in the Alsace region of France.

Estimate the intrinsic growth rate k and the duration of the epidemic.

Exercise 27 Harvesting: Suppose that the number of trees in a certain forest increases at a rate of 5 % per year, that the carrying capacity of the forest is 1,000 trees per acre and that 200 trees per acre are harvested from the forest every year. (Apply MDEP, HP GRAPHICS CALCULATOR (I can capture the screen image for you) OR MAPLE TO EXHIBIT THE SLOPE FIELD) for part b below

(a) Find the equilibria of the process

(b) Use a slope field to determine which are stable and which are unstable.

(c) Interpret your findings in (b).