Math 2280-1 Maple Project 2
Summer 2010
Directions: Hand in a single Maple document, printed and stapled neatly, which contains the
answers to the exercises below. At the top of this document, you should create a text field with an
appropriate title, the date, your name, and UID. Below this header, please answer the exercises in
order. If an exercise calls for computations by hand, you can type up your solutions in a text field,
or leave enough space so that you can hand-write your computation/explanation after printing the
document.
Maple Help: Maple has an extensive Help menu! You can search for commands to find syntax,
related commands, and extensive examples. Additionally, there are some introduction to Maple
guides posted on our class webpage (www.math.utah.edu/∼kitchen/2280Sum10). You can also get
help from tutors in the tutoring center, and follow the guidelines in the book as necessary.
Exercises:
1. Plotting Solution Families
Please re-scale your plots to fit on 1/4 page when printed!
(a)
i. Plot a family of solution curves for the differential equation
y 00 + 2y 0 + 2y = 0
satisfying y(0) = 1 (and letting y 0 (0) vary).
ii. Repeat part (i) letting y(0) vary, but fixing y 0 (0) = 1.
iii. Give a hand-written particular solution to the initial value problem y 00 +2y 0 +2y = 0,
y(0) = 1, y 0 (0) = 1.
(b) Plot a family of solution curves for
y (3) − 3y 00 + 4y 0 − 2y = 0
satisfying initial values y 0 (0) = 0 and y 00 (0) = 0. Include a hand-written solution to the
differential equation.
2. Variation of Parameters
Use Maple to implement the method of variation of parameters to find the particular solution
yp to
y 00 + y = 12x2 sin x.
3. Forced Vibrations
Investigate the solution corresponding to
25x00 + 10x0 + 226x = 2700te−t/5 cos 3t,
x(0) = 0,
x0 (0) = 0
by solving the equation with the command dsolve, then graphing the solution along with its
amplitude envelope.
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4. Logistic Modeling of Population Data
In this problem you will use census data to derive a population model, then use that model
to predict future populations. This problem follows the Section 2.1 Application in your text
on pages 90-92.
The logistic equation for a generic population in which the birth rate is linear in P and the
death rate is constant takes the form
dP
= aP + bP 2 ,
dt
P (0) = P0 ,
where a and b are constants which depend on the organism for which we are modeling the
population. How can we determine a and b? Suppose we measure the population P (ti ) at
time intervals ti , with i = 0, 1, 2, ..., n. Then, rewriting the logistic equation as
1 dP
= a + bP,
P dt
we see the points
P 0 (ti )
P (ti ),
P (ti )
all should lie on a straight line with y-intercept a and slope b.
P 0 (ti )
(a) Plot the points P (ti ), P (ti ) given by the U.S. Census data in Figure 2.1.9 of the text
on page 92. Verify their approximations of P 0 (ti ) by evaluating the formula
P 0 (ti ) ≈
P (ti+1 ) − P (ti−1 )
ti+1 − ti−1
for each i. Count time t in years from 1800. Use Maple to plot a best-fit line for these
points and determine its y-intercept a and slope b.
(b) Use dsolve to solve the logistic equation with the values a and b that you found in part
(a) and initial value P0 corresponding to the population in 1800. Compare the predicted
population for the year 2000 with the actual population listed in Figure 2.1.4.
(c) Repeat parts (a) and (b) using the U.S. Population data in Figure 2.1.4 on page 84 and
plotting only the data for 1900 and beyond (so t is time in years from 1900, and P0 is
the population in 1900). What does each model predict the U.S. Population to be this
year (the Census Bureau estimates 308.4 million)? What does each model predict the
U.S. Population to be in 2040?
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