Math 308 - Differential Equations Fall 2002
Homework Assignment 9
Due Friday, November 22.
1. Consider the system d~
Y
dt =A~
Y, where A=·−1 1/2
−1/2 0 ¸.
(a) Find the general solution.
(b) Sketch the phase portrait. Include the straight-line solutions, and a few other curves to illustrate
the qualitative behavior. Use nullclines to help draw reasonably accurate solution curves.
(c) Solve the initial value problem ~
Y(0) = ·−2
0¸, and sketch the corresponding xand ycoordinates
of this solution versus t(include positive and negative values of t).
2. Consider
d~
Y
dt =A~
Y , where A=·p1
0p¸.
Use a computer (either Maple or the “HPG System Solver” on the text’s CD) to create phase portraits
when p=−1, p=−0.1, p= 0, p= 0.1, p= 1. Include solution curves that go through the points
(2,0), (2,2) and (0,2). Hand in copies of the cases p=−1, p= 0 and p= 1, and describe (with
complete sentences) how the phase portrait changes as pchanges from −1 through 0 to 1.
3. Now consider
d~
Y
dt =A~
Y , where A=·−1p
0−1¸.
and repeat the previous problem.
Text Problems
•Section 3.6/ 8, 10, 12 (Do these without converting to a system!)
Recommended Exercises - Do Not Hand In – Check the answers in the back of the book.
•Section 3.5/ 1–7 odds, 14
•Section 3.6/ 1–11 odds (Do these without converting to a system!), 33
•Section 3.7/ 3, 7
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