Math 308 - Differential Equations Fall 2002
Homework Assignment 7
Due Friday, November 1.
For Questions 1-4,
(a) Find the general solution to the system d~
Y
dt =A~
Y.
(b) Sketch the phase portrait. Include the “straight line” solutions, and sketch several additional qualita-
tively correct trajectories in the phase plane. (I recommend that you check your answer with Maple
or with the software on the CD provided with the text. An example of using Maple to create a phase
portrait is attached.)
(c) Classify the equilibrium point of the system as either a saddle, a source or a sink. (The matrices have
been chosen so that these are the only possibilities.)
(d) Solve the initial value problem ~
Y(0) = ·2
1¸. Draw this solution in your phase portrait in part (b).
(Use a different color or use a dashed line to indicate this solution.) Also sketch x(t) and y(t) (where
~
Y=·x
y¸) for this solution, including positive and negative values for t.
1. A=·1−1
−6−4¸
2. A=·1−1
0 3 ¸
3. A=·1/2−1
−5/2−4¸
4. A=·−4−1
−2−3¸
5. Suppose that for some matrix A,A~v1=λ1~v1, and A~v2=λ2~v2, and λ16=λ2. Show that ~v1and ~v2are
linearly independent. (In other words, eigenvectors associated with different eigenvalues are always
linearly independent.)
Hint: Consider the equation
k1~v1+k2~v2=~
0.(1)
To show that ~v1and ~v2are linearly independent, we must show that the only solution to this equation
is k1=k2= 0. As a start, consider multiplying both sides of (1) by Ato get a new equation.
Then consider multiplying (1) by, say, λ1to get another new equation. Now subtract one of the new
equations from the other new equation, and see where that leads.
More on the other side!
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