Unformatted text preview: Diﬀerential Equations
Homework Assignment 7
Due on Thursday, March 24, at the start of the recitation you are registered in
Your homework should have a cover sheet with the following information: course title,
recitation, last and ﬁrst name (as they appear in the roster), number of homework. If you
use several sheets, please staple them. The problems should be written neatly and in the
order they were assigned.
Problem 1. Determine the eigenvalues in terms of α. Find all the critical values of α, where the
qualitative nature of the phase portrait for the system changes. Describe how the
phase portrait changes as α passes through each critical value.
(a)
dx
¯
=
dt 4α
x
¯
4 −3 dx
¯
=
dt α −2
x
¯
2α (b) (c)
dx
¯
=
dt −1 α
x
¯
−1 −1 Problem 2. Find the general solution of the system below. Draw the phase portrait of the system.
What is the origin called in this case? Is the origin stable, unstable or semistable?
(a)
dx
¯
=
dt 3 −4
x
¯
1 −1 dx
¯
=
dt −3 4
x
¯
−1 1 (b) ...
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 Spring '07
 Tolle
 Differential Equations, Equations, Critical Point, dt, critical value, phase portrait, Critical phenomena

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