MATH 215/255
Fall 2014
Assignment 8
due 11/19
§3.4, §3.7, §3.5, §3.9
Solutions to selected exercises can be found in [Lebl], starting from page 303.
• 3.4.6: a) Find the general solution of x01 = 2x1 , x02 = 3x2 using the eigenvalue method
(first write the system in the form ~x0 = A~x). b) Solve the system by solving each
equation and verify you get the same general solution.
• 3.4.7: Find the general solution of x01 = 3x1 + x2 , x02 = 2x1 + 4x2 using the eigenvalue
method.
• 3.4.8: Find the general solution of x01 = x1 − 2x2 , x02 = 2x1 + x2 using the eigenvalue
method. Do not use complex exponentials in your solution.
5 −3
• 3.7.2: Let A =
. Find the general solution of ~x0 = A~x.
3 −1
a a
, where a, b and c are unknowns. Suppose that 5 is a
• 3.7.104: Let A =
b c
1
is the eigenvector. Find A and
doubled eigenvalue of defect 1, and suppose that
0
show that there is only one solution.
• 3.5.101: Describe the behavior of the following systems without solving:
a) x0 = x + y, y 0 = x − y.
b) x01 = x1 + x2 , x02 = 2x2 .
c) x01 = −2x2 , x02 = 2x1 .
d) x0 = x + 3y, y 0 = −2x − 4y.
e) x0 = x − 4y, y 0 = −4x + y.
• 3.5.102: Suppose that ~x0 = A~x where A is a 2 × 2 matrix with eigenvalues 2 ± i.
Describe the behavior.
• 3.9.4: Find a particular solution to x0 = x + 2y + 2t, y 0 = 3x + 2y − 4. a) using
undetermined coefficients. b) using variation of parameters.
• 3.9.101: Find a particular solution to x0 = 5x + 4y + t, y 0 = x + 8y − t. a) using
undetermined coefficients. b) using variation of parameters.
• 3.9.102: Find a particular solution to x0 = y + et , y 0 = x + et . a) using undetermined
coefficients. b) using variation of parameters.