Extra credit! (10 points!)
1. Weve seen that a 2 2 linear system can have only periodic solutions (if we
have a spiral). Can this ever happen for a 3 3 system? What if the coecient
matrix is invertible?
1
ODE, Math S3027D Section 001
Summer 2008 Exam 1
Name:
June 12, 2007
Do all problems, in any order.
Show your work. An answer alone may not receive full credit.
No notes, texts, or calculators may be used on this exam.
Problem Possible Points
Points Earned
ODE, Review for Exam 1
June 9, 2008
1. Consider the autonomous equation x = x3 x.
(a) Sketch the phase line (the xaxis) and the corresponding direction eld on the plane.
(b) Sketch graphs of some solutions. Be sure to include at least one solution with va
ODE, Review for Exam 1
June 9, 2008
1. Consider the autonomous equation x = x3 x.
(a) Sketch the phase line (the xaxis) and the corresponding direction eld on the plane.
(b) Sketch graphs of some solutions. Be sure to include at least one solution with va
ODE, Homework 4, Last homework!
Due Monday, June 30, 2008
1. (7.5.4) Find the general solution of the system
x=
11
x
4 2
and draw some trajectories of solutions.
2. (7.5.13) Find the general solution of the system
1
1
1
1 1 x.
x = 2
8 5 3
3. (7.6.6) Find
ODE, Homework 3
Due Thursday, June 26, 2008
1. (6.3.31) We saw in class that if f is periodic with period T (ie f (t + T ) = f (t) for all t
and some xed minimal positive number T ) then
Lcfw_f (t) =
T
0
est f (t) dt
.
1 esT
Use this to compute the Laplac
ODE, Homework 1
Due June 2, 2008
1. (3.3.3) Determine whether the pair of functions f (t) = et cos t, g (t) = et sin t is
linearly independent or linearly dependent.
2. (4.2.23) Find the general solution of the dierential equation y 5y + 3y + y = 0.
3. (3
ODE, Homework 1
Due June 2, 2008
1. Draw a reasonably detailed direction eld for y = y + t. Sketch 3 solutions with dierent
behaviors. Are there any equilibrium solutions? Are there any asymptotic solutions? If so,
what are they?
2. Sketch a direction eld
ODE, Review for Final Exam
June 27, 2008
1. Find the inverse Laplace transform of X (s) =
s (s 2
4
.
+ 2s + 2)
2. What is the Laplace transform of the solution to y + 4y + 5y = sin 2t with initial conditions y (0) = 1 and y (0) = 2?
3. Compute the convolu
ODE, Math S3027D Section 001
Summer 2008 Final Exam
Name:
July 3, 2008
Do all problems, in any order.
Show your work. An answer alone may not receive full credit.
No notes, texts, or calculators may be used on this exam. You have 3 hours.
Problem Possible