Due Tuesday, September 3 (Week 2, Day 1)
Assignment 1 This is selected problems from an old final from MAT 239. Note: I took out the requests to draw phase portraits in updated HW.
Due Tuesday., Sept. 10 (Week 3, Day 1)
Assignment 2
1.1: 1acde, 2ac, 3, 4, 5. Note: 1de are not covered in the text, Perko leads you through the process of eliminating t in these cases.
Sketch the 2-D phase portrait for dx/dt = -x, dy/dt = 2y
Sketch the 3-D phase portrait for dx/dt = -x, dy/dt = -2y, dz/dt = 3z
1.2: 1ac, 2, 3ac, 4, 5, 7.
Find the matrix A such that the ODE dx/dt = A x has the general solution
x = [1 0]T e-t c1 + [1 1]T e-2t c2
(The T stands for transpose. This is a way to write column vectors.)
Due Wednesday, Sept. 17 (Week 4 Day 1)
Assignment 3
1.3: 1b, 5, 6, 7, 8
(you can use propositions from this section.)
1.4: 1, 2, 5*, 6
(* You don't need to invert any matrices for this problem.
In 5(c) you would need to compute P-1 to compute eAt, but
it's OK to find the general solution in a form that does not require P-1.)
Hints: In 1.3#1.(b)
I'd like you to find the function of t (or theta) defined by f(t) = |A.(cos(t), sin(t))|, as described in class. The calculus for finding the max of f is rather involved,
so you can use the hint in the book that ||A||op = sqrt(max eigenvalue of ATA). Then you can plot f(t)
on a calculator and the norm you computed, and see that the max of f is ||A||op.
You might want to look at this Mathematica notebook, from MAT 667:
A times Ball Mathematica notebook. The image of a ball under a linear transformation is an ellipse (or ellipsoid, possibly degenerate).
Note that the min value of f is the second singular value of A.
To run the Mathematica notebook, click on the file and save it to your computer. Open the notebook in Mathematica.
Then put the cursor somewhere in the first cell and "Shift-Enter" the cell to define the function myBall. Then put the cursor in the second cell and run myBall with the input matrix.
You can change the matrix to the one in problem 1(b) and run again.
Due Tuesday, Sept. 26 (Week 5, Day 2)
Assignment 4
1.5: 1, 3, 5, 7, 9
1.6: 1-4.
Notes for section 1.6:
On problem 1, solve the initial value problem using the book's method (that is,
find e^{At}) and also
find the general solution using my method. Also, sketch the phase portrait for
problem 1 even though Perko didn't ask for it.
On problems 2-4, you don't have to solve the IVP. Find the
general solution like I did in class. (You don't need to invert any matrices.)
On problem 2, only sketch the phase portrait
restricted to the x1 - x2 plane.
Due Tuesday, Oct 8. (Week 7, Day 1)
Assignment 5
1.7: Problems 1 b c d, 2 a b in the book.
Then do the three problems in the Mathematica Notebook SplusNhomework.nb.
Note: One does not need to use Theorem 2 or its corollary,
but I want you to do the third problem (with repeated complex eigenvalues) using both Theorem 1 (with hypotheses modified
to allow complex eigenvalues) and Theorem 2.
Due Tuesday, Oct. 22 (Week 9, Day 1)
Assignment 6
Section 1.8: 5*, 6*, 7.
In problem 5b, choose 3 of the 7 possible cases,
and write down the form of the general solution (like I did in class) for them.
In problem 6, write down the general
solution but do not solve the IVP (4). Do not invert any matrices!
You may use Mathematica or your calculator to do the row reductions for problems 6 in the book,
but you can probably do them by hand.
Solve the single ODE (D-1)3 x = x''' - 3 x'' + 3x' - x = 0 for
x(t) in two ways:
(a) Find the general solution using MAT 239 techniques.
(b) Convert the single ODE to a system x dot = Ax,
where A is a 3x3 matrix, and find the general solution using the JCF.
Note that the scalar x is the first component of x.
(c) Explain (but don't formally prove) how the
algorithm we learned in MAT 239 implies that the matrix A obtained for any single linear
homogeneous ODE with constant coefficients has a Jordan Canonical Form with only one block for each eigenvalue.
Here are two Mathematica notebooks with examples to follow for the
computer problems:
JCF example 1 and the more verbose JCF example 1 verbose.
For repeated complex eigevalues with the JCF method, but you can look here:
JCF example 2. This example does more than I ask you to do this semester.
Here is an example of doing that example 1 with matlab, JCFexample1withCheck.m. This is also on BbLearn, since you might now be able to download this
from the browser.
I will send each of you an email with some similar problems assigned in an email attachment.
Do the problems following the examples, and reply with your answers as an attachment,
in the form of a Mathematica Notebook (.nb) or MATLAB script (.m).
As in the examples, you may only use the commands Eigenvalues[] and RowReduce[]
to find the general solution.
It's no fair using ExpMatrix[]!
Check that dx/dt - Ax = 0 for your solution.
Also, compute P-1AP like I do in those example notebooks to check
your P matrix.
Due Friday, Oct. 25 (Week 9), slip under my door.
Assignment 7
1.9: 2, 5(a,b,c).
1.10: 1, 2
Due Thursday, Nov. 7 (Week 11, Day 2)
Assignment 8
2.1: 1, 2, 3, 5
Note: In problem 2.1: 1, when they ask you to compute the derivative of the function, just compute the Jacobian
matrix of partial derivatives.
Due Tuesday, Nov. 19 (Week 13, Day 1)
Assignment 9
2.2: 1, 2, 5
Do several seteps of Picard's method to solve x dot = f(x) = (-x_2, x_1), x(0) = (1, 0).
(2) Problem 1a uses the
big O notation. (Link is to Wikipedia.)
In problem 5, replace "f is in C1(E)" with "f is locally Lipshitz on E"
Extra Credit:
Experiment with Picard's method using
PicardsMethod.nb and make some interesting observations. For example, consider the IVP dx/dt = 3 x^(2/3), x(0) = 0,
which is example 1 from pg. 66. Consider the recursively defined sequence defined by uk+1 = T(uk) for various initial functions u0(t),
(not just u0(t) = 0). Does T have a unique fixed point? Is T a contraction mapping?
Erite your comments in the Mathematica notebook and send it to me as an attachment.
Due Tuesday, Nov. 26 (Week 14, Day 1)
Assignment 10
2.3: 1, 2
Due Wednesday, December 3 (Week 15, Day 1)
Assignment 11
Do problems 2, 3, and 4 of homework 1 from MAT
667. Note: in problem 4 you can approximate the square root of 1 -(c/2)^2 by 1 when you estimate c.
For next Thursday, I will want you to do some numerical integrations of the driven pendulum using the Mathematica notebook DrivenDampedPendulum.nb.
Make an animated gif, and send me the gif along with the files with the parameters that made this.
I will talk about this program on Tuesday.
Instructor Information
Jim Swift's home page
Dept of Mathematics and Statistics
NAU Home Page
e-mail: Jim.Swift@nau.edu