﻿ MAT 665 Homework, Fall 2017

# MAT 665 Homework

Due Friday, Sept. 1 (Week 1, Day 3)
Assignment 1

Due Fri., Sept. 8 (Week 2, Day 2)
Assignment 2
1.1: 1acde, 2ac, 3, 4, 5.

Due Wed, Sept. 13 (Week 3, Day 2)
Assignment 3.
Note: Harvey Buchart is the most famous NAU math professor.
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. 20 (Week 4 Day 2)
Assignment 4
1.3: 1b, 5, 6, 7, 8 (you can use propositions from this section.)
1.4: 1, 2, 5*, 6 (* find the general solution: You don't need to invert any matrices.)

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|| = sqrt(max eigenvalue of ATA). Then you can plot f(t) and the norm you computed, and see that the max of f is ||A||.
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 Monday, Oct. 2 (Week 6, Day 1)
Assignment 5
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, answer the questions asked except only sketch the phase portrait restricted to the x1 - x2 plane.

Due Monday, Oct 9. (Week 7, Day 1)
Assignment 6
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: You do 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. If you want to use MATLAB instead, you can download MATLAB scripts where the three matrices are defined. Your browser might be able to download them directly from here: SplusNmat1.m, SplusNmat2.m , and SplusNmat3.m.

Due Monday, Oct. 23 (Week 9, Day 1)
Assignment 7
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 y = y''' - 3 y'' + 3y' - y = 0 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 use MAT 665 techniques. Note that the scalar y 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 JCF example 2. (Note: I didn't give you any matrices with non-real eigenvalues, so example 2 isn't relevant for your homework 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 Wednesday, Oct. 25 (Week 9, Day 2)
Assignment 8
1.9: 2, 5(a,b,c).
1.10: 1, 2

Due Monday, Nov. 6 (Week 11, Day 1)
Assignment 9
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 Monday, Nov. 13 (Week 12, Day 1)
Assignment 10
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).
Notes: (1) I added problem 2 and took out problem 10 since the first posting.
(2) Problem 1a uses the big O notation. (Link is to Wikipedia.)

Extra Credit:
(1) Experiment with Picard's method using PicardsMethod.nb and make some interesting observations. You write your comments in the Mathematica notebook and send it to me as an attachment. This notebook has been updated since Monday's class.
(2) Essay Question: Compare and contrast Picard's method and John Neuberger's method for solving x dot = f(x), x(0) = x_0 as described in Tuesday's colloquium.

Due Friday, Nov. 17 (Week 12, Day 3)
Assignment 11
2.3: 1, 2

Due Wednesday, December 6 (Week 15, Day 2)
Assignment 12
2.4: 1(b, d), 2(b)
2.5: 2, 3, 5
2.6: 1(a, c, d), 2
2.7: 1, 5

Note: If this listing is incomplete or disagrees with what I assigned in class, please send me e-mail at Jim.Swift@nau.edu.