### Swarthmore College, Spring 2007 Math 44 - Differential Equations

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Walter Stromquist's page

4/17/07 - Finally, here is problem set 5 with attached notes. (Word, 4 pages; the problems themselves are on page four.) Also, here is the spreadsheet that should have been visible Monday, and might be visible Wednesday. It has the equation y' = y+t-1 hard-wired into it, which is a defect, but I'm not sure how to build a simple spreadsheet that doesn't have that defect.

4/8/07 - Here are rough notes on Laplace transforms that I'll hand out on 4/9. They have some embedded exercises. (Word, 6 pages, no MathType or graphics)

4/5/07 - INSTRUCTIONS FOR HOMEWORK 3: For each of the text problems from Chapter 3, do this: find a general solution, and find a solution satisfying the initial conditions. Don't bother to draw a vector field, or draw qualitatitive conclusions, at least for the turned-in version of the homework. (That means that I should have assigned problems 12-13-14 in section 3.4, which are the same as 6-7-8 but with different instructions. The web version of the homework now reflects these revisions.)

4/4/07 - "Homework 3" (numbered that way because it's actual homework, and not too different in scale from graded homeworks 1 and 2, but less formally graded) is posted below under "course documents."

3/27/07 - Here is an essay on substitutions in differential equations:
substitution.pdf -- 11-page pdf

3/25/07 - Problem Set 2 solutions

3/24/07 - Miniquiz 4 solutions

3/22/07 - CLARIFICATIONS ON THE PROBLEM SET:
Problem 3...I made this confusing by using "lambda" as the name of a constant in the problem, when you naturally want to use the same symbol for the eigenvalue. So, you should change one of the symbols. Either use "x" for the unknown eigenvalue, or change the name of the constant. Thus, it would have been better if I had written: "Suppose that the characterisitic polynomial of y''-2ky'+cy=0 has exactly one eigenvalue, for some constants c and k..." and then asked for c, the eigenvalue, etc. all in terms of the constant k.
Problem 4...In this problem, A is definitely a number, not a matrix. (A and phi are both numbers; they are the two arbitrary constants in one version of the solution.)

3/22/07 - Yes, there will be a problem session from 1:00 till 2:30 today if anyone is interested. Meet at my office; if there is a crowd we have room 105. We won't do the problems from the graded problem set, but if there are other questions or problems, this would be as good a time as any to raise them.

3/20/07 - Here's a link to Problem Set 2, due Friday. We still have a small quiz tomorrow (Wednesday) and a problem session, if anyone is interested, about 1:00-2:30 Thursday.

3/10/07 - Here are solutions to the miniquiz from Friday (Word, 1 page with MathType).

2/28/07 - Solutions for exam. SLIGHTLY MODIFIED 3/1/07 10am.

2/22/07 - Here's a draft of the first page of the exam, with the reference sheet.

2/21/07 - Here's the "What's-on-the-exam sheet" for the exam on Feb. 26 (2 pages, Word, no MathType or graphics). Today (among other things) we'll compile a reference sheet to go with the exam.

2/19/07 - Solutions to the graded problem set are posted under course documents.

Here is the assignment from 2/16 (about Sections 2.1-2.3):
(a) READ Section 2.1 (most or all), 2.2 (all), 2.3 (pp. 188-192; we'll do the damped oscillator later.)
(b) WORK problems from 2.2 (for presentation/discussion in class 2/19):
(matching) 11, 21
("routine") 1, 7, 9, 13, 15, 19
23-26
(harder) 27, 28, 29

Barring surprises, we'll have an in-class exam on Monday, Feb. 26 (NOT this week). We'll start on Chapter 3 Wednesday, I think. It starts with homogeneous linear systems with constant coefficients. For one-function systems these were absolutely trivial (y'=ay -> y=c exp(at)) but the matrix version (Y'=AY -> Y = EXP(tA) C) has much more going on; the behavior of the solutions winds up depending on the eigenvalues and eigenspaces of A.

2/14/07 - Not a snow day today; here I am already. But check out this link.

2/12/07 - I have finally posted the homework set under "course documents." There is one correction: In problem 7, you're supposed to sketch three different solutions with y3(-1)=1/2, y3(0)=1/2, and y3(+1)=1/2 respectively. (In the unrevised version, the "+1" was a "0".) The schedule is updated, too.

MATH/STAT BOWL tomorrow (Tuesday) at 4:30 in room 101!

2/4/07 - Solutions to the miniquiz are posted under "documents" below (2-page Word with graphics and MathType). Sorry about the second question; I had started to ask for y(1) and changed it to y(2) without enough checking. In fact, y(2)=317.722... and the y(t) approaches infinity when t is about 2.00315. There is no way to find that out using any variant on Euler's method and any plausible step size. I used Mathematica; maybe we can go over it in class.

On Monday we will look at linear equations. Read Section 1.8 through page 117. We'l get to the rest of that section and Section 1.9 on Wednesday, but tomorrow we'll short-circuit those sections with an explicit formula for first-order linear equations. Not necessarily an easy formula to apply, but not so hard, either. Here's a one-page description of the formula (one-page Word with graphics and MathType).

1/29/07 - FWIW, here's the Excel workbook with Euler's method from class today:
de-jan29.xls
Maybe it will get cleaned up a bit later.
Problems for Wednesday: Use s/w of your choice (eg HPGSolution or EulersMethod from disk, or the above workbook, or a workbook of your own design) to do problems (Section 1.4) 3, 4, 6, 7, and 8; and also, do #13.

1/27/07 - Solutions to mini-mini-quiz - Word with MathType, one page.

1/26/07 - Assignment for Monday:
1. Know what exciting event happened on Jan. 29, 1861.
2. Read Section 1.4, "Euler's Method."
3. Get HPGSolver to work on a machine you can use, and use it to construct slope fields for the equations in exercises 1.3 ## 1, 2, 3, 5.
4. Do 1.3 problem 15 (we'll vote in class)
5. Do 1.3 problems 16, 17 (to discuss in class)
6. Print, and bring to class, the most interesting-looking slope field you come across.

1/24/07 - Assignment for Friday:
Read: Section 1.3, pages 36-43 (slope fields)
Prepare for Friday: Section 1.2, ## 2, 3, 35, and as many as you can do of 5-32 incl. 6, 7, 10, 26.

1/23/07 - HEY, THERE'S A COOKIE FRIDAY this Friday, 3:30-4:30, in the math/stat living room. You're invited!

1/22/07 - There's a schedule, reachable by the link above.

Assignment for Wednesday 1/24/07:
Read: Section 1.1, pages 1-8 (we'll do the logistic model later)
Section 1.2, pages 20-28 (we'll do all that mixing later, too)
Prepare for Wednesday: Section 1, problems 10-11-12 (Carbon-14 decay)

There's a mini-mini exam scheduled for Friday.

12/21/06 - The text for this course is
Differential Equations, 3rd ed., by Paul Blanchard, Robert L. Devaney, and Glen R. Hall, Thompson Brooks/Cole 2006, ISBN 0-495-01265-3.
This is the newest edition of the book that has been used here for at least two years. The publisher has a website for the textbook...
http://www.thomsonedu.com/thomsonedu/student.do?product_isbn=0495012653&disciplinenumber=1
...but the authors also have their own campanion site:
http://math.bu.edu/odes/
The authors' site is organized for instructors, mainly, which might make it fun for you to peek at.

We will cover Chapters 1, 2, and 3 during the first 7-8 weeks of the semester, and then select topics from Chapters 4, 5, and 7 and from other sources.

44problemset3.doc -- homework #3, 4/4/07, due 4/9/07. 1-page Word.

substitution.pdf -- Essay with examples of substitutions in differential equations and systems. 11-page pdf file.

44miniquiz4-solutions.doc - Solutions to miniquiz 4, quiz on 3/21/07, solutions posted 3/24/07, 2-page Word with MathType.

44problemset2.doc -- Solutions to graded homework #2, posted 3/25/07. Word with MathType and graphics.
44problemset2.doc -- Graded homework #2, due 3/23/07. 3-page Word with MathType.

44exam1page1.doc -- Draft of page 1 of exam with reference material. 2/22/07, 1-page Word with MathType.
44preexam1.doc -- What's-on-the-exam sheet for exam 1. 2/21/07, 2 pages, Word, no MathType or graphics.

44problemset1.doc -- Graded homework #1, due 2/14/07. Slightly revised 2/12/07, 3-page Word with MathType.
44problemset1-solutions.doc -- SOLUTIONS, 2/19/07; 7-page WORD with MathType.

44linear1.doc -- Formula for solving first-order linear equations, 2/5/07, one-page Word with graphics and MathType.

44mq-solutions.doc -- Solutions to mini-quiz, 2/4/07, two-page Word with graphics and MathType.
44mmq-solutions.doc -- Solutions to mini-mini-quiz, 1/27/07, one-page Word with MathType.

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