### Swarthmore College, Spring 2008 Math 44 - Differential Equations

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Walter Stromquist's page

4/8/08 - Here's today's handout on Laplace transforms:
LaplaceTransforms.doc (Word, 6 pages)
Thursday's homework (not to turn in) is to know the answers to Exercises 1-6 in these notes (and maybe the rest of the exercises, too).
MathWorld on Laplace Transforms (with a good table in small print)
Wikipedia on Laplace Transforms

4/2/08 - I'll give you a take-home exam tomorrow (4/3) to be handed back on Tuesday (4/8). Here's the practice exam from class yesterday --- same content as exam 2 from last year, but with most of the empty space removed so that it fits on two pages:
2007 Math-44 exam
The real exam will be about the same, but perhaps a bit longer; also, I reserve the right to ask some different kinds of questions if I can think of any.

Also, here's the one-page handout on hyperbolic functions.

3/27/08 - Some catch-up posting:
Homework for April 1:
4.1 (page 393) 1, 7, 11, read-but-don't-turn-in 20, turn in 23.
4.2 (page 406) 1, 5, 9, 13
44homework27march.doc - problem set due March 27 ("all about y''+py'+qy=0")
44homework25march.doc - problem set due March 25 (including a magic bullet substitution and some trace-determinant-plane-like problems)
substitution.pdf - essay on substitution from last year, with a discussion of the magic bullet substitutions. 10-page-plus pdf file.

2/21/08 - Today's exam is posted under "course documents" below. So is a hasty solution sheet. When I grade the exam I'll probably find that this solution sheet is full of errors. Maybe you'll get a better score than I do.

2/20/08 - Here's a hasty but perhaps-better-than-nothing solution sheet for homework 6.
44solutions6.doc

2/19/08 - Here's a solution sheet for Homework 5 (the one that was labeled 4):
44solutions5.doc
The solution sheet for homework 6 is still pending; check back later tonight or tomorrow morning.

2/19/08 - Here is the first exam from last year's Math 44 class. It covered slightly different material and we had looked at different examples in class. Also, here's an incomplete solution sheet (through about problem 5):
44exam1-2007.doc
44exam1-2007-solutions.doc

2/14/08 - The "What's on the Exam" sheet is posted below. The exam is 2/21. Here's the homework for next Tuesday, 2/19:
To turn in:
Sec. 2.1: 1,2,3,4
Sec. 2.2: 1,7,9,13,15,19,23-26,27,27,29
Be comfortable with but don't turn in:
Sec. 2.3: 1,2,3,4
Sec. 2.4: 3,4,5,6
(This is homework #6; the handout labeled "#4" was really #5.)

2/7/08 - Homework 5,[NOT homework 4; the numbering is corrected on the web version] due Tuesday. Uniqueness theorem and Bernoulli equations! (Also posted below.)

2/6/08 - The handout from class yesterday, on linear equations, is posted under "class documents" below.

The homework for Thursday is from Section 1.8: problems 1, 3, 7, 12, 19. The first four problems are routine linear DE's (but do them carefully and check the results). The last problem presents one big idea.

Thursday we'll look at integrating factors (Sec. 1.8-1.9). They're another way to approach linear DE's, and they can be used in some other cases. Finding them is mostly a matter of luck but experience helps, too. After that we'll be done treating equations one at a time, and will be ready to jump into (a) higher-order equations and (b) systems (Chapter 2).

1/30/08 - Homework for Thursday:
Find the general solution (that is, all solutions on all intervals) to y'=y+t;
Section 1.4 problems 3, 6, 13;
To solve y'=y+t, use a substitution like in class, or guess, or treat it as a linear equation per Section 1.8, or do anything else that works.

We'll do the existence and uniqueness theorems in class on Thursday. This is the theoretical core of the subject. The existence theorem says that (given certain hypotheses) there really is a solution through every point of the (t,y) plane; the uniqueness theorem says (under stronger hypotheses) that there is only one. We'll also investigate when we can extend solutions to longer intervals and how the solutions vary when we change parameters or initial conditions. Sometimes these theorems are all you need; in certain applications it matters more to know that the function you want exists than how to compute it.

Here is the Euler's-method workbook we built in class. Feel free to use it or make one like it for the homework. Also, here is the version as we left it with the Fox-and-Rabbits system.

1/24/08 - Homework for next Tuesday:
Read Section 1.3 (or just pp. 36-43) (note the "important special cases")
Turn in Section 1.3 ## 3, 15, 16, 17.
Also, after reading pp. 2-8, turn in Section 1.1 ## 10, 11, 12.
IF you can get access to the software, especially HPGSolver, then also do Section 1.3 ## 1, 3, 4, 5 by HPGsolver
-- no need to turn these in, but optionally, bring to class the most interesting slope field you can discover.
(The text pages have been added to the list of pdf files below.)

1/22/08 - There's a Cookie Friday this week! Friday, 3:30-4:30, within sight of the math common room. You're invited!

1/22/08 - Homework for Thursday:
Read pp. 20-28 (stuff we did Tuesday)
Turn in these exercises from Section 2.1:
2, 3, 6, 7, 10, 26
(all of 5-32 should be routine by now)
Also, try Problem 1 from last year's final; pay special attention to the intervals on which the solutions are valid.

1/21/08 - The text might be hard to find at the bookstore for a few days. It will show up eventually; meanwhile, there are two copies on reserve at Cornell. Also, here are some scanned copies of selected pages:
Modeling section in section 1.1: pages2-8plus17.pdf
Definitions and Separable Equations, Section 1.2: pages20-28.pdf
Exercises for Section 1.2: pages33-34.pdf
Section 1.3 with some of the exercises: pages46-51.pdf
These are good pages to read for Thursday (or even tomorrow, if you can).

1/18/08 - 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.
It's the same book and same edition as last year. We will use the programs on the attached CD, but they will be available in various ways, so you could get away without it.The publisher has a website for the textbook...
http://www.thomsonedu.com/thomsonedu/student.do?product_isbn=0495012653&disciplinenumber=1
...and 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 half of the semester and then select topics from Chapters 4, 5, and 7 and from other sources.

Course documents:

44exam1-2008.doc - Exam 1 (2/21/2008)
44exam1-solutions-2008.doc - Solution sheet for exam 1 (also 2/21/2008)

44preexam1.doc - "What's on the exam" for Exam 1 (2/14/08 version for exam 2/21/08)

44homework4.doc - Homework 5 (not 4), 2/7/08, due 2/12/08 (two-page Word with graphics and MathType)

44linear.doc - "How to solve first-order linear equations," 2/5/08 (one-page Word document)

euler.xls - Euler's method workbook (Excel)
eulerFR.xls - same, with Fox-and-Rabbits system from class 1/29/08.