# Math 433/533, Theory of Ordinary Differential Equations, Spring 2017, Class Announcements Page

## Welcome

Welcome to the class. You should expect to find homework assignments and other important class announcements and links to class resources on this page. You should be in the habit of checking this page regularly; I will not necessarily say much about homework assignments in class -- this is the official place to find out about them!

I will generally add new material near the top of the page: a few items may always remain at the top.

## Office Hours and other schedule information

My office hours are stated on my office door card (which is also attached to my office door), which also includes the times of my classes and seminars. I am thinking of typically coming in late on Mondays and leaving early Wednesdays and Fridays: at times when I am working at home, e-mail will reach me.

## Makeup event

During the final examination period I will present a makeup event at which you can make up selected problems from Test I, Test II and recent material. You can improve designated problems from Test I and Test II by doing the Test I and Test II related problems. You can improve your homework grade by doing problems based on recent lectures. You will not harm your current standing by sitting the makeup opportunity. Do notice that you will only be able to make up specific problems which I choose from the earlier tests, no more than half the problems in either case.

## April 25th (and last) homework

This is due at the final examination period (whether you sit the makeup opportunity or not).

do exercise 28.2 i, iii, vii. I will happily do one or more other parts of this exercise cold on Thursday.

## April 19 Homework

This assignment is due on the last day of class. There will be one more assignment due at the final exam period (at which I am currently planning to offer a make-up exercise). Problems 16.3 [the point here is to write it out in system form and verify that the Lipschitz condition they describe causes the Lipschitz condition for systems to hold; you may restrict yourself to doing this for n=2, a system of two equations]; 16.4 [I think the hint works: you do need to do the work leading up to the point where the hint applies then complete the argument using the hint]; 16.6 [the hint gives an answer, but you need to explain clearly why this gives a counterexample to an 11.1 style result]. I may have more to say when I have actually worked these myself; I'm not very satisfied with this as a homework set. I reserve the right to add new problems or change some of these as late as Sunday.

## April 7 Homework

On Thursday April 6th we are stalled in the middle of discussing theorem 16.3. Your homework due April 13 is 16.1 (set up and carry out enough steps of the Picard iteration to get the pattern); 16.2 (he really does mean 1.6, 1.8 in chapter 1); doing it for n=3 is enough; you may optionally attempt 16.4 (I may lecture it when I figure it out). Set up spreadsheets to estimate the solutions to the two problems in 16.1: you may turn in the spreadsheets electronically -- do some experiments and discuss the magnitude of the error in your estimates for different step sizes (with actual numbers) on your papers. In response to a request from the class, we consider the equation u1' = x + u2^2; u2' = x + u1^2; u1(0)=1; u2(0) = -1. Show that this system satisfies a Lipschitz condition and determine an interval around 0 on which we are sure that it has a unique solution. You may use b=1. This should involve some fairly evil bookkeeping, and you need to remember what his measure of distance is (taxicab metric). Showing that it has a Lipschitz condition should not be hard.

## Week 10

• Tuesday March 14 (writing on the morning of the 15th): Finished lecture 11, started lecture 12. Homework (including things in lecture 12 which I'll cover on Thursday): 11.1 (give a different example), 11.2, 11.3, 11.4 (looks straightforward using the theorem and corollary in the section), additional chapter 11 exercise: prove the other inequality y1 < y claimed in corollary 11.2 (this should be closely analogous to the proof at the top of p. 78), 12.1, 12.4, 12.6 (they give an answer: find the equation that gives it), 12.8.

This is not the promised test review, just homework. It is of course all material that will be on the test [no, 12.6 and 12.8 are outside the scope of the test, though interesting. 12.5, not assigned, is I think a good test review problem]. It is officially due at the first class period after the exam, but if you turn it in on Tuesday the 4th it will be graded before the exam on the 6th.

## Week 8

• Thursday March 2 (really writing on the morning of the third): we are embedded in chapter 10, right after the proof of the K. K. theorem, which I will probably revisit Tuesday. Homework: 10.3, 10.6* (optional), 10.7, 10.8, 10.9, 10.11, 10.12. These problems look distinctly tricky. Although I star 10.6, you can probably do it by recapitulating the proof of Peano's theorem (using my reformulation to avoid the error in the book, I should post a write-up of this) and reversing signs and inequalities as appropriate. I undertake to do these problems myself before Tuesday so that I can give sensible advice (and maybe withdraw ones I can't do ;-) When I have done them I will supply hints.

## Week 7

• Thursday 23 Feb (really posting this on the 24th): we have just started chapter 10. Homework due March 2: 9.1 (should be straightforward); 9.2 (challenge, extra credit for success), 9.3, 9.5, 9.7. 9.9 (leaving even problems for class discussion), 10.1 (I do not know how easy or hard 10.1 is!), 10.2 (this is easy if you notice the right facts).

## Week Five

• Tuesday, Feb 7 (really posting this on the 8th): I started lecture 9 today but the homework does not go that far.

The homework is this worksheet intended for test review. It is also graded homework: if you turn it in on 2/14, I promise to mark it and have it available sometime on 2/15; if you turn it in on 2/21, after the test, which is also allowed, you will not get this benefit.

## Week Three

• Thursday, January 26: Lectured the proof of theorem 8.1 in lecture 8.

Second homework assignment, due next Thursday: 7.5, 7.7, 7.10, 7.11, 8.2 all parts, 8.3 all parts, 8.6, 8.10.

## Week Two

• Tuesday, January 17: more discussion of lecture 7.

First homework assignment, due Thursday, Jan 26th: p. 49 7.1 (use the same method we used to prove Theorem 7.1, but integrate/differentiate twice; there is a solution or hint toward a solution given, but you should attempt it yourself); 7.2 i, iii, vi [in 7.2 you may want to bound x and/or y to get things to work; you can look at their hints to see what kind of bounding is needed, but you need to show verifications that the constants you come up with work.]; 7.3 i, iv (I did iv in class, but this is a good one to be reminded of!); 7.4 ii. review from early sections (these have answers, but try them on your own: 3.3i, 3.5, 4.5, 4.6* [I can do the general substitution described in 4.6, but I can't see how to use it to solve the specific equation in 4.6; maybe you can]; 5.1, 5.3i, 6.12ii.

## Week One

• Thursday, January 12: I lectured material from Lecture 1 and Lecture 7 in the book, as planned for Tuesday. The first homework assignment is delayed: it will be assigned on Tuesday, or maybe even on Thursday.

• Tuesday, January 10: this was a snow day!

The historical material in Lecture 2 might be interesting to read. The techniques of solving differential equations in Lectures 3 through 6 should be familiar to you from Math 333, and you should review them (I do mean that you should review them now!) I am not lecturing them (at least not in full; I may end up making some remarks about them) but I am very likely to assign some problems from these sections for review in the first homework assignment.

## Test II

Test II will occur on Thursday, March 30th (the Thursday after the break, sorry about writing April 6th earlier...). Test III (which will not be cumulative but will contain makeup opportunities) will occur during the final examination period. Test II will cover all material covered before Spring Break. I will provide review material making it clear what sorts of things I intend to ask about either late this week or during the break.

I will have to do some creative work to come up with test questions that work for you, and I will do my best to make sure that they do not come as a surprise!

Here is the review sheet (just an outline of topics as I enter it at noon 3/17/2017: I am planning to add worked examples and sample problems over the break: as of Monday 3/27 I have added some examples, some with solutions).