The following book will be used:

[BN] F. Brauer and J.A. Nohel: The qualitative theory of ordinary differential
equations. An introduction.

We shall go through Chapters 1-5 with some omissions.

Supplementary notes will be made available, on the modern
formulation of existence and uniqueness statements, on Sturm-Liouville
problems, and on phase space analysis in the basic two-dimensional case.

A reference that sheds more light on some of the subjects in this course
is the textbook (in Swedish): K. G. Andersson and
L.-C. Böiers: Ordinära Differentialekvationer, Studentlitteratur,

Prerequisites are: the local courses MatIntro, LinAlg and An1, or
equivalent courses followed elsewhere. We shall in particular need the material
on differential equations from the book of Tom Lindstrøm:
Kalkulus, 3rd edition, Sections 10.1-6 (first
part), and the formulas for nonhomogeneous equations
you have learned to find via Maple. Also convergence rules and metric spaces
will be important, as well as the treatment of matrices.

Weakly scheduled hours:

Lecture 1 Monday from 10:00 to 12:00 in Auditorium 10

Lecture 2 Tuesday from 15:00 to 16:00 in Auditorium 10

Lecture 3 Friday from 10:00 to 12:00 in Auditorium 8

Exercise 1 Tuesday from 13:00 to 15:00 in A102

Exercise 2 Friday from 09:00 to 10:00 in A 110

It is customary at the university to start 15 minutes
into the hour. So, add 15 min.s to each of the
starting times above.

The first lecture begins Monday August 31 at 10.15
in Aud. 10 at the HCØrsted Institute, Universitetsparken 5.

**Teaching period: **August 31 - November 1 2009

There is homework every week, and the
homework in weeks 38, 40 and 42 will be obligatory (counting with 20% each in
the final result). The dates for handing in these three obligatory homeworks are so far planned to be: September 15, September
29 and October 13, 2009. (These dates may be modified if some outer
circumstances make it necessary.)

In week 44, on the last day of the course October 30,
there will be an in-class test of 2
hours counting with 40% (it will preferably be placed in the period 9-12
o’clock).

You are welcome to collaborate with others in
solving the homework problems, but you must write your own formulation of the
answer (we do not accept copying). Write in Danish or English.

*Pensum** - the syllabus of the course.*

From [BN], the book of Brauer
and Nohel:

Chapter 1, Sections 1.3-1.7, where Theorems 1.1-3 are
replaced by Theorem S1 in the notes by GG.

Chapter 2, Sections 2.1-2.8.

Chapter 3, Sections 3.1-3.5, where p.132 l.1 - p.134 l.4 is replaced by the
text in the notes.

Chapter 4, Sections 4.1-4.4 until the middle of p.164.

Chapter 5, Sections 5.2-5.3.

Appendix 1, except for the proof of Theorem 2 and the
proof and statement of Theorem 3.

Notes by G. Grubb, Sections S1-S4.

From Chapter 12 on Sturm-Liouville
problems in the book by R. K. Nagle and E. B. Stief,
“Fundamentals of Differential Equations and Boundary Value Problems”: Section 12.2, Section 12.3 until line 7 of p.
629, Section 12.6 until line 6 from below on p. 650.

CORRECTION:

In the notes, page 2, lines 10-12 from below, replace
the sentence “In fact, the mapping … isomorphism.” by:

“In fact, the mapping T_{t_0} : C^n -> V that sends **eta**
in C^n over into the solution **phi** for which **phi**(t_0)=**eta**, is linear and sends a basis **sigma**_j,
j=1,…,n, of C^n over into a basis of V; hence it is a
vector space isomorphism.”

PLAN OF THE COURSE. Past lectures will be described in detail,
and tentative plans for future lectures will be posted here.
The exercises for each week will be posted here, on or before the Wednesday
of the preceding week. (Prepare for ca. 2/3 of the exercises for the Tuesday
session.) Written homework is to be handed in on Tuesdays.

PAST LECTURES:

Week 36, Aug. 31, Sept. 1 and 4: From Chapter 1 we went through Sections
1.3-1.7. Theorems 1.1-3 are not well formulated and are replaced by
Theorem S1, together with the definition of maximal solutions, from the
supplementing text: notes\sdl09.pdf . In Chapter
2, we covered the material up to and including Theorem 2.2. The proofs of
Theorem 2.1 and its corollary were skipped, since they build on the unclear
statements in Theorems 1.1-3. (They will be treated in Chapter 3 later.)
- These introductory sections contain much material that should be known
from earlier courses (e.g. linear algebra); some of it was recalled at the
lectures.

Week 37, Sept. 7, 8 and 11: We
have covered the material from Theorem 2.2 until Example 2.5.3 on page 61. Both
Theorem 2.4 and the remark in lines 4-7 on page 49 were deduced from Theorem S2
in the notes, which implies these facts directly without referring to Abel’s
formula (Theorem 2.3). Theorem S2 was also used in the proof of Theorem 2.7
(instead of referring to Abel’s formula).

Week 38, Sept. 14, 15 and 19: In the rest of Section 2.5, we used
the results of Appendix 1, on the properties of generalized eigenspaces, to construct the general solution. We
also went through the material from the supplementing note on real solutions.
In Section 2.6 we used the information from Appendix 1 on the

Week 39, Sept. 21, 22 and 25: We discussed autonomous systems (Section
2.8) and in particular the linear 2-dimensional cases and their phase
portraits. Section 2.9 was skipped. Next, we covered the existence results in
Chapter 3 up to page 120.

Week 40, Sept. 28 and 29, Oct. 2: We continued with existence and
uniqueness results from Chapter 3. For Theorem 3.4 we used the sharper
formulation in the notes (Theorem S4.1). Pages 132-134 were replaced by the
rest of Section S4 of the notes. Theorem 3.5, as well as the supplementing
explanation to it in the notes, were skipped.

Week 41, Oct. 5, 6 and 9: This week’s topic was Sturm-Liouville problems, based on excerpts from Chapter 12
of the book by R. K. Nagle and E. B. Stief:
Fundamentals of Differential Equations and Boundary Value Problems. Copies were
distributed at the lectures. The lectures were focused on Sections 12.2-3
and 12.6 there. In the construction of Green’s function we made use of the
formula from Exercise 2.4.6 in [BN].

Week 42, Oct. 12, 13 and 16: The lectures returned to [BN]. First
Section 3.5 was done, on the continuous dependence of the solution on the data.
Then we did parts of Chapter 4: Sections 4.1-4.4 until the middle of page 164.
In Theorem 4.2, the hypothesis on B was replaced by that in Exercise 4.3.15,
allowing a considerably simpler proof exhibiting the main strategies. In the
proof of Theorem 4.3 we used Corollary S4.3 as explained here: th-4-3.pdf

Week 43, Oct. 19, 20 and 23: The topic was the Lyapunov
theorems in Chapter 5, where we went through Sections 5.2 and 5.3 with many
examples.

Week 44, Oct. 27 and 30: Since the last week is kept free of new
subjects, there were no lectures on Monday Oct. 26, and the lecture on Tuesday
Oct. 27 was used to give an overview of the course and to answer questions from
the participants. On Friday Oct. 30 there is the in-class test, see further
information below.

EXERCISES:

Week 36: Brush up on material known from earlier courses. 1.3.1-9,
1.3.12-13, 1.3.17, 1.4.7-8.

Week 37: 1.4.9-11, 1.5.1, 1.6.(2, 4, 8, 14, 16), 1.7.1, 2.1.(2,
8), 2.2.4, 2.3.8. Written homework: **1.3.11,
1.6.10** (to be handed in on Tuesday Sept. 8).

Week 38: 2.1.6, 2.3.(17, 21, 22, 23, 26, 27, 28), 2.4(3, 5,
6), 2.5.4. In 2.3.21, replace the second sentence by: "Express det(Phi) in terms of solutions of the n-th
order equation; it is called the Wronski determinant
(the Wronskian)." You do not have to look up the
references. **Obligatory **written
homework: **1.6.18,** **2.3.10, E11 (a)-(c) . **(To be handed in to
Philip not later than Tuesday Sept. 15 at 15 o'clock.) Comments: In Exercise
1.6.18(a) you are not asked to find an explicit solution. E11 is an exercise
from the notes; replace question (c) by: Find a solution with

Week 39: 2.5.(11, 12, 14, 16, 18, 23, 24, 25), 2.6.3, 2.7.2, and
E1. In 2.5.12, assume that the diagonal elements are distinct (what can go
wrong if two of them are equal?). Written homework: **2.4.4, 2.5.(8, 9).**

Week 40: 2.7.(5, 6), 2.8.(2, 3, 4, 5, 11, 12, 14, 16, 17), and, from
Miscellaneous Exercises to **Obligatory **written homework: **2.5.27, Misc. 2.5, Misc. 2.9. **(To
be handed in to Philip not later than Tuesday Sept. 29 at 15 o'clock.) - A
Maple working sheet is made available here: notes\sdlex2.mws
, to be used next week as a model in some exercises, where you will be asked to
draw vector fields and solutions. If you have a computer without Maple, you can
download the program via Punkt.KU. It is also
possible to work on one of the computers available in the computer terminal
rooms A111 and A112.

Week 41: Misc. 2.3(a, b, e), using Maple to draw phase portraits (with
vector fields and some solutions). 3.1.(2, 5, 7, 8, 12, 14, 17), 3.2.3.
Written homework: **Misc. 2.2 (d) (****determine
the type of the critical point (0,0), as in Misc. 2.3****), 2.7, 2.8.** - There are
useful instruction files for Maple available at
http://www.maplesoft.com/documentation_center/

Week 42: From the chapter on Sturm-Liouville
problems, Exercises 12.2.(13, 14, 16, 18, 19), 12.3.(17, 18, 20, 22, 23),
12.6.5. A hint to exercise 12.2.19: One can use the substitution x=e^t. **Obligatory **written
homework: **3.3.1, E4, E5(a,b,c). **In 3.3.1 one can apply the
original Gronwall Theorem 1.4 if one uses the boundedness of (t-s). (To be handed in to Philip not later
than Tuesday Oct. 13 at 15 o'clock.)

Week 43: 4.2.(2, 4, 6, 9(d,e,g)), 4.3.(2, 4,
6, 10, 17). E5(d), E11(d). Written homework: **E9, E10, E13(a).**

Week 44: (only October 27) 5.2.(4, 5, 8, 10), E2, E6, E7, E12.

A solution sheet for the exercises E8-E10 is available here:
http://www.math.ku.dk/~grubb/sdl074.pdf

The in-class test will take place on Friday October 30, 2009, in the
rooms A103 and A105 from 10.00 (precisely) to 12.00. Come 5-10 minutes before
to install yourself at a table (one for each participant). The text for the
exercises will be given both in English and in Danish.

**NB!** The in-class
test is **obligatory, **in the same way
as the homeworks Sept. 15, Sept. 28 and Oct. 13. I
have consulted our study director (studieleder) about
this. The complete set of obligatory tests must cover the 6 points listed under
“Expected competences” in the description of the course in the SIS system.
Since you have not yet been tested in the last point (on stability etc.), you
must participate in the last test that covers this. - G. Grubb

*About the in-class test on Friday October 30, 2009:*

It is allowed to bring written material of all kinds (books and printed
papers, personal notes etc.). Electronic equipment is not allowed. Answers can
be formulated in Danish or English, as preferred.

Results from the textbook by Brauer and Nohel, the supplementing notes by G. Grubb, and the chapter
on Sturm-Liouville problems, can be used, when
the place they are taken from is precisely indicated.

Please bring an envelope with your corrected obligatory homework
along, and hand it in together with the answer to the classroom test.
(This is for practical reasons, to allow an overall evaluation if we need more information
than the percentage values give.) Write your name and address on the envelope,
so that we can return the material to you in it when the whole evaluation
procedure is over.

**Added November 23, 2009:**
Here is the test test09 and here is a solution solution09.