Sædvanlige differentialligninger – Ordinary Differential Equations 2009

Welcome to the course SDL, Fall 2009.

Course material

 The following book will be used:

[BN]  F. Brauer and J.A. Nohel: The qualitative theory of ordinary differential equations. An introduction. Dover Publications 1989. It can be bought at the polytechnical bookstore in Biocentret, Ole Måløesvej 5.

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, Lund 1992. It will be referred to as [AB]. 

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.


Course description

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. 




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.



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 Jordan canonical form.  We did Section 2.7, on how growth properties of solutions can be deduced from the size of the real parts of the eigenvalues. Finally, we did cases (i)-(iv) of the classification of real 2x2-matrices on page 90, both as a preparation for the discussion of autonomous systems next time and as an illustration of how to find a real transformation to achieve the Jordan canonical form. This used Section S3 of the notes (which replaces Appendix 2).

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.



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 eta=(0,0).

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 Ch. 2 p. 103: Misc. 2.2(a, b, e), Misc. 2.4(b). 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.