Math 266A: Applied Ordinary Differential Equations, Winter 2011

Announcements

• March 17: The final grades for the course have been uploaded.
• March 16: The grades for the final exam have been uploaded, the final grades for the course will follow soon, probably tomorrow or on Friday. Here are the solutions to the final exam.
• March 11: Next Monday, the day before the exam, I will have my last office hours for this course at the usual time 10:30 am -12:00 noon.
• March 11: People have asked me for good sources of practice exercises to help study for the exam. You can of course always take a book on ordinary differential equations from the library and try to make the exercises corresponding to the topics we have covered. Most books will contain sections on existence, uniqueness, stability (via linearization or Lyapunov). Topics like the Poincaré-Bendixson theorem or linear boundary value problems are also commonly treated in textbooks, but are perhaps not as ubiquitous as the others. Another good source of exercises are the homework exercises of professor Ralston which can be found on his Mathh 266a website.
• March 11: We didn't get to it in class, but you might want to read Section 6.2 yourself. The computation of the period as function of the amplitude which is done in the first part of Theorem 6.1 is a classic computation that you might have seen before. The eventual result of the section in Corollary 2 is also quite nice: it shows that the period either increases or decreases with the amplitude, depending on the sign of g''. This will not be part of the exam, but it is a well known topic that is good to have seen.
• March 4: Because Poincaré-Bendixson is not treated in Brauer & Nohel, here are scans of my notes on the subject. Keep in mind that they were not written with the explicit goal in mind of distributing them. Also, in some places unfortunately the margin of the page does not fit entirely on the scanned page, but together with your own notes, I hope these are of help to you. Also note that in the past week I have added many references to other notes and books in the "Calendar and classes" section below to the relevant lectures.
• February 24: The extra lecture on Thursday March 3, 12:00-12:50 pm, will be held in MS 7608.
• February 23: Unfortunately last Friday's lecture had to be cancelled. Next week, Thursday March 3 12:00-12:50 pm there will be an extra lecture to make up for the missed one. Keep an eye on this website for the announcement of the location.
• February 14: The TA has written a handout on asymptotic stability addressing among other things the solution of the correct formulation of Ex. 3 (p.182) from the homework. If anyone wants more information on Gronwall inequalities and bootstrap arguments, his suggestion is to have a look at chapters 1.2 and 1.3 of professor Tao's book on dispersive equations, which can be found here.
• February 11: Since I will be away the second half of next week, I have put the next set of homework on the website early.
• February 11: I have updated the solutions to the midterm slightly, clarifying the assumptions under which we could use Theorem 2.10 in question 3.
• February 7: Here are the solutions to today's midterm. Questions 1 and 2 were generally done well, question 3 gave problems for many of you.
• February 2: The midterm next Monday can include material from everything we have done up to and including all of Chapter 4. The midterm will be written in class at the usual location and time.
• February 2: There has been an unfortunate typo on the website all this time. My office hours are on Mondays and Thursdays (not Wednesdays). The Course Organization Handout and the sign next to my door have the right times. If you have a question but cannot come to the office hours, email me and we will try to make an appointment.
• January 26: During today's lecture there was a bit of confusion over the proof of Lemma 4.1. Here is a confusion free version.
• January 15: Typo corrected in the corollary to Theorem 3.6.
• January 14: As mentioned in class I wasn't happy with the way Theorem 3.6 turned out on the blackboard last Wednesday. Here is a better version. If you find any ambiguities or typos, let me know. Please disregard the version I put on the blackboard in class.
• January 13: Updated version of the Course Organization Handout with the TA office hours.
• January 10: The 'proof' of Theorem 3.5 I gave in class today was, as some of you correctly pointed out, not completely correct. Thank you! Here is a correct version. The proof in the book is quite confusing. The answer to the question in class is of course that there is no special significance to the specific time t_1 I used in class. We only need an interval (t_1, t_2) on which we know u>0 with u(t_1)=0. Continuity of u combined with the fact that u(t_0)=0 gives us this.
• January 3: Per today the lecture room has changed to MS 5233, so starting with Wednesday's lecture the lectures will be in a different room than the one this morning. The discussion sections will still be in MS 5127.
• January 3 2011: After consulting the class I have decided to have my office hours on Monday 10:30 am--12:00 noon and Thursday 2:30--4:00 pm.
• December 30: During the first lecture on January 3 I will finalize the date of the midterm and my office hours. Check your agenda and have your preferences ready.
• December 30 2010: This space will be used for announcements. Check back regularly to see if there is news. The newest items will appear at the top of the list. Enjoy the course!

Homework

• Due Jan. 12: Excercise 14 (p.14), Ex. 2, 3 (p.32), Ex. 2 (p.110), Ex. 12 (p.118), Ex. 7 (p.140), Ex. 9 (p.141). Because not everyone has a textbook yet, I have typed out this week's exercises.
• Due Jan. 19: Read Chapter 2 (Linear systems), Ex. 8 (p.105), Ex. 1 (p.137), Ex. 2 (p.138), Ex. 10 (p.141), Ex. 8 (p.151), Ex. 9 (p.151) parts (a), (c), (d), (f), Extra exercise 1. (Note: There is a typo in the book in exercise 10, p.141: The second = sign in the formula for phi_2' should be a + sign.)
• Due Jan. 26: Ex. 6 (p.100; for part (a) read and understand Example 1), Ex. 2 (p.152), Ex. 7 (p.154), Ex. 12 (p.154), Ex. 15 (p.158)
• Due Feb. 02: Ex. 3 (p.164), Ex. 6 (p.170), Ex. 7 (p.170), prove Theorem 4.6 completely (p.171; the book gives an overview of the proof, but leaves many holes), Ex. 4 (p.177), Extra exercise 2.
• Due Feb. 09: Ex. 16 (p.158), Ex. 3 (p.182), Ex. 2 (p.185), Ex. 4 (p.185), Ex. 5 (p.186), Ex. 6 (p.186). There is a typo in the statemennt of Ex. 3 (p.182) in the book. See the announcement of February 14.
• Due Feb. 16: Ex. 5 (p.191), Extra exercise 3, Ex. 2 (p.193), Ex. 3 (p.193), Ex. 5 (p.200), Ex. 11 (p.204), Ex. 12 (p.205).
• Due Feb. 23: Ex. 1 (p.208), Ex. 1 (p.214), Ex. 2 (p.214), Ex. 3 (p.214), Extra exercise 4.
• Due Mar. 02: Extra exercises 5 and 6, Ex. 1 (p.217), Ex. 2 (p.219), Ex. 5 (p.222), Ex. 6 (p.222). (Although the exercises from Brauer & Nohel are from Section 5.5, they are mainly applications of the results in Section 5.4, so they should be doable even if we haven't reached Section 5.5 yet. You might want to read the first 8 pages of Section 5.5 for extensive examples of such applications.)
• Due Mar. 09: Ex. 8 (p.225), Ex. 3 (p.230), Ex. 4 (p.230), Ex. 8 (p.231), Ex. 12 (p.234).

Calendar and classes

• Monday Jan. 03: Discussed the concepts of solution to an ODE and IVP from Chapter 1, as well as Gronwall's inequality. Started with local existence, Section 3.1.
• Wednesday Jan. 05: We continued our discussion of local existence for the scalar IVP. We ended in the proof of Theorem 3.1 where we had just shown that our successive approximations converge uniformly. Next time we will continue by showing that their limit is a solution of the IVP.
• Friday Jan. 07: We completed the local existence proof for the scalar IVP (Section 3.1) and also gave the corresponding result for systems of first order ODE with initial condition. (Section 3.2). There was a question about the assumption we have been using that the y-derivatives of f are continuous. For what we have done so far this assumption can be replaced by the weaker assumption that f is Lipschitz continuous in y.
• Monday Jan. 10: We discussed uniqueness of solutions (Section 3.3) and proved a preliminary lemma we will need for the continuation of solutions (Lemma 3.3)
• Wednesday Jan. 12: We proved a theorem on continuation of solutions (Section 3.4) and a theorem about the continuity of solutions with respect to the initial condition (Section 3.5). There is another continuity theorem in that section (Theorem 3.8) about continuity with respect to the right hand side f. Proving that theorem is part of this week's homework.
• Friday Jan. 14: I wrote down Theorem 3.8 (proof is homework). We discussed definitions of stability (Sections 4.1 and 4.2). I ended with the statement of a lemma which can be used to simplify the definition of stability. The proof follows next time.
• Monday Jan. 17: No lecture: Martin Luther King Day
• Wednesday Jan. 19: I gave the proof to last time's lemma. After that I summarized a lot (but not all!) of the important results from Chapter 2 on linear systems.
• Friday Jan. 21: We discussed stability of linear systems (Secton 4.3), in particular we proved Theorem 4.1 and stated Theorem 4.2.
• Monday Jan. 24: We finished Section 4.3 with the proof of Theorem 4.2. Note that I changed the statement of the corollary to Theorem 4.2 from the way the book presents it. We started with Section 4.4 on almost linear systems. I stated Theorem 4.3. The proof follows next time.
• Wednesday Jan. 26: We proved Theorem 4.3 and Lemma 4.1 which we will need to prove Theorem 4.5 in Section 4.4.
• Friday Jan. 28: We proved Theorem 4.5 and discussed conditional stability in Section 4.5. The proof of Theorem 4.6 in that section is part of the homework.
• Monday Jan. 31: Section 4.6 on asymptotic equivalence: some examples and Theorem 4.7.
• Wednesday Feb. 02: I mentioned Theorem 4.8 from Section 4.6, whose proof is very similar to the proofs of Theorems 4.6 and 4.7. After that we discussed Section 4.7 and a quick review of some necessary results from Section 2.9 on linear systems with periodic coefficients. We got to the point where we can write our periodic system as a perturbed linear system with constant coefficients. I didn't yet get to stating Theorem 4.9, whose proof will be in the homework.
• Friday Feb. 04: Statement of Theorem 4.9 for completeness. After that we started Chapter 5 on Lyapunov's direct (or second) method for stability. Section 5.1 and a bit of Section 5.2 (up to Lemma 5.1).
• Monday Feb. 07: Midterm, in class during class time, the material includes everything we have done so far up to and including Chapter 4 (so everything except for the little bit of Chapter 5 we discussed on Feb. 4).
• Wednesday Feb. 09: We continued Section 5.2. We proved some more results about orbits and then we stated the Lyapunov theorems. We saw a geometric interpretation of the theorems and some examples (up to and including Example 4 in the book).
• Friday Feb. 11: We discussed some more examples from Section 5.2 and proved the Lyapunov theorems from Section 5.3 for stability and asymptotic stability. Addendum: The style of my proofs differed somewhat from those in the book and was motivated by a note by professor Ralston on the Lyapunov theorems.
• Monday Feb. 14: We proved the instability Lyapunov theorem and started with Section 5.4. We ended with stating Lemma 5.5, the proof will follow next time.
• Wednesday Feb. 16: I was absent today. Your substitute instructor professor Micheli continued with Section 5.4 up to and including the two corollaries at the end (whose proofs are homework).
• Friday Feb. 18: I was still absent and unfortunately due to illness the planned substitute instructor could not make it. Class cancelled.
• Monday Feb. 21: No lecture: President's Day
• Wednesday Feb. 23: Finished Section 5.4 by applying Corollary 1 to the Liénard Equation. Started the Extra Section on the Poincaré-Bendixson theorem (this is not in the book; take notes!) up to and including Lemma PB2. Addendum: My treatment of this subject was based mainly on Chapter 16 in "Theory of Ordinary Differential Equations" by Coddington and Levinson.
• Friday Feb. 25: Continued with Poincaré-Bendixson, up to and including Lemma PB4.
• Monday Feb. 28: Finished Poincaré-Bendixson (Lemma PB5 and the proof of the theorem itself), then continued with Section 5.5 from Brauer & Nohel up to and including Example 2.
• Wednesday Mar. 02: Finished Section 5.5 and discussed Section 5.6 (I gave no proofs of the theorems in this section. The proofs of Theorems 5.7 and 5.8 are homework, the proof of the instability theorem I gave (which is not in Brauer & Nohel can be found in for example Chapter 5 of "Ordinary Differential Equations - Introduction and qualitative theory", 3rd ed., by Jane Cronin. I briefly mentioned Theorem 5.9 from Brauer & Nohel. They do not give a proof, but refer to reference 25, Chapter 3.
• Thursday Mar. 03: Extra lecture to make up for the cancelled class on Feb. 18. Time: 12:00-12:50 pm. Location: MS 7608. Started (an adapted version of) Section 6.4: proved part (I) of the existence of a periodic solution to a Van der Pol/Liénard equation. My treatment follows closely the note by professor Ralston about the Van der Pol equation.
• Friday Mar. 04: Finished the proof for existence of the periodic solution. I did not cover the uniqueness.
• Monday Mar. 07: Started with linear boundary value problems. This subject will probably take up this whole last week. I will base my treatment of this subject mainly on the notes by professor Ralston about Green's functions and adjoint problems. I also handed out the course evaluation forms at the end of this lecture.
• Wednesday Mar. 09: Continued with linear boundary value problems, most notably proved the Fredholm alternative and started proving the Green's function theorem.
• Friday Mar. 11: We finished linear boundary value problems. I proved the Green's function theorem and combined it with the Fredholm alternative into Theorem BVP 3. The final results about the Green's function for the adjoint equation I only mentioned. There was no time anymore to give the proofs. You can find the proofs on pages 4-5 of professor Ralston's notes about adjoint problems.
• Tuesday Mar. 15: Final exam, 11:30 am -- 2:30 pm, location: MS 5233. The material for the final exam includes everything we have done this quarter.