Math 286 (Differential Equations Plus): (Fall 2008)

Instructor: Alexander Yong

Lectures: 229 Natural History Building: M,T,W,Th: 12:00-12:50 PM

Office hours: M,W,Th: 2:00-3:0O PM

Syllabus: Contains outline, grading and other information about the course. NOTE: We've moved Exam 3 from December 1 to THURSDAY, December 4, 2008 (please note typo in syllabus, which says Monday). The final exam is a non-combined final and therefore is determined by this table. Hence our final is 1:30-4:30PM on December 17th, 2008. I've been informed that our exam room has changed. One of the exam rooms will be 229 NHB (Natural History Building) [For LAST NAME beginning with A-K inclusive, e.g., Zack Benson] and the other is 161 Noyes Lab [For LAST NAME beginning with L-Z inclusive, e.g., Allan Parker]. I will discuss seating assignments nearer to the exam.

Sections we'll cover: See here.

Practice Homework list: These reflect problems I want you to be able to solve, but not hand in.

Current Homework: Homework 8: Section 9.3: Q4,8,18,24; Section 9.4: Q8,10,14; Section 9.5: Q4,6,14. This is due Monday December 1, 2008.

FREE TUTORING: Will be available through the math department. See here.

Lecture 1: Email me some information about yourself (mathematical background and interests). Complete the following survey (if you haven't already done so in class). It will help me better prepare the course material to suit your needs. Also, read Polya's method of solving problems, either in his book, or the summary found here at wikipedia as "four principles".

Today we discussed some basic high level questions in trying to understand differential equations, including existence, uniqueness, as well as other suggested by the audience. I ended with an exercise to find more. HW: Section 1.1: Q1-6, 12, 17-20, 27-28, 45-46.

Lecture 2: We talked about Section 1.2, thinking of integration in one variable as a technique of solving DE's of the form dy/dx=f(x). Since this is mainly review, we took the opportunity to check on the skill set of preparing a solution. The HW (not stated in class) is Section 1.2 Q:3,9,13,21,25,27. I have posted the list of all _practice_ homeworks above. I will post the _first homework to be turned in_ on Thursday (each homework is due in one week).

Lecture 3: We went through some problems and presented "model solutions". We discussed Section 1.3 on the slope field method. The key point there is that you shouldn't draw it mechanically, but rather take advantage of the form of the equation dy/dx=f(x,y). We also discussed the possible failure of existence and uniqueness of DE's. We then stated a theorem guaranteeing existence and uniqueness for equations of the form dy/dx=f(x,y). We ended by asking the audience to go home and think about any terms in the theorem that are unfamiliar.

I want to repeat that all practice HW's for the entire course are posted above. From now on, I won't post the problems in class.

Office hours for Thursday Aug 28th are cancelled. Please send any questions by email.

Lecture 4: We finished our discussion of Section 1.3 and worked on Section 1.4. We left with a reading assignment to understand the warnings on page 35, 36 about singular solutions. (We'll take this up Tuesday.)

Assignment 1 (due next Thursday at beginning of class): Section 1.1: Q10, Section 1.2: Q 28, 44; Section 1.3: Q8,26, Section 1.4 Q 12, 28.

Note: please acknowledge all sources you use to solve your assignment problems. I encourage you to talk to me, fellow students etc. I would rather you not look up solutions online etc. However, in all instances, you must write up your solutions in your own words.

Lecture 5: We went over the active learning problem given at the end of the previous class. One of the themes of the class today was that many operations one does to a DE can alter the problem, by, e.g., introducing or removing a solution. One small example that arose as a result of an in class question was dy/dx=-1 vs (dy/dx)^2=(-1)^2=1. We then did our first example of _deriving_ a DE from an application (population growth under constant birth/death). We then moved on to talk about first order linear ODE's. The basic trick here is to multiply by the integrating factor. We ended by claiming (without proof, yet) that for these DE's one avoids any issues about generating false solutions, or eliminating valid ones.

NOTE: Exam 3 was moved to December 4th, 2008 in order to make more time to study after the Thanksgiving break.

Lecture 6: We went over the proof of existence and uniqueness of a solution to an ordinary first-order linear DE. What we showed was that any solution to the original DE can be expressed in the final form given by the "integrating factor trick". Then we argued by substitution that any function satisfying that final form solves the original problem. Lastly, a single initial condition fixes the constant "C". Then we talked about Section 1.6, and various types of substitution and transformation tricks one can employ. We're moving away from spending time on computation in class (which I expect you'll work on at home), and focusing on high level ideas, including proofs of various results.

Lecture 7: We went over the necessary and sufficient condition for a DE M(x,y)+N(x,y)(dy/dx)=0 to be exact. It was a long proof, but I encourage you to train yourself to follow such exact reasoning.

About half the class knew what was meant by "necessary and sufficient", so we spent a bit of time discussing this, since it'll come up again. A statement is necessary for an assertion to be true if the assertion implies it holds. For example, it is a necessary condition that an NBA basketball player know how to shoot a ball. The assertion is that John is an NBA basketball player. You know he therefore must be able to at least shoot a ball, otherwise how could he get into the NBA?

A sufficient condition is like it sounds: it's enough to know the sufficient condition holds to be guaranteed the assertion is true. For example, (as pointed out in class), if you are THE Shaq O'Neal, then that sufficient to know you're an NBA basketball player.

Some things to note: obviously necessary and sufficient conditions can be far from one another. This is the case above: the necessary condition of being able to shoot a ball is a pretty weak thing to know is true about an NBA basketball player. On the other hand, a sufficient condition of being Shaq is a very strong condition: only one person is Shaq, and yet there are many NBA basketball players.

Ideally, as in the theorem in class, you have a condition that's both necessary and sufficient. That is, a _useful_ condition that _characterizes_ the assertion you want to understand. Here's a nonuseful one: "A person is an NBA basketball player if and only if he has signed a contact to play on an NBA team." Duh.

By "useful", in the basketball case, if you're an aspiring NBA player you want to know "exactly what it takes to make it to the NBA". Sadly, I'm not qualified to give such a useful characterization, since there's counterexamples to most statements like "must be taller than 6 foot 6 inches". But that's like math, sometimes its _hard_ to find a useful characterization.

Knowing about the simple idea of necessary and sufficient has some useful consequences that you can exploit when thinking about math problems. Usually one needs to prove something of the form "if A is true, then B holds". That is A is a sufficient condition for B. (Think A="Shaq", B="NBA player.) In that case you _cannot_ think that also "if B is true, then A holds". This is a classic fallacy, or logical misstep. However, what is equivalent is that "if B is false, then A is false." The latter statement is the _contrapositive_ of the first, and often it's easier to show that.

Also, it's often useful to try to understand to what extent a necessary condition fails to be sufficient. Or to weaken a condition to see how it changes the problem (in the sense of trying to find easier problems to solve). Often these mental tricks help you get a grip on the problem, so that you can solve the problem you really wanted.

From this point of view, this is what happened in class. Given an exact DE we deduced a necessary condition. But then (...imagine some playing around before this conclusion...) we found it was sufficient.

As another illustration, suppose we want to solve a serial murder mystery, e.g., the famous Jack the ripper case. Really you want to prove that so and so was it. That's pretty hard. But you can change the problem a bit. You can assume that the killer is visibly mentally insane and lives in the area. What does that extra _possibly false_ assumption do? It helps narrow the possibilities. Now you just narrow your search to finding such a person (by asking around). The point is that you now have introduced an avenue to think about the problem (ask people for anyone who appears insane) not available to you without that extra assumption. It's a natural assumption to make -- if in the problem you study you make to strong or unnatural an assumption, then yes, this could be a blind alley -- but that's the point of experience (e.g., doing homework), and good sense.

Lecture 8: We talked about Sections 2.1 and 2.3 of the textbook. The main common themes we're to pick a general framework to understand some phenomena, study specific managable (i.e., solvable using DE techniques we have) examples, and to think of the t->infinity limit.

I also handed out Assignment 2, due at the END OF CLASS next Monday: Section 1.5: Q16,24,40; Section 1.6: Q28,58; Section 2.1: Q:16,34.

I've been asked about the first exam, which is September 18, 2008. We'll cover all material up to and including Wednesday's class, which I anticipate will be parts of Section 3.1. The test will consist of four problems, possibly with parts. I will mainly ask problems that are similar to homework. You might be required to derive a differential equation, as we did in class for the population model. I won't ask for you to reproduce an entire proof from class, but it is possible that I'll ask you to show some fact about a differential equation like one in a proof (e.g., a solution to some DE is given by a certain explicit expression, or to explain the limit behavior). My desire is to make the test straightforward if you've done the HW's and have been following along in class.

Practice exam: well, more generally, basically here's how I'll create a test: I'm going to look through each section and consider problems like the HWs (both practice and those to hand-in), as well as exercises and examples from class. I might modify or combine two or more questions, keeping in mind the time constraints of only 50 minutes. Obviously I'd like to ask questions that will involve techniques from more than one section, since I'd like to achieve coverage. One practice test is: Section 1.4 Q 12; Section 1.5 Q33, Section 2.1 Q13, Section 2.3 Q4, Section 3.1 Example 5.

Lecture 9: We went over more examples concerning air resistance and vertical motion of a mass. We also discussed gravitational problems. I ended with a question about why in the "retrorocket" problem, we simply set up an equality between the two equations we derived. We'll take this up next class.

In view of the upcoming exam, next week I'll hold an extra office hour Tuesday 2-3PM and also extend my Monday office hour by one hour, until 4pm.

Lecture 10: We went over the principle of induction, since that's needed for one of the homework problems. We also finished off our discussion of the "retrorocket" problem. One thing that I didn't make clear in my explanation of the two equalities is the following: let t_{on} be the time you need to turn on the rockets. Then the substitution t_{on} into the T=0 equation gives you an identity between r_{on}=r(t_{on}) and some unknown velocity v_{on}. Doing the same thing to the T=4 equation gives you a different identity for r_{on} and v_{on}. The point is that you have two unknowns and two variables, so you can solve for r_{on} (achieved by "setting the RHS of the two equations equal"). I personally find this explanation more _personally_ convincing. That said, once I'm convinced, I wouldn't need to write it down in an argument, because "it's so trivial". The larger point being that for a claim you make, you can decide to do three things: (1) assert it's obvious, (2) assert it's nonobvious but admit you don't know why it's true, or (3) argue it. (1) is fine so long as you don't make a mistake. (2) is at least intellectually honest. For (3): if you don't think it's obvious (even if you suspect others might), you should argue it, especially because maybe it's not obvious, or worse yet, false (which hopefully the process of writing the argument will uncover)!

The key here is to come to a definitive conclusion about every claim you make. By definitive, there should not be any "probably" or "..." or "something like". (Here's a guidepost, in most of our solutions, the arguments are a sequence of pointed statements like "A => B", or "A=B", where A and B are two succinctly stated mathematical objects, such as a mathematical expression.)

Relatedly, try to come to a definitive conclusion about _something_ during every period that you work on a problem. Obviously, ideal is "the solution is this", but also helpful are "this approach does not work", or "the DE I needed is this, but there's still more to do". What you want to avoid is "I'm just stumped, because, well, I don't know why."

We also went over a bunch of exercises: one was a Bernoulli equation, whose point was that you should be able to derive most of the solutions, without resorting to memorizing the book formula. Another was an application of the logistic equation to solutes in solvent analysis. The third was a straight-up partial fractions computation (arising from a separable DE).

Lecture 11: We started chapter 3 on higher order linear DE's. One thing that I think is really important to learn at least the principles of is that of an abstract vector space. The solutions to the homogeneous forms of these DE's form a vector space. We discussed mechanical systems, which are an important application of the 2nd order case.

Lecture 12: Continued discussion of Section 3.1 up to and including the proof that two linearly independent solutions span the solution space. Handed back assignment 1.

note 1: Bring your ID card to the Exam. No calculators.

note 2: If you did problem 38 rather than 28 on the first assignment, turn it in to me by the end of the exam on Thursday.

Lecture 13: We finished up Section 3.1 and worked though a bunch of Section 3.2. NOTE: the test on Thursday will only cover up to and including Section 2.3.

Lecture 14: Review session for Test 1.

Lecture 15: Test 1. Here are the solutions and the grading scheme.

Lecture 16: We finished Section 3.2. Two linear algebra concepts we discussed are that of _the determinant_ and _Gaussian elimination_.

Lecture 17: We discussed Section 3.3. One thing that I'd like to clarify, from the end of the lecture is that the coefficients can be complex, hence I shouldn't have written that we were breaking it into the real and complex parts (we would if they were real). I'll point this out tomorrow.

Lecture 18: Assignment 3 is Section 3.1 Q28,32, 40, Section 3.2 Q24, 34, 36 and Section 3.3 Q4, 22, 34, due next Wednesday at the end of class. Starting with this assignment, I will change the grading scheme a little: two or three of the problems will be graded as before. (Unfortunately, the marker's time constraints makes it impossible to do more with 72 students.) However, if an honest effort has been made on the remaining problems, he will give a score of 5 points.

Lecture 19: Continued talking about Section 3.5. We discussed what specifically it means for f(x) to be 'simple', and did some more examples of the method of "undetermined coefficients".

Lecture 20: Finished Section 3.5, including discussion of variation of parameters.

Lecture 21: Section 3.6, and discussion of the homework problem on Vandermonde determinants.

Lecture 22: We talked about resonance and then about electrical systems, and the mechanical-electrical analogy. Assignment 4 (due next Wednesday at the end of class) is Section 3.3 Q48, Section 3.5 Q8,16,26,50,60, Section 3.6 Q4,20,22,28.

Lecture 23: Return of Test 1. The average was 32.34/40 or roughly 81%; great job!

Lecture 24: Finished up Chapter 3 and did Section 4.1 for the most part.

Lecture 25: Started Chapter 5. It was pointed out that there was a typo above in the assignment list. In Section 3.6 please do question 4, not 40 (which doesn't even exist). The list is corrected above.

Lecture 26: Homework 5, due next Wednesday: Section 3.7 Q2,10; Section 3.8 Q2,6,14; Section 4.1 Q22,24,26; Section 5.1 Q22,32.

We continued Chapter 5.

Lecture 27: Continued Chapter 5, discussed eigenvalues.

Lecture 28: Today we recapped how to handle systems of the form X'=PX via the eigenvalue method, and in particular the case where P has distinct eigenvalues (=> LI eigenvectors => LI solutions built out of those eigenvectors).

After class a student brought up a question about how the initial conditions that force a unique solution for X'=PX look like. They are of the form X(a)=[b_1 b_2]^T. But sometimes they might be expressed as x_1(a)=b_1 and x_2(a)=b_2, by which we mean x_1(a) and x_2(a) are the two components of the 2x1 matrix X(a). (This brings up a mild notational inconvenience, as we sometimes like to talk about two matrix valued functions X^{(1)} and X^{(2}), which I like to distinguish from the components x_1 and x_2 of X by using superscripts.)

The second exam is coming up Thursday, October 23rd, in class. It will cover Section 3.1 to Section 5.2 inclusive.

Lecture 29: We covered two variations on the x'=Ax problem. First, we consider the case that A has complex eigenvalues. The main surprise here is that for each pair of complex conjugate roots, you only need to determine the eigenvector of one of them and use the real and imaginary parts as LI solutions. (WHY??) We also analyzed the problem of x''=Ax.

Here is a selection of review problems, some of which I will take up next Wednesday. (NOTE: I don't promise the test will only have questions such as these, but they give a sense of what I'm thinking about.) You should also look through the practice and assigned homeworks, and examples from class: Section 3.1 Q26,40; Section 3.2 Q13,25,30; Section 3.3 Q30,32,38,48,49; Section 3.4 Q11,22; Section 3.5 28,30,50,56; Section 3.6 Q20,22; Section 3.7 Q10,12; Section 3.8 Q4,8,14; Section 4.1 Q22,24,26; Section 5.1 Q28,43,44; Section 5.2 Q22,26,28,44,48.

Lecture 30: We worked on Section 5.3, discussing applications of systems of second order DE's on mechanical systems.

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Lecture 31: We worked on Section 5.4. Started the test 2 review.

Lecture 32: Test 2 review.

Lecture 33: Test 2. Here are the solutions and grading guide.

Lecture 34: We continued our discussion of generalized eigenvectors. It's worthwhile to remember that while AX'=X seems like a very specific class of DE's to solve, more general linear systems of DE's reduce to this case.

Lecture 35: We'll finish up on generalized eigenvalues and discuss the method of solving AX'=X by matrix exponentiation.

Lecture 36: Finished up on solving AX'=X by matrix exponentiation.

Homework 6, due Thursday November 6, 2008: Section 5.2 #30,46; Section 5.3 #14,16; Section 5.4 #4,16,30,32; Section 5.5 #8,30,31,32.

Lecture 37: Finished Section 5.5 with an example (emphasizing first principles, not the unmemorable formulas presented there). We covered Section 5.6 (excluding variation of parameters) and introduced Fourier series. The main questions we consider: what are the Fourier coefficients (and why?) and when can we expect (pointwise) convergence?

Lecture 38: We'll continue our discussion of Chapter 9.

Lecture 39: We discussed what happens if the period of a function is general (how to obtain it's Fourier series), convergence issues, and did an example.

Lecture 40: Exam 2 will be returned. The average score was 25.34/40, or about 63.3%. The grading scale is roughly A=34-40, B=28-33, C=22-27, D=18-21, F=0-17.

We discussed the Fourier series of even and odd functions, as well as even and odd extensions of non-periodic functions. We also explained when term-by-term differentiation of the Fourier series of f converges to f'. I want to correct a typo from class. The odd extension of a function f defined on [0,L] should have f(t)=-f(-t) on [-L,0] (so that f is an odd function). The picture I drew didn't represent this! (The definition of an even extension is correct.) I'll discuss this more during an example in the next class.

Assignment 7: Section 5.6: 10,12,16; Section 9.1: 4,20,24,30; Section 9.2: 8,14,16,20,24.

Addendum about exam grading: I wanted to emphasize that if there are any questions about the grading of your exam, I encourage you to contact me for a discussion (unfortunately, this was not possible to do in any reasonable way during the process of returning exams on Thursday). Often (e.g., in my own experience) objective evaluation of our mistakes tends to be difficult. It is often the case that with some help analyzing the mistake one can learn to avoid not only the specific error again, but similar (more general errors) in the future.

Lecture 41: We took up some exam and homework problems.

Lecture 42: More exercises; applications of Fourier series to mechanical systems.

Lecture 43: Exercises, started Section 9.5. I will post the new HW on Monday.

I've been informed that our exam room has changed. One of the exam rooms will be 229 NHB (Natural History Building) and the other is 161 Noyes Lab. I will discuss seating assignments nearer to the exam.

Lecture 44: We'll continue talking about the heat equation, and in particular solve the two boundary problems we posed last time. Homework 8: Section 9.3: Q4,8,18,24; Section 9.4: Q8,10,14; Section 9.5: Q4,6,14. This is due Monday December 1, 2008.

Lecture 45: I want to put in a reminder that Test 3 is on THURSDAY, December 4th, 2008. (Please note typo in the syllabus which says MONDAY rather than THURSDAY). Today we'll continue covering the heat equation, and in particular discuss the method of separation of variables.

Lecture 46: We covered the string equation.

Lecture 47: More on the string equation, and separation of variables.

Test 3 will cover sections 5.3 to 9.6 inclusive. I will post some specific review questions in the next few days, but "basic" problems on the test will reflect homework, practice homework questions and examples, exercises from class. The test will also have problems that test your understanding of the core ideas emphasized in class.

Some review questions: I'll expect all the basic competencies concerning techniques taught. In addition, here are some more "advanced" problems: Section 5.3 Q:14, 16,17; Section 5.4 Q:20-22, 33. Section 5.5: 28, 29, 33; Section 5.6: 11-15; Section 9.1: 23,27; Section 9.2: 17-22, 25; Section 9.3: 18, 21, 22, 23; Section 9.4: 5-8; Section 9.5: 8-12, 15, 17; Section 9.6: 6-10, 16, 20. Be sure with Sections 9.5 and 9.6 that you understand the method of separation of variables.

Lecture 48: We covered Section 9.7.

For the final exam, please note the assignment of exam room by LAST NAME above (see notes to syllabus).

Lecture 49: We covered Section 10.1

For Test 3, remember to bring your ID etc.

Lecture 50: Review for test 3.

Lecture 51: Test 3. Here are the solutions and grading guide. I'll try to have it graded by the upcoming Tuesday.

Lecture 52: Finished up section 10.1, concluding the new material for the course. Noted that the exam will be comprehensive. On Wednesday we'll review for the final. In light of the amount of material, I think it is most convenient if you suggest topics for discussion of importance (we won't go through specific examples at length, due to time constraints however).

Lecture 53: I'll return Test 3 and discuss the solutions at length, emphasizing common errors and misconceptions. The average score was 51%. The test was on the hard side. Consequently a number of students have expressed their concerns about the final grading curve. Without getting technical, I want to re-emphasize that my sincere desire is for you to get a good grade, while also maintaining suitable standards for our school. In particular, for all of the tests, and for the final grade I expect the median grade to be B- (thus 1/2 of all students will have a grade at or above this).

Notes for the final: The comprehensive final will emphasize problems that we did not cover already on tests. That is not to say that there will not be problems related to those from the test, but I hope the previous statement will be helpful to you. For example, we never had a mechanical system question on the midterms.

The exam will contain _no_ definitions or statement of theorems to remember. However, the questions do intend to test the fundamental principles that underlie the course. I believe one useful exercise is the following: can you name the important ideas in the course? Usually in a semester long class, in effect there are not more an 10 or 15 _key_ ideas. Feel free to come and talk to me about your list.

Lecture 54: Review for final. Here is the FAQ for the final . Also here is an exam for a related class that I wrote. WARNING: our exam may look nothing like this!! But I include it in case you find it useful.

Another practice exam can be found here.

Notes on the final: Here are some brief remarks about the problems:

Q1: this is a homework problem we discussed in class.

Q2: this is something we discussed in class explicitly for n=2, and in similar contexts: see Section 5.3 (the main idea is to guess the solution x=ve^(\lambda t) into x^(n)=Ax and see v is an eigenvector of A and \lambda^n is the corresponding eigenvalue).

Q3: This is a problem from the textbook, and we've discussed a very similar problem in class. Compute the fourier series for t (as we've done a bunch of times) and guess the fourier series solution, and equate coefficients. See page 606 #14.

Q4: Also in the textbook, pg 595 #20. The main idea to simplify things is to compute the series for t^2 and relate it to the given quadratic.

Q5: Another textbook problem, and we also discussed a similar problem in class. See pg 652 #8.

Q6: As the hint suggests, consider a linear homogeneous DE, i.e., a_n x^(n)+ a_{n-1} x^(n-1)+...+a_0=0 whose roots are \alpha_1,...,\alpha_n. The assumptions guarantee that n LI solutions are of the form x_i=e^{\alpha_i t}. The general solution x_gen is given by superposition. The main theorem of existence and uniqueness says if we know what x(0)=\beta_1,x'(0)=\beta_2,...,x^(n-1)(0)=\beta_n then we can pin down x_gen. How do you determine the coefficients in the expression for x_gen? You solve the nxn system given in the question.

Q7: The problem asks you to check that BAB^{-1} has the same eigenvalues of A. From class this means you check the roots of det(BAB^{-1}-I\lambda)=0. But this equals det[B(A-I\lambda)B^{-1}]=0. Since the determinant of a product of matrices is the product of determinants and det(B)=1/det(B^{-1}), the roots of the latter equation are the same as those of det(A-\lambda I)=0. Now apply the main theorem on LI solutions to a system of the form X'=MX.

Q8: This was a homework problem, that was turned in. See pg 329 #14.

Q9: This is a textbook question, pg 652 Q8. We also discussed a similar problem in class.

FINAL NOTES: Except for questions 6 and 7 we've discussed all the problems explicitly in class, and other than Q2, all other problems were from the text or even the homework. The solutions to Q6 and Q7 are based on the core ideas we've emphasized.

Final distribution of grades by raw score (followed by number of students in range): A+=95-100 (1), A=76-95 (4), A-=71-75 (9), B+=65-70 (4), B=60-65 (16), B-=54-59 (6), C+=49-53 (7), C=39-48 (10), C-=30-38 (7), D=20-29 (4), F=0-19 (2). Hence 40 students (or about 55% achieved a score of B- or better).

The median for the final was 30/90. I've submitted (almost all) the grades to the registrar. Have a nice holiday!