## Math 2243 (Linear algebra and differential equations): (Fall 2007)

Instructor: Alexander Yong ayong@math.umn.edu

Lectures: Mondays, Wednesdays and Fridays 11:15AM-12:05PM. 1701 University 143

Outline: From Course guide: Overview: The course is divided into two somewhat related parts. Linear algebra: matrices and matrix operations, Gaussian elimination, matrix inverses, determinants, vector spaces and subspaces, dependence, Wronskian, dimension, eigenvalues, eigenvectors, diagonalization. ODE: Separable and first-order linear equations with applications, 2nd order linear equations with constant coefficients, method of undetermined coefficients, simple harmonic motion, 2x2 and 3x3 systems of linear ODE's with constant coefficients, solution by eigenvalue/eigenvectors, nonhomogenous linear systems; phase plane analysis of 2x2 nonlinear systems near equilibria. Audience: Part of the standard 2nd year calculus course for students outside of IT.

Grading: Quizzes (15%), (three) 50 minute in class midterms (45%) and a final (40%). Your final grade (A,B,C, etc) will be determined by your relative performance to others in the class.

Textbook: Differential equations and linear algebra, 2nd edition, by Edwards. We will cover Sections 1.1-1.5, 2.1, 2.2, 2.4, 2.5, 3.1, 3.2, 3.4-3.7, 4.1-4.3, 5.1-5.6, 6.1-6.2, 7.1-7.4, 10.1-10.3.

Homework: The list is here.

Office hours: The discussion section leaders, together with their office hours are:

Jiaqi Yang -- TTh 1:25pm-2:15pm -- email: jqyang@math.umn.edu

Xin Shen -- TTh 9:00am-10:00am; tutoring time is T 15:35-16:30 and W 13:25-14:15 -- shenx125@math.umn.edu

Also see the following free tutoring information. My office hours are MF 12:05PM-1:05PM after class.

Exam notes: No cheat-sheets during midterms or exams. Calculator policy: scientific/non-graphing ones are allowed. There will be no make up tests. If you miss an hour exam for a *documented* medical reason, your grade on it will be the prorated grade of your final exam. The pop quizzes will be 10 minutes in duration and given during the discussion sections. Although they will be unannounced, they will cover recent homework material.

The one-hour exam dates are: Thursday September 27, 2007; Tuesday October 30, 2007; Tuesday November 27, 2007.

Final Exam date: December 14, 2007 (1:30pm-4:30pm); see here. Any conflicts are handled through the Undergraduate Math office. The room assignments for the exam may be found here. (Look under final exams.) Sections 021, 022, 023 =AmundHB75, Section 24=AmundH 240.

Homework: Problems will be assigned for each section.

Lecture 1: Read Polya's method of solving problems, either in his book, or the summary found here . We covered Section 1.1 (HW #23-27) and began on Section 1.2. We discussed the basic goals for the course and the dichotomy of theory vs numerical approximation. We also discussed the classification of differential equations. We are concerned with _ordinary_ DE's (only one independent variable). Presently we played with linear DE's, meaning only a first order derivative appears.
It's stated above, but I want to emphasize that there are tutors available much of the day, during M-F, in LindH 150, see
here.

Lecture 2: We covered all of section 1.2. We started by explaining the easiest form of differential equation: the integral of a single variable function. I reminded you of the fundamental theorem of calculus and used this to set up a way to approximate the solution of the integral, e.g., by Monte Carlo methods (but this is special to this kind of DE). We then ramped up this point to obtain a second order DE, solved by integrating twice. This situation is useful for studying motion of a particle (e.g., a ball) in terms of velocity and acceleration (e.g., due to the effects of gravity). We ended off with a richer example, the "swimmer's problem". I think the key point in the example is to convince yourself of how the horizontal velocity of the swimmer and the vertical velocity of the current affects the _instantaneous_ trajectory of the swimmer. A few students asked me an interesting question after class: why we couldn't just divide the width of the river by the swimmer's horizontal velocity, and then multiply that by the velocity v_0 at the center of the river of the current. This is, however, a common mistake: the point is that the velocity of the current changes! (Try this calculation and compare the answers to the one in the textbook.) Such analysis is correct only at the _infinitismal_ level. HW: Section 1.2: 2,3,6,7,13-18,20,23,24,26,29,30.

Lecture 3: We covered section 1.3. We discussed differential equations of the form y'=f(x,y). In general, there is no systematic way to obtain an _exact_ solution. However, slope fields do provide a way to obtain an approximate solution. Moreover, they help describe the complete _family_ of solutions. We also stated a theorem giving sufficient conditions for y'=f(x,y) to have a unique solution (the existence and uniqueness theorem). The key words in that theorem that you should understand are "open interval", "partial derivative" and "continuous". HW: Section 1.3: Q12,13,16-26, 29, 31.

Lecture 4: After quickly recalling the existence and uniqueness theorem of the previous lecture, we went through an example that shows it in action. Reading the hypotheses and disclaimers of the conclusion is important! In particular, the only guarantee that the theorem gives is that there is a _local_ solution near the point that defines the initial condition of the first order DE in question.

Another thing I should emphasize, since a few questions were unclear about this: if the hypotheses of the theorem fail, you cannot necessarily deduce that there is a nonunique solution, or that no solution exists. It just means you can't conclude anything. (Logically, the comparison is that theorem is that (hypothesis) this person is my dad (conclusion) his hair is grey does not mean that if some person is not my dad (failure of the hypothesis) then the person's hair is not grey -- this is clearly ridiculous.)

After all that we went onto separable differential equations. A number of questions in class concerned the combining of constants into a single constant during the intergration procedures. This is OK, since the constants were arbitrary anyway. A technicality that is crucial however is that our cross-multiplication procedure to solve these DE's can (and often does) lose solutions, giving rise to "singular solutions". I ended with an example: note-- I erroneously said that when b>0 we have a unique solution; I meant two solutions.

Lecture 5: HW (section 1.5) is 1-25(odd), 33, 36, 37, 38, 39, 42. Today we talked about first order linear DE's. The main idea is to introduce an integrating factor, the exponential of the integral of P(x). We also discussed the mixture problem, which led out first derivation of a differential equation modelling a situation.

Lecture 6: HW (Section 2.1) is 1-9, 12, 13, 15, 19, 21, 22, 23, 26, 27, 30-33. Today we talked about population growth models. Generalizing the models that use constant birth and death rates, we explained the derivation in the setting where there is _non_constant birth and death. (Note, be prepared on exams to be able to actually construct, with justification the appropriate differential equation, using the kind of reasoning I shown you in this class and the previous one.) In particular, we studied the logistic equation.

Lecture 7: HW (Section 2.2) is Q1-12, 13-24,29. Today we started to introduce some qualitative aspects of differential equations that we can talk about, even if we do not have explicit solutions (although presently, we begin with having such a crutch). We focused on autonomous first order DE's, namely dx/dt=f(x). The _critical points_, i.e., the solutions x=c of f(x) give rise to equilibrium solutions of the DE. We classified the equilibrium solutions into two classes: stable and unstable. REMINDER: the first test will be on Sept 27th in your discussion section. It will cover up to and including Section 2.5 of the textbook.

Lecture 8: HW (Section 2.4) Q-1-10, 17-24. We finished off the discussion of section 2.2, by analyzing the effects of perturbing the parameters of a differential equation and asking how that affects the qualitative nature of the solutions. This gave rise to "bifurcation points", i.e., points where the structure of the solutions change markedly. Admittedly this is somewhat vague, but is certainly clear in the example of the logistic equation with harvesting.

A question was asked about "semistable" equilibrium solutions. I know this appears in the textbook's answers -- but NOT the chapter itself(!). For me, either an equilibrium solution is stable (with the delta-epsilon definition we stated) or otherwise not stable.

We also talked about Euler's method, which is essentially a recap of our discussion on slope fields (the subtle difference being that with slope fields, we emphasize approximating all solutions at once, whereas now we emphasize approximating one specific solution). I spent some time explaining the programming mechanics (in pseudo code) for writing a Euler's method program. Let's say that the step size is h=.5 and you have 20 steps; dy/dx=x^2+y^2 and y(0)=10. The code might be

euler:=proc(x,y,h):

local i;
for i=1 to 20 do
y=y+h*(x^2+y^2);
x=x+h; # update y before updating x!
print((x,y));
od: # end do

end;

or something like that depending on what system you're using. Then you run "euler(0,10,.5)".

Lecture 9 Some notes about the tests. There will be five questions (some with at 2 parts). Some of the questions will be variants on homework, or examples from class/text. There will be no "trick questions" or "challenge problems", and I will focus on comprehension of basic material. However, I do expect you to be fast and accurate with the material covered. Also, I expect you to give clear explanations of what you are doing, e.g., if you think a DE is to be solved by separation of variables, etc, say so. Finally, I expect you to know the theorems we discussed in class.

Put another way, here's my algorithm for constructing tests for this course (there's no secret really, I expect most instructors do the same thing): I look at the homeworks, examples, and think about what I emphasized in class. Then I try to find questions that test comprehension of these matters. There are so many things that one _could_ test, but sadly, there's only 5 questions to do that with and I can only expect 10 minutes from you for each question! So I'm pressed to find the most salient things to ask about. I need to know that you can actually carry out the mechanics of the material; I also would like to know that you understand the theoretical justification behind things as well. Fortunately, I know that you've done the homeworks and paid attention in class, so I can ask questions that have some depth and expect you to recognize how to solve them quickly.

Today we talked about the Euler and improved euler methods and the Runge-Kutta method for approximating solutions to dy/dx=f(x,y). I also stated the Theorem on Error for Euler's Method. It shows that Euler's method is a "linear error" algorithm. The improved version is quadratic (better) while Runge-Kutta is quartic (better still). I described a "Sample test": Section 1.4 Q15, Section 1.5 Q22, Section 2.1 Q12, Section 2.2 Q 22, Section 2.4 Q10. As I said above, I expect you to know definitions and theorems, and how to "derive" a DE. Warning: the sample test should give you a sense of how long the actual test is, but I don't claim the problems or even the problem types will be the same!

Lecture 10: We went over the sample test.

Lecture 11: We had the first test. Lecture 12: Professor Calin Chindris substituted for me today. He covered Section 3.1, HW #1-28 odd, 31-34. (Note: the class on Monday October 1, 2007 is cancelled, due to a flight cancellation to Minneapolis St. Paul Airport Sunday afternoon. I will cover section 3.2 in the next class, the HW is Q 1-20, 28.)

Lecture 13: I went over Section 3.2 on Gaussian elimination; HW is Q1-20, 28. We also discussed the theorem and proof that systems of linear equations with row equivalent augmented matrices have the same solution set. We mainly focused on the 3 equations in 3 variables, but also discussed how the methods generalize.

Regarding Test 1, here are the solutions . The average on the first test was 28.35/50. Here is a grading guide to let you know how things are going: A=45-50; A-=42-44; B+=38-41; B=35-37; B-=31-34; C+=25-30; C=22-24; C-=19-21; D=14-18; F=0-13.

Warning: this is a rough guide. Your final grade will be based on your total raw score based on all exams and quizzes. I will also rescale based on differences in grading in the sections. However, the above guide is arranged so that a little more than half of the class had a C+/B- or better.

Lecture 14: We covered Section 3.4 on Matrix Operations. HW is Q1-12, Q30-34.

Lecture 15: We covered section 3.5 on the inverse of a matrix. HW is Q1-46. Much of today was spent covering some foundational facts with proofs.

Lecture 16: Today we discussed the determinant of a matrix. In the 2x2 case we saw that a matrix is invertible if and only if its determinant is nonzero. We work to generalize this statement, as well as a new idea: Cramer's rule, which expresses the solution of a system of linear equations as quotients of certain determinants. Building from the 2x2 case, we recursively define determinants of nxn matrices via _cofactor expansion_. We then discussed some crucial (and I personally think, fun) properties of determinants. HW Section 3.6 was 3,4,9,10,15,20,21,22,43.

Lecture 17: We finished off chapter 3 with section 3.7, on polynomial interpolation of data points, as well as fitting circles to three points. HW in Section 3.7: 1-10, 35.

Lecture 18: We started Chapter 4 and discussed the (abstract) vector spaces and their definition (recall the closure and properties (V1)-(V8)). R^3 is an example of a vector space, but so is the set of all (say, continuous) real valued functions on the reals (check me!). The HW is Section 4.1 Q1-4, 19-24, 29-32, 39.

Lecture 19: We'll cover section 4.2. HW is Q1-14, 15-18, 19-22, 29.

Lecture 20: We mostly wrapped up Chapter 4. Notice that Chapter 3 and 4 can be summarized into a few key ideas, linking inverses, determinants, uniqueness of solutions to Ax=b, and linear independence of the column vectors forming a matrix A. Expect to verify subspaceness on the midterm. Section 4.3 Q-1-8, 17-20.

Lecture 21: Finished chapter 4 with a discussion of how to determine if vectors v_1,...,v_k (k less than n) are linearly independent. Then we moved to Chapter 5, concerning higher order DE's. Our first stop is to look at second order DE's. But really, we want to think about _linear_ ones, and their associated homogeneous DE's. We then explained how solutions to the latter form a subspace of the vector space of all real valued functions (the example of a "non R^n" vector space from last chapter). We also stated a uniqueness and existence theorem for a second order linear DE (normalized to have coefficient 1 in front of the y''). Homework: Section 5.1 Q-1-40 odd.

Here are some sample questions that caught my eye for the purposes of our forthcoming midterm: Section 3.2: 11, Section 3.5: 19, 35; Section 3.6 27,53; Section 4.2: 6, 13, 28; Section 4.3: 9,27; Section 5.1: 37. There will question(s) about theorems and definitions. Warning: Do not assume the above questions are all, or even part of the test, they are only a guide, just like for the first midterm. You should encorporate them as part of your overall study. That being said, you should be able to answer such questions for the test.

Lecture 22: HW 5.2: # 1-15 (odd), 17, 18, 24, 25.

Lecture 23: Professor Chindris lectured on Section 5.3.

Lecture 24: Professor Chindris lectured on Section 5.4. Note that I've put _all_ homeworks in a link found above (near the top of this page).

Lecture 25: We discussed the first half of Section 5.5 on inhomogeneous linear DE's, up to "Rule 1" in the textbook.

Here is the grading scheme for Midterm 2: 0-20=F; 21-24=D; 25-27=C-; 27-29=C; 30-31=C+; 32-34=B-; 35-37=B; 38-40=B+; 41-45=A-; 46-50=A.

Lecture 26: Discussed the second half of Section 5.5.

Lecture 27: Discussion of section 5.6. The mathematical techniques in this section are from 5.5, but the "modelling" ideas about springs are really inspiring, and there's a lot to learn from these examples, since many things can be modelled in terms of springs.

Discussion about grading: I had some interesting discussions with students during office hours about grading. I want to share them since I hope it might be of some help.

There's always a question about whether "too many marks" were stripped for a mistake. What's I demand of grading is that it's consistent (procedurally just), since at some level, that's all we can ask: each student is graded in the same way for any given solution.

That being said, I also want to _not_ discuss the "fundamental justice" (is this grade "fair" for the work I did) issue, but rather the related question of "what makes a good solution" since it needs to addressed (and is perennial).

Your job is to present a clear solution to the grader. Often, your grade is not a sum of individual parts, but rather an evaluation of the whole: is the argument sensible, and supported, on the whole?

Since you do not have an opportunity to sit by them while they grade and say "well, I _really_ meant...", you must make it as easy as possible to understand the logic of our argument, whether it be a derivation of a solution to a DE, or a proof of a linear algebra problem. This sometimes means full sentences, whenever possible. I know there are time constraints, so key phrases such as "this is a theorem from class", "it follows from ... that ..." help to clarify the flow of your ideas. Remember, it you must _convince_ the grader that you know what you're talking about. Ask yourself honestly, if I were a third party, would I know what this solution is saying? This kind of introspection often pays dividends.

One of the problems that had the most issues was the proof that a matrix with two equal rows has no inverse. The argument is that if A is such a matrix, then det(A)=det(A') where A' is the matrix obtained by subtracting one row from another (we showed this in class). But if we do this using the equal rows, then A' is a matrix with one row being all 0's. Now use cofactor expansion on that row and conclude det(A'), and thus det(A) are 0. Since from class we know det(A)=0 iff A inverse does not exist, the claim follows.

I want to discuss some mistakes students had made:

(1) one student said, let A=[[1,2,2],[0,1,1],[0,1,1]] (a 3x3 matrix with first row [1,2,2] etc). Now he showed that det(A)=0 and A inverse doesn't exist, which is the desired conclusion, right? -- The reason why this is erroneous is that the problem asks you to prove it for ALL matrices, not just one you choose that fits the hypotheses. I encourage you to use examples to see what's going on, but examples need to lead to a general solution to get full marks.

(2) Another student said "...if two rows are equal, cancel one of the equal rows to get an (n-1)xn matrix. Since the matrix is not square, it has no inverse." -- The operation of cutting out a row is as valid a thing to do as anything else (nothing sops you from doing anything you want to a matrix). BUT the question to ask here is "what things am I claiming are equal?". In the solution I suggest, I say det(A)=det(A'), where A and A' are related in a _precise_ way. Here, is the claim that the inverses of the nxn and (n-1)xn matrix are the same? If that is the claim, why? (In fact, this statement is not true, and we, of course, never proved the false statement in class.) Don't forget the "what things am I claiming are equal?" self-check, it comes up all the time. In many instances, if you cannot answer this question, it's because what you're writing is possibly not correct. Another one is the "why is this true" question (answers could be "by the previous sentence" or "because it's a result from class").

Mathematics is about precision: I often see sentences that speak of, e.g., the "equations in a matrix". Matrices per se do not have equations. They are used to form equations, but that's not the same thing. Ah, you might say, that's just nitpicking! --- But not so! Clear understanding of what the objects your dealing with have and don't have, go hand in hand with true understanding of the math. Imagine if you went to a mechanic and (s)he started talking about putting a new battery in your gas tank, and suggesting you buy four "winter windows" to prevent accidents. You'd probably back off and go find another mechanic!

(3) Finally, another student said "...(for the 3x3 case), subtract one row from another to get a matrix of all 0's (actually all that was said was "R3-R2"). This matrix is not row equivalent to the identity. A matrix is row equivalent to the identity iff it has an inverse. -- This time, every sentence is CORRECT, but still full credit was not granted, why? I'll admit this is harder to explain, and the student has a point: Let my intially ignore the fact that the argument was done only for the 3x3 case and the obscure "R3-R2" (why R3 and R2?), which while material (and some credit should be deducted for these unexplained assumptions), is not the main concern.

Main concern: one needs to explain why the second sentence (not row equivalent to the identity) follows from having a row of all 0's. This was not something we covered in class. BUT, as was pointed out to me, it is a remark on page 195 of the textbook to the theorem (third sentence) that we didn't cover either. (Also, for results we didn't cover in class (expecially remarks to theorems and footnotes), it's helpful to cite that you are indeed using a theorem from the textbook, on top of clearly stating the hypotheses and conclusions. Otherwise, as an admittedly extreme example, if the question itself happened to be in the textbook, why can't I just write the question and expect full marks?)

In view of the "evaluation of the whole" principle stated above, I believe that the grader took off marks in case (3) because the entirety of the argument looked specious (starting with the unstated assumptions of using 3x3 matrices). Also, while the main concern might look minor, consider the following sequence of statements: "John Doe has credit cards. The person who stole my credit cards has them in hand. Conclusion: John Doe is the thief". Analogous main concern: where did you say John Doe has _your_ credit cards?

Another sequence of statements: "The person who stole my credit cards must be the person who entered my room while I was away. John Doe was in my room. Therefore John Doe is the thief." Now, it might be that every statement is actually true, but how does each stand up in court? The first is sensible (note it sets up an "equality"). The third follows logically from the first two. So everything rests on the second statement. _Why_ is it true? If it was common knowledge since it was introduced into evidence many times by many witnesses, then no more needs to be said. But if this was never discussed, then evidence needs to be introduced (e.g., "Because Jane Doe saw him in my room."). Otherwise how do I know to take this as a fact, rather than mere speculation on your part?

I encourage you to speak to me about these kinds of issues, as I think it's an important skill that's secretly part of the point of the course.

Here are the solutions to the Second Midterm . (I noticed one typo to the soln to Q4, it should ..."iff A^{-1} does not exist". I'll change this soon.)

Lecture 26: Covered section 6.1 on eigenvalues.

Lecture 27: Covered section 6.2 on diagonalizing matrices. One correction: in the picture relating arbitrary matrices to the "beautiful diagonal matrices" I might have said accidently "diagonalizable" when I just meant "diagonal". Matrices that are "friends" with a diagonal matrix are "diagonalizable".

Lecture 28: Finished off section 6.2 and began Section 7.1 on systems of DE's. In particular, we gave a nice example of how a connected spring system is naturally described as such a system. We also showed that arbitrary systems can be reduced to first order systems. Midterm 3 is coming up after Thanksgiving. It will cover Sections 5.3 to 7.1 inclusive. As usual, there will be a mix of computational and theoretical questions (the theory questions will be on things we emphasize from class, as has been the case all along). Here are some sample questions to study: Section 5.3 Q 19, 31, 49; Section 5.4: Q4, understand the derivation of the DE for the mass-spring-dashpot system (see figure 5.4.1); Section 5.5: Q37, 47; Section 5.6: understand the derivation of the DE for the cart-with-flywheel system (figure 5.6.1); Q20, 22; Section 6.1: Q21, 36, 37; Section 6.2: Q23, 31, 34, 36, 37; Section 7.1: 7, 24, understand the derivation of the system of DE's for a mass-spring system as done in class. Bring your photo ID to MT3, only non-graphing calculators are allowed.

I recommend that you spend the time to try to solve the sample MT questions yourself, using, e.g., Polya's method (see lecture 1). Even if one of the problems takes you some time, it will pay great dividends in your understanding. In my opinion, this actually will _save_ you time studying and lead to better test scores, but this is just a recommendation. We will cover the problems before the exam.

Lecture 29: Note: I'll take up the sample MT questions on Monday, the day before the exam, in keeping with our tradition. Today I'll continue our discussion of chapter 7. At the beginning of the class, as a result of some student requests, we took a brief review of eigenvalues and eigenvectors and the relations to diagonalization. I know there are technicalities in this topic (which can be worked out with some practice problems), but the main thread of ideas (at a "high level") are

(1) Fix \lambda, and solve Av=\lambda*v. If there is a _nontrivial_ solution then that's an eigenvector associated to \lambda. Finding all eigenvalues and eigenvectors amounts to determining for which \lambda Av=\lambda*v has a nontrivial solution. From the theory of homogeneous linear equations, that amounts to computing the characteristic equation det(A-\lambda I)=0.

(2) Two matrices are _similar_, "A~B" if A=P^{-1}BP for an _invertible_ matrix P. If moveover A is diagonal, then we say B is _diagonalizable_.

(3) Relation between (1) and (2): if the eigenvectors v_1,v_1,...,v_n of B are linearly independent, and associated to eigenvalues \lambda_1,...,\lambda_n, then B is diagonalizable in the sense of (2). Moreover P=[v_1 v_2 ... v_n] and A is the diagonal matrix with \lambda_1, \lambda_2,...,\lambda_n stuck on the diagonal (and zero elsewhere).

Excellent question from class: don't just compute the eigenvalues of B and say that B diagonalizes to the matrix A from (3)! You need to establish that B does actually diagonalize by checking the eigenvectors are linearly independent. One exception: _if the eigenvalues are all different_, then indeed you can immediately deduce B diagonalizes to that A.

We also discussed more about what makes an argument clear. The main point was that an orderly argument helps the grader (and more importantly, you) find mistakes in your thinking.

One other thing we talked about was the issue of "interpreting a theorem". In this case it was the result that linear combinations of solutions to a system X'=P(t)X are solutions. There's what this theorem _literally_ says: if X_1,...,X_n are solutions, then so is c_1 X_1+...+c_n X_n. But in terms of the theory of vector spaces, it says "the set of solutions of X'=P(t)X is a subspace of the vector space V^{(n)}={(f_1,...,f_n): where each f_i is a continuous function}.

I care that you be able to understand how to interpret these theorems because I realize there's a lot to learn in this course, and this is typically how one can save time in learning the concepts. Think of this as a mnemonic for this topic.

I also emphasized that this midterm will necessitate facility with matrix algebra, and that understanding the spring-mass system derivations will be important.

One thing I should emphasize, which has come up in discussions with students, is that the grading scale is set relative to the performance of your sections. My expectation is that a score at the 50% percentile (i.e., half the class does better than you, half, not as well) is at least a B-, which is the standard scale.

Hints for the sample MT problems: (Posted 11/23/07) I want to give some suggestions for the problems, in case you get stuck. As time progresses, I'll update these to be more specific, until Monday, when I'll take them up all together.

Section 5.3 Q19: This is much like a problem we did in class. Try factoring the characteristic equation (a cubic) to find the roots. [Update: The characteristic equation is (r+1)^2(r-1)=0, with roots (r=-1, r=-1, r=1), apply Theorem 2, pg 313.]

Section 5.3 Q31: Use the factor theorem from class, try the divisors of the constant term 8. [Update: characteristic equation is (r-1)(r^2+4r+8)=0; this has complex roots, use Thm 3, pg 316.]

Section 5.3 Q49: This is essentially a more technically demanding version of the previous two problems. Same ideas, find the roots of the characteristic equation. I put this on here as a way to have you check you skills. If you can do this, all other problems of this form will be a piece of cake. If you get stuck, try related and simpler problems in the section. [Update: the characteristic equation factors as (r+1)(r-2)(r^2+1).]

Section 5.4 Q4: Study Figure 5.4.1 and the accompanying analysis. Don't try to do everything at once. Can you determine the effect of Hooke's law? How about what Newton's 2nd law says. Then try to piece them together. [Update: Ignore the requirement that it be put it in the form x(t)=Ccos(w_0 t -a), and just set up the second order linear DE. See example 1, page 325 to see how Hooke's constant k is determined. To put it in the final form, review pg 324. But don't worry: _if_ I ask you to put it in this form, I'll remind you of the needed formulas.]

Section 5.5 Q37: Follow the method from class: determine the y_c solution to the associated homogeneous DE, then find y_p, a pariticular solution, using Rule 1 or 2. y=y_c+y_p; Finish it with the initial conditions. [typo correction: I wrote Section 5.4 before, but hopefully the hint clarified that I meant section 5.5. This problem is standard].

Section 5.5 Q47: I'll admit we didn't talk about this one much in class. Use the formula on page 345. [Update: I won't require you to remember the formula for the test.]

Section 5.6: I'm mostly interested that you understand the derivation in class. Test your understanding with Q20, 22. Again, try to piece together the DE by examining separately the effects of forces involved first. [Update: for Q20 compute omega=(k/m)^.5 which is by definition the natural frequency. Do this by determining the force exerted by gravity on the spring. You may skip Q22.]

Section 6.1: Q21 is standard. Q36, Q37: can you prove it for 2x2, 3x3 matrices first? Then generalize your argument. [For Q36, compute the characteristic polynomial -- it's particularly simple here. For Q37, \lambda=0 picks off c_0.]

Section 6.2: Q23 is standard. Q31: demands that you know how to work with the definition of similar. Write down what it means for A and B to be similar, then manipulate using matrix algebra. Q34: apply the diagonalization procedure to the 2x2 matrix; except instead of having numbers to work with, you have expressions in a,b,c,d. Q36: isn't that bad! (I emphasize this to help you: often students just "psych" themselves out and this a problem is hard -- I know this problem can be solved by each and every one of you). Write down what it means for matrix A to be similar to D, and for matrix B to be similar to D, where D is diagonal. Now manipulate using matrix algebra. Q37: as usual, try to prove for special cases (2x2, 3x3), where you can control the analysis more easily. Then ask how the argument might extend. [Update: the main point is that you must know how to work with the notion of similar, for each of these problems.]

Section 7.1: Q7 is standard, use the method we discussed in class. Q24 is important: as before, break down the problem into pieces. What is the effect of the forces on the first block, the second block, and the third block, separately. Then combine. [Update: This is a variation from what we did in class. I'll explain in class Monday.]

Lecture 30: Took up the sample MT. I de-emphasized the importance of the problem on "variation of parameters" and Section 5.6, Q20, 22.

Lecture 31: Covered section 7.3 and most of 7.4 of the textbook. The solutions/grading guide for Midterm 3 may be found here. . Note concerning Q3 of the Midterm. Some students have wondered why it is not correct to multiply all of y_p by x^2. In class we did a similar problem (Section 5.5 Q37) where we applied "Rule 2" of page 341, and it happened to be the case that one multiplies all of y_p by the "correcting factor". That's because in the latter question, each term of the original trial solution duplicated a term in the complementary solution. In the Midterm question, this is not the case. However, both in class and in the solution to Q3 given above, we use "Rule 2" correctly. Analogy: sometimes when you go to a restaurant you get a free side of wings with your dinner. That's because you went to a certain restaurant with a special deal. But you can't expect to always get free wings. So you need to follow the "Rule" which is that you pay the price on the menu (even if the "price" might say "free" for some items).

Not sure what to ask? Graders see a lot of kinds of mistakes. Ask questions like "What might be some common misconceptions in this technique?" is one way to get the ball rolling. Also, you can ask "Do you think this is a key problem to study? If not, what might be a related, but better one?"

Sample problems from Sections 7.2-10.3: Section 7.2 Q13; 7.3 Q18; 7.4 Can you do things like example 1?; 10.1: 17; 10.2: Q9, 21. Combine these with sample problems from MT1, MT2 and MT3, as well as the problems from those midterms themselves and you'll have a reasonable idea of the areas and kinds of problems I'm interested in for the final.

Curve for midterm 3: 43-50 A [10 students], 38-42 A- [14 students], 35-37 B+ [12 students], 32-34 B [13 students], 28-31 B- [19 students], 26-27 C+ [12 students], 20-25 C [19 students], 16-20 D [10 students], 0-15 F [7 students].

68 students (>50 percent of the class) scored B- or above. Reminder: this curve is intended merely to give you a rough idea of your performance. Your final grade will be computed using your total raw score in comparison to your section. Please submit any regrading requests to me, with a note describing the issue (this includes any issues with question 3, which I will be happy to study).

Lecture 33: We'll continue with 10.1 and 10.2. Note: 10.3 will _not_ be on the final. The room assignments for the exam may be found here. (Look under final exams.) Sections 021, 022, 023 =AmundHB75, Section 24=AmundH 240. Friday's class will be given by professor Reza Pakzad.

We did some more examples of computing Laplace and inverse Laplace transforms. We also went over manipulations using the Gamma function. Be sure to know the Laplace transforms (and their inverses) of elementary functions.

We also did some more exam review. In particular, we went over MT Q3. In particular, it is very similar to a problem from class.

I discussed another review issue with a student after class. The question was dealing with the case of complex eigenvalues when solving first order systems of linear DE's. This is a subtle point that needs to be explained again: Given x'=Ax (x being an n-vector, A=real nxn matrix), one computes the eigenvalues \lambda and eigenvectors v=n-vector. Now a solution is x(t)=ve^(\lambda t)=v(cos(\lambda t) +isin(\lambda t)) . But if \lambda is complex (and v is complex) we need to take the real and imaginary parts. This demands some technique in that you need to do some FOIL between cos(\lambda t) +isin(\lambda t) and the components of v. If you know what I mean, then go no further. Otherwise please see me or the TA's.

Lecture 34: Professor Reza Pakzad covered section 10.2, up to Theorem 2 in the textbook.

Lecture 35: Review session. Here is an old exam (from a different instructor). Some of the material covered that semester may differ from that covered this semester.