Homework Assignments
MAP 2302, Section 3228—Elementary Differential Equations
Fall 2010

Homework problems and due dates (not the dates the problems are assigned) are listed below. This list, especially the due dates, will be updated frequently, usually in the late afternoon or evening the day of class or the next morning. Due dates and assignments more than one lecture ahead are estimates; in particular, due dates may be moved either forward or back, and problems not currently on the list from a given may be added later (but prior to their due dates, of course). On a given day there may be problems due from more than one section of the book.

Exam-dates and some miscellaneous items may also appear below.

If one day's assignment seems lighter than average, it's a good idea to read ahead and start doing the next assignment, which may be longer than average.

Unless otherwise indicated, problems are from our textbook (Nagle, Saff, & Snider, Fundamentals of Differential Equations and Boundary Value Problems, 5th edition). It is intentional that some of the problems assigned do not have answers in the back of the book or solutions in a manual. An important part of learning mathematics is learning how to figure out by yourself whether your answers are correct.

Read the corresponding section of the book before working the problems. Don't just read the examples, and don't just try the homework problems and refer to the text only if you get stuck.

Date due	Section # / problem #'s
W 8/25/10	Read the syllabus and the web handouts "Taking notes in a college math class" and "What is a solution?". Read Section 1.1 and do problems 1.1/ 1-16.
F 8/27/10	1.2/ 1, 3-5, 14, 15, 19-22. "Explicit solution" is synonymous with "solution". I will say more about the terminology "explicit solution" and "implicit solution" (which we have not used yet in class) at a later time. Do non-book problem #1.
M 8/30/10	Read section 2 (pp. 2-19) of these notes. 1.2/ 2, 10, 11, 30 Do non-book problem #2.
W 9/1/10	1.2/ 18, 23-29. (These should have been part of the previous assignment.) 2.2/ 1-5, 8, 9, 11, 13, 17-19, 23. "Solve the equation" means "Find the general solution of the equation". Read section 2.3 (pp. 19-28) of the notes.
F 9/3/10	2.2/ 27abc, 28-31, 33, 34. Do non-book problems 3–6. In case you've already printed these out, there was a minor typo in the answer for the domain in 5c (an omitted parenthesis) that has been corrected. Reminder: reading the notes I've written for the class is not optional (except for sections that I say you may skip, and the footnotes that say "Note to instructor"). I expect each reading assignment to be completed by the due date I give you. I've taught this course more than twenty times, and what I'm putting in the notes are things that are not adequately covered in our textbook (or any modern textbook that I know of). There is not enough time to cover most of these in class; we would not get through all the topics we're supposed to cover.
General comment	In doing the exercises from Section 2.2 or the non-book problems 3–6, you may have noticed that, often, the hardest part was doing the integrals. I intentionally assign problems that require you to refresh your basic integration techniques. Remember my warning from the first day of class (which is also on the class home page): You will need a good working knowledge of Calculus 1 and 2. In particular, you will be expected to know integration techniques ... .If you are weak in any of these areas, or it's been a while since you took calculus, you will need to spend extra time reviewing or relearning that material. Mistakes in prerequisite material will be graded harshly on exams. Your integration skills need to be sufficient for you to get the right answers to problems such as the ones in the homework assignments above, not merely to go through the motions. One type of mistake I penalize heavily is mis-remembering the derivatives of common functions. For example, expect to lose A LOT of credit on an exam problem if you write ``∫ lnx dx=1/x'', or ``(d/dx)(1/x)= lnx'', even if the rest of your work is correct. (1/x is the derivative of lnx, not an antiderivative; lnx is an antiderivative of 1/x, not its derivative.) This does not mean you should study integration techniques to the exclusion of material you otherwise would have studied to do your homework or prepare for exams. You need both to review the old (if it's not fresh in your mind) and learn the new, even if this takes a lot of time.
W 9/8/10	Read pp. 29-33 of the notes. You do not have to read Example 2.27 yet; that's part of your next assignment. Also read section 2.5 of the notes, pp. 38-43. I've made some minor wording changes and corrected some typos in the earlier part of the notes. If there is something you read that confused you, or that you thought was a mistake, see whether that's something that I've changed.
F 9/10/10	2.3/ 1-6, 7-9, 13-15, 17-20, 33. When you apply the method we learned today (which is in the box on p. 51) don't forget the first step: writing the equation in "standard linear form". Be especially careful to identify the function P correctly; its sign is very important. For example, in 2.3/17, P(x)=–1/x, not just 1/x. One of the things you'll see in #33 is that what you might think is only a minor difference between the DE's in parts (a) and (b)--a sign-change in just one term--drastically changes the nature of the solutions. When solving differential equations, a tiny algebra slip can make your answers utter garbage. For this reason there is usually no such thing as a "minor algebra error" in solving differential equations. Read the remainder of Section 2.4 of the notes (pp. 33-38).
M 9/13/10	2.3/ 22,23 (re-read Example 2 on p. 52 first), 24, 27a, 28, 30-32, 35. Do non-book problem #7. Read Section 2.4 of the textbook. Read Section 2.5.1 of the notes (pp. 43-50). Optional: Read Section 2.5.2 of the notes (pp. 50-53) and/or Section 2.5.3 (pp. 53-55). Update: A small bit was added to the end of Section 2.5.3 Sunday night, 9/12/10, 11:19 p.m.
W 9/15/10	Read Section 2.5.4 of the notes (pp. 56-67). Update, Tues. 9/14, 11:15 p.m. : I rewrote this section extensively on Tuesday, after posting it on Monday. (Sorry!) The major changes since Monday's post are (i) I modified Definition 2.48 to make it more encompassing, so that I would not have to give you yet another definition and more terminology for cases it previously didn't cover. (ii) This change allowed me to shorten and simplify Examples 2.51 and 2.53. (iii) Some of what was formerly in Example 2.51 is now in Example 2.54 (with a little more explanation). Certain important material that had been in Example 2.51 Monday night is temporarily gone; it will reappear in the next section of the notes (not yet ready for posting). (iv) I've added several more examples. In addition to the changes above, I added a paragraph on p. 45; it's the second paragraph after Definition 2.41. 2.4/ 1-8. Classify an equation in differential form as linear if at least one of the related derivative-form equations (the ones you get by formally dividing through by dx and dy, as if they were numbers) is linear. It is possible for one of these derivative-form equations to be linear while the other is nonlinear. This happens in exercises 1,2,5,6 and 7. For example, #5 is linear as an equation for x(y), but not as an equation for y(x). 2.2 (not 2.4)/ 6, 16. An equation in differential form is called separable if, by the operations of addition/subtraction of differentials, and multiplication by functions (other than the constant function 0), you can arrive at an equation of the form g(x)dx=h(y)dy (assuming the variables are x and y). This is equivalent to the condition that the derivative-form equation obtained by formally dividing the original equation by dx or dy is separable. 2.2/ 22. Note that although the differential equation doesn't specify independent and dependent variables, the initial condition does. Thus your goal in #22 is to produce an explicit solution "y(x)= ...". But this exercise is an example of what I call a "schizophrenic" IVP. In practice, if you are interested in solutions with independent variable x and dependent variable y (which is what an initial condition of the form ``y(x0)=y0'' indicates), then the differential equation you're interested in at the start is one in derivative form (which in exercise 22 would be x2 +2y dy/dx=0, or an algebraically equivalent version), not one in differential form. Putting the DE into differential form may be a useful intermediate step for solving such a problem, but differential form is not the natural starting point. On the other hand, if what you are interested in from the start is a solution to a differential-form DE, then you should not express a preference for one variable over the other by asking for a solution that satisfies a condition of the form ``y(x0)=y0'' or ``x(y0)=x0''. Instead, you should ask for a solution whose graph passes through the point (x0, y0), which in exercise 22 would be the point (0,2). If you feel well-enough prepared from reading the book's section 2.4, start on the exercises that are due Friday, so that you don't have all of them to do at once.
F 9/17/10	2.4/11, 12, 14, 16, 17, 19, 20-22, 27a, 28a, 32, 33ab (note that #22 is the same DE as #16, so you don't have to solve a new DE; you just have to incorporate the initial condition into your old solution). Note that exercises 21-26 are "schizophrenic IVPs". In all of these, the goal would be to find an explicit formula for a solution—if algebraically possible—with the choice of independent/dependent variables indicated by the initial condition. However, if the equation ``F(variable1, variable2)=0'' that you get via the exact-equation method (in these exercises) is impossible to solve for the dependent variable in terms of the independent variable, you have to settle for an implicit solution. Do non-book problems 8 and 9. Note: problem 9 has parts (a) through (j).
M 9/20/10	Read the online handout A terrible way to solve exact equations. Where it says "(we proved it!)" substitute "see your textbook for a proof", since I did not go through the proof in class. Read The Math Commandments. Update: no new notes have been posted since last week, and none will be before Wednesday's exam. Further update: Ignore the last 2.5 pages of the currently posted version of the notes, from "Now let us turn ..." on p. 73 to the end. They did not accomplish what I wanted them to, and there are mistakes. My current (private) version of the notes has fixed most of that, but I'm not posting that version yet because it also includes extensive revisions of earlier parts of the notes, as well as additions the later parts. It would not have been fair to give you all that to read with so little time before the exam. For students who want a supply of exercises to practice with: If you have done all your homework (and I don't mean "almost all"), you should be able to do all the review problems on pp. 81-82 except #s 9, 11, 12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35, 37, and the last part of 41. Reminder: the syllabus says, "Unless I say otherwise, you are responsible for knowing any material I cover in class, any subject covered in homework, and all the material in the textbook chapters we are studying." I have not said otherwise, the homework has included reading the notes I've posted, and the textbook chapters/sections we've covered are 1.1, 1.2, 2.2, 2.3, and 2.4.
W 9/22/10	First midterm exam (assignment is to study for it) . If you want to see how the class that took your "practice exam" as a real exam, go here and here.
F 9/24/10	Read section 4.1. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)
M 9/27/10	4.1/ 1-10. Typo correction: In #10a, 2nd line, and in #10d, 2nd line, the function written after "B" should be "sin", not "cos".
W 9/29/10	4.2/ 1,3,4,6-9,11,12,13-17,21-25
General info	The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution. Exams will be returned in class on Wednesday 9/29/10. Students not in class that day should pick up their exams during one of my office hours as soon as possible. Points may be deducted from your exam score if you wait longer than a week, unless you make prior arrangements with me and have a valid excuse. After a week, I may toss into the recycling bin any exams that have not been picked up. Failure to read this notice on time will not count as a valid excuse, since you are expected to check this homework page at least three times a week to get the homework assignment that's due by the next class. No students received any points for the extra-credit problem on the exam. This problem was closely related to Example 2.31 in the current version of the notes (pp. 35-38), the reading of which was part of your homework. The answer involves two arbitrary constants, as in equation (78) of the notes. (Note: this version of the notes will not remain posted much longer; in the updated version, "Example 2.31" may have a different number and a different location, and "equation (78)" may have a different number.)
F 10/1/10	4.2/ 2,5,10,18-20,26,27-32,33,35,36,46ab. Note: The answers to #s 27 and 35a in the back of the book are wrong. 4.3/ 1,2,4,6,7,9-12,17,18 Re-do all the exam problems on which you did not get a perfect score. For all but #4b, you should be able to check by yourself whether your new answers are correct, but you are welcome to check your answers against my solutions during office hours. (As a matter of policy, I do not post exam-solutions on the internet or on my door.)
M 10/4/10	4.3/ 3,5,8,13-16,21-26,28,32,33 (students in electrical engineering may do #34 instead of #33),36
W 10/6/10	Read Section 4.4.
F 10/8/10	4.5 (not 4.4)/ 3-8. Use the "y=yp + yh" approach outlined at the end of Wednesday's class. We have not yet discussed in class how to find yp's yet. If you feel that you understand how to find at least some yp's based on your reading of Section 4.4 (the previous homework assignment), you should get started on the homework that's due Monday. Over the next few classes, I will be assigning almost all the exercises in sections 4.4 and 4.5.
M 10/11/10	4.4/ 9 (note that -9 = -9e0 t), 10,11,14,17 Add parts (b) and (c) to 4.4/ 9-11,14,17 as follows: (b) Find the general solution of the DE in each problem. (c) Find the solution of the initial-value problem for the DE in each problem, with the following initial conditions: In 9,10, and 14: y(0)=0=y'(0). In 11 and 17: y(0)=1, y'(0)=2. 4.5/ 25,26 Do non-book problems 10 and 11.
A glimpse into the future	According to Nostradamus: Fri. Oct. 22 is still the date for your next midterm. The extra-credit assignment that I've mentioned in class, to be worth 17 points (max) added to the score of your first midterm, will be a modified version of Project E, p. 88 (Clairaut Equations and Singular Solutions), in your textbook. I have not yet settled on just how I want to modify the book's instructions, and other logistical details such as what I will want you to hand in and when. But you will not be harmed by getting an early start on this project as it's written in the textbook. Oct. 31: Small humanoids will prowl the land in packs. The end of civilization as we know it?
W 10/13/10	4.4/ 12,13, 18, 23,25 4.5/ 1,2,17-22,23,24,27-30, 33,34,41 4.5/ 45. This is a nice problem that requires you to combine several things you've learned.     The strategy is similar to the approach in Exercise 41. Because of the "piecewise-defined" nature of the right-hand side of the DE, there is a sub-problem on each of three intervals: Ileft = (-∞, -L /(2V)], Imid = [-L /(2V), L /(2V)], and Iright = [-L /(2V), ∞). The solution y(t) defined on the whole real line restricts to solutions yleft, ymid, and yright on these intervals.     You are given that yleft(t) is identically zero. Use the terminal values yleft(-L /(2V)), y'left(-L /(2V)), as the initial values ymid(-L /(2V)), y'mid(-L /(2V)). You then have an IVP to solve on Imid. For this, first find a "particular" solution on this interval using the Method of Undetermined Coefficients (MUC). Then, use this to obtain the general solution of the DE on this interval; this will involve constants c1, c2. Using the IC's at t=-L /(2V), you obtain specific values for c1 and c2, and plugging these back into the general solution gives you the solution ymid of the relevant IVP on Imid.     Now compute the terminal values ymid(L /(2V)), y'mid(L /(2V)), and use them as the initial values yright(L /(2V)), y'right(L /(2V)). You then have a new IVP to solve on Iright. The function yright is what you're looking for.     If you do everything correctly (which may involve some trig identities, depending on how you do certain steps), under the simplifying assumptions m = k = 1 and L = π, you will end up with just what the book says: yright(t)=A sin(t), where A=A(V) is a V-dependent constant (i.e. constant as far as t is concerned, but a function of V). In part (b) of the problem you are interested in the function | A(V) |, which you may use a graphing calculator or computer to plot. The graph is very interesting.     Note: When using MUC to find a particular solution on Imid, you have to handle the cases V≠ 1 and V = 1 separately. (If we were not making the simplifying assumptions m = k = 1 and L = π, these two cases would be πV/L ≠ (k/m)1/2 and πV/L = (k/m)1/2.) In the notation used in the last couple of lectures, using s for the multiplicity of a certain number as a root of the characteristic polynomial, V≠ 1 puts you in the s = 0 case, while V = 1 puts you in the s = 1 case.     The book's answer to part (a) is correct only for V ≠ 1. The value of A for V=1 is π/2. (I'm just telling you this so you can check your answer; you're still supposed to figure out this answer yourself.)
M 10/18/10	4.4/ 1-8,15,16,19-22,24,26, 27-32 4.5/ 9-16,31,32,35,36,42,46. Problem 42b, if done correctly, shows that the particular solution of the DE in part (a) produced by the Method of Undetermined Coefficients actually has physical significance.
W 10/20/10	4.7/ 9-20, 24ab. (We will cover section 4.6 after the midterm. The portion of Section 4.7 that is fair game for Friday's exam is only the part up to and including Example 3, p. 210.) Note on terminology. I've said in class that "characteristic equation" and "characteristic polynomial" are things that exist only for constant-coefficient DEs. The term I used in class for Equation (7) on p. 209, "indicial equation", is what's used for Cauchy-Euler DEs in most textbooks I've seen. What we saw in class Monday is that the indicial equation for the Cauchy-Euler DE is the same as the characteristic equation for the constant-coefficient DE obtained by the Cauchy-Euler substitution t=ex. In my experience it's unusual to hybridize the terminology and call the book's Equation (7) the characteristic equation for the Cauchy-Euler DE, but I won't consider it a mistake for you to use the book's terminology for that equation. Non-book problem 14. In part (b), "the method of 4.7/ 23,24" just means the method I used in class.
F 10/22/10	Second midterm exam (assignment is to study for it)
M 10/25/10	No homework assigned.
W 10/27/10	Non-book problems 12 and 13. #13 has just been revised, so if you have printed it out before, you will need to re-print it. Read Sections 6.1 and 6.2.
F 10/29/10	6.2/ 9,11,13. The characteristic polyomial for #9 is a perfect cube (i.e.(r-r1)3 for some r1); for #11 it's a perfect fourth power. #13 is of the form of the last DE that I put on the board in class Wednesday.
M 11/1/10	6.2/ 1,14,15-18 Read section 6.3 Read section 4.6
Extra Credit Project	Instructions for your extra-credit project are now posted.
Spooky info	The grade scale for the second midterm has eerily appeared on your grade-scale page, with a link to the list of scores so that you may see the grade distribution. Exams will be returned to the living on Monday 11/1/10. Students not in class that day should pick up their exams during one of my office hours as soon as possible. Points may be deducted from your exam score if you wait longer than a week, unless you make prior arrangements with me and have a valid excuse. After a week, I may toss into the recycling bin any exams that have not been picked up. Failure to read this notice on time will not count as a valid excuse, since you are expected to check this homework page at least three times a week to get the homework assignment that's due by the next class.
W 11/3/10	4.6/ 1, 3-18, 19 (first sentence only) 4.7/ 24cd, 37-44. (In all of these, assume the domain interval is {t > 0}.) Remember that to apply Variation of Parameters as presented in class, you must first put the DE in "standard form", with the coefficient of the second-derivative term being 1 (so divide by the coefficient of this term, if the coefficient isn't 1 to begin with).     Note that it is possible to solve all the DEs in 24cd and 37-43 either by the Cauchy-Euler substitution applied to the inhomogeneous DE, or by using Cauchy-Euler just to find a FSS for the associated homogeneous equation, and then using Variation of Parameters for the inhomogeneous DE. Both methods work. I've deliberately assigned exercises that have you solving some of these equations by one method and some by the other, so that you get used to both approaches. Redo 4.7/ 44 by starting with the substitution y(t)=t–1/2u(t) and seeing where that takes you.
F 11/5/10	Read sections 7.1 and 7.2.
M 11/8/10	4.7/ 25,26,27. In #27e, "Counterexample to the theory in this section" means "Counterexample to Theorem 6, p. 211". Read the review of partial fractions on pp. 396-400. We will be using this material soon, and there will be no time to review it in class (and you will have plenty of other homework to do). For now, you need not worry about what "(curly L)-1[F]" means in Examples 5, 6, and 7; just go over the parts of these examples that involve finding partial-fractions decompositions.
W 11/10/10	7.2/ 1-8, 10, 12 (note: "Use Definition 1" in the instructions for 1-12 means "Use Definition 1", NOT the box on p. 384 or any other table of Laplace Transforms), 13-20 (for these, use the table on p. 384; we'll finish deriving the entries in this table on Wed.), 21-28, 29a-d,f,g,j, 30
General info	New date for third midterm exam: Friday, Nov. 19.
F 11/12/10	7.3/ 1-10,12,14,19,25,31. In class, we have not finished deriving Table 7.2 yet, but I want you to start getting practice using all the properties listed in the table. Read Section 7.4.
M 11/15/10	7.4/ 1-10,11,13,15,16,20,21-24,26,27,31 Read Section 7.5 through Example 3. 7.5/ 15,21,22
W 11/17/10	Note: You will be given a Laplace Transform table to use on your exam; you don't have to memorize any transforms or inverse transforms. (But you'll be expected to know how to use the information in the table.) The table you'll be given was photocopied from the inside back cover of your textbook, with lines 27, 28, and 33-37 whited-out. I suggest that when you do the exercises below, and other practice problems for the exam, you start using this table instead of the smaller ones in Chapter 7, so that when you're taking your exam you'll know where on the table to find the formulas you want. You will be allowed to use only those lines on the table that we've derived in class, plus line 5 (whose derivation I had you read in the book, but didn't have time to do in class). In particular, it's very important that you not use line 8 of the table. The symbol "*" does not mean multiplication; it means something more complicated called convolution, which we haven't covered yet and probably won't have time to cover. 7.5/1-8,10,29. To learn some shortcuts, you may want first to read the web handout "Partial fractions and Laplace Transform problems" (pdf file).
F 11/19/10	Third midterm exam (assignment is to study for it). Fair-game material for this exam includes everything we've covered since the last exam (where "covered" includes classwork and homework, and "homework" includes reading the relevant portions of the textbook), with the following exceptions regarding Chapter 6: In Section 6.1, you are not responsible for knowing what the Wronskian is except when n=2. In Theorem 2, p. 345, replace the condition involving the Wronskian with "If the solutions {y1, ..., yn} are linearly independent on (a,b)." (Linear independence is defined on p. 346.) In Section 6.3, you are not responsible for the terminology "annihilate" or "annihilator". What you are responsible for is what I showed in class: the way that knowing FSS's of homogeneous higher-order constant-coefficient linear DEs allows us to see why the MUC for non-homogeneous higher-order constant-coefficient linear DEs works. Regarding Section 7.5, I will not ask you to solve any DEs using Laplace Transform on this exam. However, what I could do is give you inverse-Laplace-transform problems in which the function you have to inverse-transform is one that comes up when you do the already-assigned homework from Section 7.5 (or is very similar)—so it would be a bad idea to wait till after the exam to do those exercises. I also consider it fair game to give you problems of the form in 7.5/15-24 (three of which were assigned for homework). The problems from the practice exam that, as written, are most relevant to your third midterm are numbers 2, 4, and 5. (When I gave you the handout before the second midterm, I overlooked that #3 pertained to material I'd already covered in your class.) However, you can still use #3 as practice for your midterm, by treating it the same way as HW problem 7.5/29 (due Wednesday Nov. 17). Then #3 becomes a pair of problems, the first of which is of the same type as 7.5/15-24, and the second of which is of the type, "Find the inverse Laplace transform of the following function." Both of these types of problems are fair game for your midterm.
M 11/22/10	No new homework. The grade scale for the second midterm is now posted on your grade-scale page, along with the usual link to the list of scores.
W 11/24/10	Read Section 7.6 through Example 5. 7.6/ 1-4. Note that there will be quite a bit of homework due the Monday after Thanksgiving; you may want to get started on this early, based on your reading. I'm sorry I have to give you work over Thanksgiving, but there are so few class-days left that I don't have much choice. I will be covering most of Section 7.6 in class on Wednesday. Missing Wednesday's class will put you at a disadvantage; this is the section of the Laplace Transform material that gives students the most trouble. FYI: On Friday, Provost Glover sent the following message to deans, who then forwarded it to department chairs, who then forwarded it to faculty: "It has come to our attention that numerous faculty are cancelling classes for Thanksgiving week. Would you please remind chairs and faculty that it is not their prerogative to do so. Thank you." The math department's chair added: "It is the policy of the Department of Mathematics that classes not be cancelled. The office will be opened Monday through Wednesday of next week. Classes scheduled for those days should be met." But for those of you who will still be here in 2012, take heart: the university is on the verge of making the Wednesday before Thanksgiving an official UF holiday, starting in 2012. The Faculty Senate will probably vote on this next month, and I expect that the proposal will pass overwhelmingly, even though I think it's a terrible idea. You can hear some of my objections, expressed not very eloquently, at the webcast of last Thursday's Faculty Senate meeting. My portion starts at about 1:58:20 into the meeting (although the camera doesn't pan to me till later).
M 11/29/10	7.6/3-10,11-18,29-32,33-40. For all the problems in which you solve an IVP, write the final answer in "tabular form". (For those of you who missed class the day before Thanksgiving, by "tabular form" I mean an expression like the one given in Example 1, equation (3), p. 411, for f(t). I.e. do not leave your final answer in the form that's at the end of Example 1, after you've inverse-transformed Y(s). The unit step-functions should be viewed as convenient gadgets to use in intermediate steps to work through a problem efficiently.) Read Section 8.1.
W 12/1/10	Read sections 8.2, 8.3, and 8.4. If you feel well-enough prepared just from your reading, get started on the homework due Friday. Note that there are already a lot of problems I'm planning to assigning for Friday and Monday, and this number could increase. There could also be problems due Wednesday of next week.
F 12/3/10	8.2/1-6,7,8,9,10,11-14,17-20,23,24,27,28,37. In 9-14, determine the open interval of convergence on which your formal manipulations are valid. (For power series, "formal manipulations" means "algebra/differentiation/integration done as if power series were polynomials". To use this principle to compute the product of two power series, you have to group terms intelligently, as at the bottom of p. 459. When you re-read p. 459, I suggest you first take x0 to be 0 in order to understand where the pattern of terms is coming from. If you still don't see where the pattern comes from on p. 459, first consider the case in which all the coefficients an,bn are 0 for n≥2; then the case in which all the coefficients are 0 for n≥3; then the case in which all the coefficients are 0 for n≥4, etc., until you see what's going on.) Note: "convergence set" in the book is what I initially called "domain of convergence" and, later, "interval of convergence". Any time the book's problems tell you to find the convergence set, find only the open interval of convergence; I don't want you to spend time trying to decide whether the series converges at the endpoints. For this class, 100% of the way we'll apply power-series ideas to solving DEs involves only the open interval of convergence.
M 12/6/10	8.3/1,3,5-10. Although I haven't yet defined "singular point" in class, you should be able to do these just from the definition and Example 1 on pp. 467-468. Based on your reading, try to start on the problems that are due Wednesday Dec. 8. Ideally, I would start covering Sections 8.3. and 8.4 in class before assigning exercises from them. But even though these exercises will become easier after I've gone over this material and done some examples in class, I think you'll have a better chance at mastering it if you start now, rather than trying to cram it all into your brain between Monday and Wednesday.
Review session for final	As announced in class, I will still be covering new material (hopefully, just doing examples) on Wednesday. I will hold a review session for you on Friday Dec. 10, 2:00-3:00, in our usual classroom. I'll also have my usual Friday office hour 3:00-4:00 that day. My usual office-hour schedule does not apply during finals week. Don't worry, I will have several office hours for you that week. I'll post them on this page when I've worked out my schedule.
W 12/8/10	Do your best to finish the problems below, based on reading the book. I will do as many examples on Wednesday as time allows. 8.3/ 11-14,18, 20-22,24,25,32,33,34. In #34 note that n is not a summation index; it has a different meaning in this problem. This problem can be done without using "Σ-notation"; you shouldn't need a summation index. 8.4/15, 20 ("Equation (16)" is the first un-numbered equation after Equation (15)),21,23,25, 29. In #29, just as in 8.3/34, be careful not to use the letter n as a summation index since it already has an assigned, different meaning in this problem).
Between now and final exam	Eat differential equations for breakfast, lunch, and dinner. Have a differential equations snack before you go to bed, and dream about differential equations. If you talk to anyone, talk only about differential equations. If you watch TV, watch only the Differential Equations Channel. Decide what differential equation you want to be when you grow up. Prove that if everyone in the world understood differential equations, there would be no war, no poverty, and no disease.
Final Exam Thurs. 12/16/10	The final exam will be given on Thursday, December 16, starting at 5:30 p.m., in our usual classroom.

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