Homework Assignments
MAP 2302, Section 5603—Elementary Differential Equations
Fall 2016

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 section may be added later (but prior to their due dates, of course). Note that 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 (if posted), which may be longer than average.

Unless otherwise indicated, problems are from our textbook (Nagle, Saff, & Snider, Fundamentals of Differential Equations, 8th 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. The advice below from James Stewart's calculus textbooks is right on the money:

Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises.

Date due	Section # / problem #'s
W 8/24/16	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 8/26/16	1.2/ 1-6, 9-12, 19-22. See Notes on some book problems. Do non-book problems 1 and 2. Note: Many of your homework assignments will be a lot more time-consuming than the ones I've given so far. I don't want anyone to feel after Drop/Add that he/she wasn't warned. Often, most of the book problems in a section aren't doable until we've finished covering practically the entire section, at which time I may give you a large batch to do all at once. Heed the suggestion above the assignment-chart: "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."
M 8/29/16	1.2/ 14-27, 29-30, 31bc
W 8/31/16	Read Section 1.3. 1.3/ 2,3 2.2/ 1-5, 8-11, 14, 17-19, 21, 24, 30. "Solve the equation" means "Find all solutions of the equation". Read Section 2.1 (pp. 2-4) and at least half of Section 2.2 (pp. 4-25) of these notes.
F 9/2/16 (effectively M 9/7/16 because of storm)	2.2/ 7, 12, 13, 27abc, 28, 29, 31 . See if you can figure out why I didn't include problems 7, 12, and 13 in the previous assignment. What feature do they have that the others in the 7–14 group don't have? Finish reading Section 2.2 of these notes. Do non-book problems 3–5 (updated 8/31/16). 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. What I'm putting in the notes are things that are not adequately covered in our textbook (or any current 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 3–4, you may have noticed that, often, the hardest part was doing the integrals. I intentionally assign problems that require you to refresh most of your basic integration techniques (not all of which are adequately refreshed by the book's problems). 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. I don't wan't you merely going through the motions of how you'd do these problems, either doing the integrals incorrectly or stopping when you reach an integral you don't remember how to do. Your integration skills need to good enough that you can get the right answers to problems such as the ones in the homework assignments above. 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.
F 9/9/16	Do non-book problem 6 (added to list on 9/7/16). Answers to some of the non-book problems are now posted here. In the book, read "Formal Justification of Method" on p. 42. Read at least the following parts of the notes: Beginning of Section 2.3 on p. 25, through the next-to-last paragraph. p. 28, starting with Example 2.23, through the end of Section 2.3 (p. 34). (Reading the in-between material on pp. 25–28 is optional.) Section 2.4 (pp. 28-37). Section 2.5, up through p. 40. Throughout my notes, skip anything that says "Note to instructors". They're not secrets; you just won't understand them, and some of them are very long.
M 9/12/16	2.3/ 1-6, 8, 12, 18, 28. Read Section 2.3 of the textbook. (You should be able to do the exercises above just based on our classwork up through Friday 9/9/16, however.) Read the remainder of Section 2.5 of the notes (pp. 41–46).
W 9/14/16	2.3/ 7, 9, 13-15 (note which variable is which in #13!), 17, 19, 20, 22, 33. When you apply the method we learned in Monday's class (which is in the box on p. 48, except that the book's "∫P(x) dx" is my ∫specP(x) dx), 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. Read the following parts of the notes: Section 2.6, through p. 50. Section 2.6.1, through the end of the first paragraph on p. 54. The rest of Section 2.6.1 is optional reading. If you don't read the rest of Section 2.6.1, ignore the word "inextendible" wherever it comes up later in the notes. Note: One of the things you'll see in exercise 2.3/33 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. This is a fact of life you'll have to get used to. The severity of a mistake is not determined by the number of pencil-strokes it would take to correct it, or whether your work was consistent after that mistake. If a mistake (even something as simple as a sign-mistake) leads to an answer that's garbage, or that in any other way is qualitatively very different from the correct answer, it's a very bad mistake, for which you can expect a significant penalty. A sign is the only difference between a rocket going up and a rocket going down. In real life, little details like that matter. I urge you to develop (if you haven't already) the mindset of "I really, really want to know whether my final answer is correct, without having to look in the back of the book, or ask my professor." Of course, you can find answers in the back of the book to many problems, and you are always welcome to ask me in office hours whether an answer of yours is correct, but that fact won't help you on an exam—or if you ever have to solve a differential equation in real life, not just in a class. Fortunately, DEs have built-in checks that allow you to figure out whether you've found solutions (though not always whether you've found all solutions). If you make doing these checks a matter of habit, you will get better and faster at doing the algebra and calculus involved in solving DEs. You will make fewer and fewer mistakes, and the ones that you do inevitably make—no matter how good you get, you'll still only be human—you will catch more consistently.
F 9/16/16	2.2/ 27a, 30-32, 35. Do non-book problem #7. Read at least the following parts of the notes: Section 2.6.2 (pp. 55–58). Section 2.6.3 is optional reading. If you do not read this section, then in the remainder of the notes ignore any reference to "singular points". Section 2.6.4 (pp. 60–63).
M 9/19/16	Read Section 2.4 of the textbook. 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)/ 5, 15, 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 h(y)dy=g(x)dx (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. Equations of the form h(y)dy=g(x)dx can be solved by integrating both sides. 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 it's illogical to 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''. What's logical to ask for is a solution whose graph passes through the point (x0, y0), which in exercise 22 would be the point (0,2). 2.4/ 9, 11–14. If you feel well-enough prepared from Friday's class and reading the book's section 2.4, you can also get head-start on the exercises that are due Wednesday. Read at least the following parts of the notes: Section 2.6.5 through Definition 2.61 (pp. 63–65). The remainder of Section 2.6.5 is optional reading. Section 2.7, minus Example 2.63 (this example occupies most of pp. 67–70). Do read Example 2.64 on p. 70. Example 2.63 is optional reading. Section 2.8 (starting on p. 70) through the middle of p. 71 (the end of the paragraph that has "Pretend" in boldface), then from the middle of p. 75 (paragraph beginning "To have a name ...") through the end on p. 76. The remainder of Section 2.8 is optional reading.
W 9/21/16	2.4/ 16, 17, 19, 20, 21, 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. Read Section 2.9 of the notes, through Example 2.67 on pp. 79–80. Also read Example 2.69 (pp. 87–88).
F 9/23/16	Do non-book problem 8. Read the online handout A terrible way to solve exact equations. The example in this recently-revised handout is rather complicated; feel free to read the example in the original version instead. The only problem with the example in the original version is that ∫ sin(x) cos(x) dx can be done three ways (yielding three different antiderivatives, each differing from the others by a constant), one of which happens to lead to the correct final answer even with the "terrible method". Of course, if the terrible method were valid, then it would work with any valid choice of antiderivative. However, I've had a few students who were unconvinced by this, and thought that because they saw a way to get the terrible method to work in this example, they'd be able to do it in any example. I've constructed the more complicated example to make the failure of the terrible method more obvious. Read The Math Commandments. Optional: If you are interested in seeing a differential that passes the exactness test My=Nx on a region R but is not exact on R, do non-book problem 9. This example does not contradict anything we've learned, because the region R has a hole (so, in particular, it's not a rectangle).
M 9/26/16	First midterm exam (assignment is to study for it). 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 p. 77 except #s 9, 11, 12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35, 37, and the last part of 41. A good feature of the book's "review problems" is that, unlike the exercises after each section, the location gives you no clue as to what method(s) is/are likely to work. Your exam will have no such clues on exams either.     A negative feature of the book's exercises (including the review problems) is that they don't give you enough practice with a few important integration skills. This is why I assigned my non-book problems 3, 4, and 7. 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 (just the portions I've assigned), and the textbook chapters/sections we've covered are 1.1, 1.2, 2.2, 2.3, and 2.4. If you want to see how the class that took your "practice exam" as a real exam did, go here and here.
W 9/28/16	Read section 4.1. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)
F 9/30/16	Read Section 4.2. Unfortunately, hardly any of the exercises are doable until the whole section has been covered. If you feel ready based on your reading, start on the exercises that are due Monday. There are a lot of them. 4.7 (yes, 4.7)/ 30. No reading is necessary for this one.
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 at the end of class on Friday 9/30/16. Students not in class that day should pick up their exams during one of my office hours as soon as possible. After a week, I may toss into the recycling bin any exams that have not been picked up, unless you make prior arrangements with me and have a valid excuse. 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.
M 10/3/16	4.2/ 2–5, 7, 8, 10–17, 26, 27-32, 35, 46ab
W 10/5/16	4.2/ 1, 6, 9, 18–20. 4.7/ 1–8, 25, 26. First read from the top of p. 194 through Example 1 on that page. I've stated and used Theorem 5 in class several times recently, so now is a good time for you to do these problems. In problems 1–4, interpret the instructions as meaning: "State the largest interval on which Theorem 5 guarantees existence and uniqueness of a solution to the differential equation that satisfies [the given initial conditions]." Read Section 4.3.
M 10/10/16 (postponed from F 10/7/16 because of storm)	4.3/ 1–18, 21–26. For purposes of getting these done by Monday, you may assume that, for the complex-conjugate-root case (for a 2nd-order, linear, constant-coefficient, homogeneous DE), the set of functions I said was a FSS, is indeed a FSS. (This fact is effectively what's in the box at the bottom of p. 169, in different words.) The complex exponential function is not needed for these problems; complex exponentials will simply be our tool for deriving the fact that the functions in that box on p. 169 are solutions of the relevant DE.
W 10/12/16	4.3/ 28, 30, 32, 33 (students in electrical engineering may do #34 instead of #33). 4.7/ 30. 4.5 (not 4.4)/ 1–8. Use the "y=yp + yh" approach discussed in class. We have not yet discussed in class how to find yp's. Start reading Section 4.4.
M 10/17/16	4.4/ 9 (note that \(-9=-9e^{0t}\) ), 10,11, 13, 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/ 20, 25, 26.
W 10/19/16	4.4/ 18, 23 4.5/ 1, 2, 17, 18, 20, 22, 24–30, 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-expressed" nature of the right-hand side of the DE, there is a sub-problem on each of three intervals: \(I_{\rm left}= (-\infty, -\frac{L}{2V}\,] \), \(I_{\rm mid} = [-\frac{L}{2V}, \frac{L}{2V}] \), \(I_{\rm right}= [\frac{L}{2V}, \infty) \). The solution \(y(t)\) defined on the whole real line restricts to solutions \(y_{\rm left}, y_{\rm mid}, y_{\rm right}\) on these intervals.     You are given that \(y_{\rm left}\) is identically zero. Use the terminal values \(y_{\rm left}(- \frac{L}{2V}), {y_{\rm left}}'(- \frac{L}{2V})\), as the initial values \(y_{\rm mid}(- \frac{L}{2V}), {y_{\rm mid}}'(- \frac{L}{2V})\). You then have an IVP to solve on \(I_{\rm mid}\). 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 \( c_1, c_2\). Using the IC's at \(t=- \frac{L}{2V}\), you obtain specific values for \(c_1\) and \(c_2\), and plugging these back into the general solution gives you the solution \(y_{\rm mid}\) of the relevant IVP on \(I_{\rm mid}\).     Now compute the terminal values \(y_{\rm mid}(\frac{L}{2V}), {y_{\rm mid}}'(\frac{L}{2V})\), and use them as the initial values \(y_{\rm right}(\frac{L}{2V}), {y_{\rm right}}'(\frac{L}{2V})\). You then have a new IVP to solve on \(I_{\rm right}\). The solution, \(y_{\rm right}\), is what you're looking for in part (a) of the problem.     If you do everything correctly (which may involve some trig identities, depending on how you do certain steps), under the book's simplifying assumptions \(m=k=F_0=1\) and \(L=\pi\), you will end up with just what the book says: \(y_{\rm right}(t) = A\sin t\), where \(A=A(V)\) is a \(V\)-dependent constant (i.e. constant as far as \(t\) is concerned, but a function of the car's speed \(V\)). 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 \(I_{\rm mid}\), you have to handle the cases \(V\neq 1\) and \(V = 1\) separately. (If we were not making the simplifying assumptions \(m = k = 1\) and \(L=\pi\), these two cases would be \(\frac{\pi V}{L}\neq \sqrt{\frac{k}{m}}\) and \(\frac{\pi V}{L}= \sqrt{\frac{k}{m}}\), respectively.) 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, \(V\neq 1\) puts you in the \(s= 0\) case, while \(V = 1\) puts you in the \(s= 1\) case.
F 10/21/16	4.4/ 1–8,15,16,19–22,24–26, 27–32. In the instructions for 27–32, the word "form" should be replaced by "MUC form". This also applies to 4.5/ 31–36 below. 4.5/ 9–16 (read instructions carefully or you'll do a ton of extra work!), 19,21, 31–36, 42. 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.
M 10/24/16	4.4/ 12, 33–36. Note that #12 can also be done by Chapter 2 methods. The purpose of this exercise here is to see that it also can be done using the Method of Undetermined Coefficients. 4.5/ 23 (same comment as for 4.4/12 applies), 37–40. Do these non-book exercises on the Method of Undetermined Coefficients. The answers to these exercises are here.
W 10/26/16	Second midterm exam (assignment is to study for it). For students who want an extra supply of exercises to practice with: you should be able to do the review problems 1–36 on p. 233–234, except those problems in which the left-hand side of the DE does not have constant coefficients or has order greater than 2, and #28. (The latter would have non-constant coefficients on the left-hand side if rewritten in standard linear form). You should also be able to figure out how to solve the fourth-order DE in #18.
F 10/28/16	No new homework.
M 10/31/16	4.7/ 9–20. Prepare for small humanoids to prowl the land in packs.
W 11/2/16	4.6/ 2, 5–8, 9, 10, 11, 12, 15, 17, 19 (first sentence only) 4.7/ 41–43. Remember that to apply Variation of Parameters as presented in class, you must first put the DE in "standard linear 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). The book's approach to remembering this is to cast the two-equations-in-two-unknowns system as (9) on p. 190. This is fine, but my personal preference is to put the DE in standard form from the start, in which case the "\(a\)" in the book's pair-of-equations (9) disappears.
F 11/4/16	4.7/ 24, 37–40, 44. In #24, "the method of Problem 23" just means the change-of-variables method I presented in class (with my \(x\) and \(t\) being the reverse of what the textbook uses in this method for Cauchy-Euler equations).     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 \(y(t)=t^{1/2}u(t)\) and seeing where that takes you. Read Section 6.1.
M 11/7/16	6.1/ 1–6, 7–14 (do not use Wronskians), 19, 20, 23. Read Section 6.2.
W 11/9/16	4.7/ 27 (this was accidentally omitted from an earlier assignment). 6.2/ 1, 9, 11, 13, 15–18. The characteristic polynomial for #9 is a perfect cube (i.e. \( (r-r_1)^3\) for some \(r_1\)); for #11 it's a perfect fourth power. 6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the annihilator method (which we are skipping for reasons of time); just use what we've done in class with MUC and superposition.
M 11/14/16	Read sections 7.1 and 7.2. 7.2/ 1–4, 6–8, 10, 12 (note: "Use Definition 1" in the instructions for 1-12 means "Use Definition 1", NOT the box on p. 359 or any other table of Laplace Transforms), 13–20 (for these, do use Table 7.1 on p. 359; we'll derive most or all of this table in class), 21–23, 26–28, 29a–d,f,g,j. Find where the entries of Table 7.1 are located in the table on the first page of the book (right after the front cover). On your third midterm and on your final exam, you will be given a version of this longer table, so you'll want to familiarize yourself with where things are on it. The table you'll be given will be essentially this table with most or all of the entries beyond line 27 removed.
W 11/16/16	Read Section 7.3. For some of the problems below, you'll be using Theorem 6 on p. 364, which is summarized in the last line of Table 7.2 on p. 360. 7.3/ 1–10, 12–14, 20, 25, 31. 7.4/ 11, 13, 14, 16, 20. You should be able to do these with or without reading Section 7.4 first (recall the "Prerequisites" paragraph on the class home page), but there's additional review in Section 7.4 if you need it. Read Section 7.4.
General info	New date for third midterm exam: Monday, Nov. 28.
F 11/18/16	7.4/ 1–10, 21–24, 26, 27, 31. Read Section 7.5 through Example 3. 7.5/ 15, 17, 18, 21, 22. Note that in these problems, you're being asked only to find \(Y(s)\), not \(y(t)\).
M 11/21/16	7.5/1–8, 10, 29. To learn some shortcuts for the partial-fractions work that's typically needed to invert the Laplace Transform, you may want first to read the web handout "Partial fractions and Laplace Transform problems".
M 11/28/16	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, "homework" includes reading the relevant portions of the textbook, and "everything" includes material that did not appear on last year's third exam), with the following exceptions in the textbook: In Section 6.1, you are not responsible for knowing what the Wronskian is except when \(n=2\). In Theorem 2, p. 321, replace the condition involving the Wronskian with "If the solutions \(y_1, \dots, y_n\) are linearly independent on \( (a,b) \)." In Section 6.3, you are not responsible for the terminology "annihilate" or "annihilator", or for the "annihilator method". You are responsible for begin able to use the Method of Undetermined Coefficients, as presented in class, to solve DEs such as the ones in the homework problems assigned from this section. In Section 7.5, on this exam you are not responsible for the material involving non-constant-coefficient IVPs, such as Example 4. You are responsible for the definition of "fundamental set of solutions" that I gave in class. Note: Last year's class was taught using a textbook with topics arranged in an order different from what's in this year's textbook. Consequently, at the time of last year's third exam, we had covered some material that we have not covered yet this year (problems 1(c) on last year's exam, and the best way to do #2, represent such material), and we have covered some material that I had not yet covered at corresponding time last year. It would be very unwise to think that just because some topic we've covered doesn't appear on last year's exam, you don't need to be prepared for it on your exam.
W 11/30/10	Read Section 7.6 through Example 5. 7.6/ 1–4. If you feel ready to do more based on your reading, start on the problems due Friday (there are a lot).
F 12/2/16	7.6/ 5–10, 11–18, 29–32, 33–40. For all of the above problems in which you solve an IVP, write your final answer in "tabular form", by which I mean an expression like the one given for \(f(t)\) in Example 1, equation (4), p. 385. Do not leave your final answer in the form of equation (5) in that example. On an exam, I would treat the book's answer to problems 29–40 as incomplete, and would deduct several points. The unit step-functions and "window functions" (or "gate functions", as I call them) should be viewed as convenient gadgets to use in intermediate steps, or in writing down certain differential equations (the DEs themselves, not their solutions). The purpose of these special functions is to help us solve certain IVPs efficiently; they do not promote understanding of solutions. In fact, when writing a formula for a solution of a DE, the use of unit step-functions and window-functions often obscures understanding of how the solution behaves (e.g. what its graph looks like).     For example, with the least amount of simplification I would consider acceptable, the answer to problem 31 can be written as $$ y(t)=\left\{\begin{array}{ll} t, & 0\leq t\leq 2, \\ 4+ \sin(t-2)-2\cos(t-2), & t\geq 2.\end{array}\right.$$ The book's way of writing the answer obscures the fact that the "\(t\)" on the first line disappears on the second line—i.e. that for \(t\geq 2\), the solution is purely oscillatory (oscillating around the value 4); its magnitude does not grow forever.     In this example, using trig identities the formula for \(t\geq 2\) can be further simplified to several different expressions, one of which is \(4+ \sqrt{5}\sin(t-2-t_0)\), where \(t_0=\cos^{-1}\frac{1}{\sqrt{5}} = \sin^{-1}\frac{2}{\sqrt{5}}\). (Thus, for \(t\geq 2\), \(y(t)\) oscillates between a minimum value of \(4-\sqrt{5}\) and a maximum value of \(4+\sqrt{5}\).) This latter type of simplification is important in physics and electrical engineering (especially for electrical circuits). However, I would not expect you to do this further simplification on an exam in MAP 2302.
M 12/5/16	Skim Section 8.1. Carefully read Section 8.2. Most of the material in Section 8.2 is review of prerequisite material from Calculus 2. Since there is so little class time remaining, I do not want to spend any of it on anything that's purely review of prerequisite material. However, chances are that you do not have most of this material at your fingertips, so it is important that you do the review on your own time. I plan to start Section 8.3 right away on Monday Dec. 5. 8.2/1–6, 7, 8, 9, 10, 11–14, 17‐20, 23, 24, 27, 28, 37.     Notes: (1) In these problems, anywhere you see the term "convergence set", replace it with "open interval of convergence". In the notation of Theorem 1 on p. 428, "open interval of convergence" means the set \( \{x: |x-x_0|<\rho\}\). (2) The instructions for problems 23–26, as well as for Example 3 on p. 431, somewhat miss the point. The point is to re-express the given power series in \(x\) as a power series in which the power of \(x\) is exactly equal to the index of summation, not to use any particular letter or name for that index. The index of summation is a dummy variable; you can all it \(k, n, j\), Sidney, or almost anything else you like, including a name already used as the summation-index of another series in the same problem. In class you will see me using the letter \(n\), not \(k\), for such re-indexed series, just as in Example 4 on pp. 431–432 and exercises 8.2/ 27, 28.
W 12/7/16	Read Sections 8.3. Read Section 8.4, ignoring statements about radius of convergence (in particular, you should skip Examples 1 and 2). I will not hold you responsible for the part of Theorem 5, p. 446, that makes a statement about radius of convergence. That part of Theorem 5 is actually the only piece of information in Section 8.4 that's not in Section 8.3; however, Examples 3 and 4 in Section 8.4 are of types not presented in Section 8.3. Some facts related to radius of convergence that I will hold you responsible for are: The power series centered at 0 for \(e^x, \sin x\), and \( \cos x\) (given on p. 433) have infinite radius of convergence. If a power series centered at a point \(x_0\) has infinite radius of convergence, then the function represented by that power series is analytic everywhere, not just at \(x_0\). A corollary of these two facts are that the exponential, sine, and cosine functions are analytic everywhere. I used this in class when I deduced that the function \(q(x)=\frac{\sin x}{x}\), with its removable singularity at \(x=0\) removed, has no other singular points. 8.3/1, 4, 6–8, 10. Based on your reading and the not-yet-completed example of a power-series solution in Monday's class, do as many of the problems below as you're able. Any that you can't do by Wednesday, you should try to do before Friday's review. But even though these exercises will become easier after I've done more examples in class on Wednesday, I think you'll have a better chance at mastering this material if you start now, rather than trying to cram it all into your brain between Wednesday and the final exam. 8.3/ 11–14, 18, 20–22, 24, 25. 8.4/ 15, 20, 21, 23, 25.
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. 9, 10:40-11:30, in our usual classroom. I will also have an office hour that day, starting at 3:30 (not 3:00). My usual office-hour schedule does not apply after Dec. 7. My office hours for the rest of the semester are posted below. Reminder: Your final exam starts at 7:30 a.m. on Friday Dec. 16, in our usual classroom.
Office hours for remainder of semester	Below are the office hours I plan for the rest of the semester. Circumstances may arise that force me to alter some of these, so before you come to an office hour, please check to see whether I've sent you email revising these hours. During each office-hour block, I will probably take one or more short breaks, so if you come and I'm not there (and I have not emailed a change of plans), I'll be back shortly. Fri. 12/9: 3:30-4:30 Mon. 12/12: 1:30-3:30 Tues. 12/13: 11:00-12:00 and 2:00-3:00 Wed. 12/14: 2:00-4:00 Thurs. 12/15: 12:30-2:30

Class home page