Homework Assignments
MAP 2302, Section 1G33—Elementary Differential Equations
Fall 2018

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, 9th 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
F 8/24/18	Read the syllabus and the web handouts "Taking and Using Notes in a College Math Class" and "What is a solution?". Read Section 1.1 and do problems 1.1/ 1–16.     Note: the sentence on p. 4 that contains equation (7) is not quite correct as a definition of "linear". An ODE in the indicated variables is linear if it has the indicated format, or can be put in this format just by adding/subtracting expressions from both sides of the equation (as is the case with the next-to-last equation on the page). Do non-book problem 1. In my notes on first-order ODEs, read the first three paragraphs of the introduction, all of Section 3.1.1, and Section 3.1.2 through the third paragraph on p. 13 (the paragraph beginning with "Graphically"). In this and future assignments from these notes, you should skip all items labeled "Note to instructors". (In particular, you should skip the footnote that begins on p. 11 and occupies all but the first three lines of p. 12.)
M 8/27/18	1.2/ 1, 3–6, 19–22. See Notes on some book problems. Do non-book problem 2. In my notes, read from where you left off on p. 13 through the end of Section 3.1.2, and do the exercise on p. 17.     I update these notes from time to time during the semester, so you should always download the notes anew each time you're going to read them, to make sure you have the most up-to-date version. Note: The exercise portions of many (probably most) of your homework assignments will be a lot more time-consuming than in the assignments to date; I want to give you fair warning of this before the end of Drop/Add. 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 near the top of this page: "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."
W 8/29/18	Read the remainder of Section 3.1.3 of my notes, but not all at once. Read through the end of Example 3.10 on p. 19 before doing the exercises below, then do all the exercises below, then return and do the rest of the reading. 1.2/ 2, 9–12, 14–17, 30. In #30, ignore the book's statement of the Implicit Function Theorem; use the statement in my notes. The theorem stated in problem 30 of the printed edition (as of 8/25/18) is much weaker than the Implicit Function Theorem, and should not be called by that name. This will be fixed in the next printed edition of the textbook (I communicate with the primary living author, Edward Saff). The change will show up in the electronic version of the book sooner, but I'm not sure whether that means in a few days, weeks, or months. The fixed version, once it appears, may not be worded identically to the way I worded the Implicit Function Theorem in my notes, but should be equivalent.
F 8/31/18	1.2/ 18, 23–28, 31. For 23–28, the instructions should end with "... has a unique solution on some open interval." Similarly, in 31a, "unique solution" should be "unique solution on some open interval". In the textbook, read Section 1.3, and read the first two pages of Section 2.2 (through the end of Example 1). 1.3/ 2, 3 2.2/ 1–4, 6. In my notes, read Section 3.1.4 up through Example 3.29 (the rest of this section will be in the next assignment), and read Section 3.1.5 up through Definition 3.37. You are allowed to skip a portion of this material; see e.g. the last two sentences of the first paragraph on p. 32. In my notes, also read Section 5.1 (the appendix, pp. 126–127) except for the last paragraph. In class on Wednesday I stated this theorem with "open rectangle" instead of "open set", since I'd not yet given you the definition of "open set" (which is in the portion of Section 3.1.5 assigned above). With the term "open rectangle" instead of "open set", Theorem 5.1 is what the textbook's Theorem 1 on p. 11 should have said; see the paragraph in my notes after Theorem 5.1. (The theorem is nicer with "open set" instead of "open rectangle", but most DE textbooks don't define "open set" in the plane.) This error in the textbook will be fixed in the next edition, and sooner in the e-book.
W 9/5/18	2.2/ 7–14, 17–19, 21, 24 . "Solve the equation" means "Find all (maximal) solutions of the equation". Although I haven't finished discussing various subtleties, or justified the separation-of-variables technique yet, the two-part method discussed in class does find all the solutions of the DEs you're being asked to solve in this assignment. In my notes, read the remainder of Sections 3.1.4 and 3.1.5. Reminder: reading my notes is not optional (except for portions that I [or the notes] say you may skip, and the footnotes or parenthetic comments that say "Note to instructor(s)"). Each reading assignment should 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 carefully in class; we would not get through all the topics we're supposed to cover. Do non-book problems 3, 4, and 6. Answers to these and some other non-book problems are posted on the "Miscellaneous handouts" page. General comment. In doing the exercises from Section 2.2 or the non-book problems 3, 4, and 6, you may have found 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 in the syllabus: 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. Don't just go 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 "\(\int \ln x\, dx =\frac{1}{x} +C\)", or "\( \frac{d}{dx}\frac{1}{x} = \ln x\)'', even if the rest of your work is correct. (The expression \(\frac{1}{x}\) is the derivative of \(\ln x\), not one of its antiderivatives; \(\ln x\) is an antiderivative of \(\frac{1}{x}\), 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 to both review the old (if it's not fresh in your mind) and learn the new.
F 9/7/18	2.2/ 27abc Do non-book problems 7, 8. Several parts of #7 were done in class, but it won't hurt you to do those parts over again. Problem 8 was newly added to the list of non-book problems the evening of 9/5/18, so you may need to refresh your browser to get an up-to-date copy of this list. (If the #8 you're looking at has parts (a), (b), and (c), you're looking at the old problem 8.) In my notes, read Sections 3.1.6 and Section 3.1.7. (Remember that the blue portions of my notes are optional reading; you are allowed to skip them even if they're within the sections I'm assigning. However, answers to some students' questions are in the blue portion of Section 3.1.7, so I encourage interested students to do this optional reading.) I fixed a typo on 9/5/18, so again, refresh your browser to make sure you're downloading the current version.        Theorems 3.43 and 3.44 in my notes are closely related to the "Formal Justification of Method" on p. 45 of the textbook. You will find the book's presentation simpler than mine, but this simplification comes at a price: (1) the book's argument does not actually establish what it purports to (because it puts no hypotheses on the functions \(p\) and \(g\), without which several steps in the book's argument cannot be justified), and (2) the conclusion it purports to establish skirts an important issue. (The question of whether the method gives all the solutions, or even all the non-constant solutions, is never mentioned, let alone answered.)
M 9/10/18	2.3/ 1–6 Read Section 2.3 of the textbook. In my notes, read from the beginning of Section 3.2 (p. 67) up to, but not including, the beginning of Section 3.2.1 (p. 72).        I will assign the earlier Section 3.1.8 after we've spent another lecture on linear DEs. With the exception of the notes' Definition 3.56, the Section 3.2 material in the reading above is basically not discussed in the book at all, even though differential-form DEs appear in (not-yet-assigned) exercises for the book's Section 2.2 and in all remaining sections of Chapter 2. (Very little of what's in Sections 3.2.1–3.2.4, 3.2.6, or 3.3 of my notes is discussed in the book either.) I'm assigning the early portion of Section 3.2 now since the rest of the current assignment is pretty light. This will (partially) spare you from having to do a lot of reading and exercises at the same time once we get to the book's Section 2.4.
W 9/12/18	2.3/ 7–9, 12–15 (note which variable is which in #13!), 17–20.    When you apply the method we learned in Monday's class (which is in the box on p. 50, except that the book's imprecise "\(\int P(x)\,dx\)" is my "\(\int_{\rm spec} P(x)\,dx\)"), don't forget the first step: writing the equation in "standard linear form", equation (15) in the book. (If the DE started with an \(a_1(x)\) multiplying \(\frac{dy}{dx}\)—even a constant function other than 1—you've already divided through by it. Be especially careful to identify the function \(P\) correctly; its sign is very important. For example, in 2.3/17, \(P(x)= -\frac{1}{x}\), not just \(\frac{1}{x}\). Do non-book problems 9ab. Read Section 3.1.8 of my notes.
F 9/14/18	2.3/ 22, 27a, 28, 30– 33, 35. See Comments, part 1, below. In the textbook, read Section 2.4 through the boxed definition "Exact Differential Form" on p. 59. See Comments, part 2, below. In my notes, read from the beginning of Section 3.2.2 (p. 77) through the end of Example 3.75 on p. 85. Remember, you are allowed to skip anything in blue. The term "regular parametrization" is defined in Definition 3.63 on p. 74 (part of Section 3.2.1). With the exception of Definitions 3.63 and 3.64, you should have seen the material in Section 3.2.1 in Calculus 2, so I'm not requiring you to read Section 3.2.1. However, in the sections I am requiring you to read, you may occasionally come across terminology you're not familiar with because it was defined in Section 3.2.1 or somewhere in blue text that you skipped. If that happens, it's generally safe to skip over the sentence(s) containing such terminology, but you may want to find that definition, read it briefly to get the general idea, and then go back to where you were.     See Comments, part 3, below. Comments, part 1. One of the things you'll see in exercise 2.3/33 is that (as indicated in the last assignment) 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, 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 and IVPs 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. Comments, part 2. There are some terminological problems in Section 2.4 of the book, most notably an inconsistent usage of the term "differential form". Most students will probably not even notice the inconsistency, but some may—especially the students with a deep interest in mathematics—and I don't want anyone to come out my class with an improper education. Every instance in this chapter in which the term "differential form" is used for anything that is not an equation—a statement with an "=" sign in it—the word "form" should be deleted. In particular this applies to all instances of "differential form" in the definition-box on p. 59 (including the title). The definition-box's use of the term "differential form" is not technically incorrect, but at the level of MAP 2302 it is a very confusing use of the word "form", and the less-misinterpretable term "differential" is perfectly correct. The usage of "differential form" within the definition-box, after the title, is inconsistent with the usage outside the definition-box. The usage in the title is ambiguous; it is impossible to tell whether the title is referring to an exact differential, or to an equation with an exact differential on one side and zero on the other.     In my notes I talk about "derivative form" and "differential form" of a differential equation. The meaning of the word "form" in my notes is standard mathematical English, and is the same as in each of the two occurences of "form" on on p. 58 of the book. In this usage, "form of an equation" refers to the way an equation is written, and/or or to what sort of objects appear in it.     But when a differential itself (as opposed to an equation containing a differential) is called a "differential form", the word "form" means something entirely different, whose meaning cannot be gleaned from what "form" usually means in English. In this other usage, "differential forms" are more-general objects than differentials. (Differentials are also called 1-forms. There are things called 2-forms, 3-forms, etc., which cannot effectively be defined, let alone understood, at the level of MAP 2302. You won't see them in this course, or in any undergraduate courses at UF—with the possible exception of the combined graduate/undergraduate course Modern Analysis 2, and occasional special-topics courses.) Since, with this meaning of "differential form" the only differential forms that appear in an undergraduate DE textbook are differentials—1-forms—there is no good reason in a such a course, or textbook, to use the term differential form for differential.     There is also a pronunciation-difference in the two usages of "differential form". The pronunciation of this term in my notes is "differential form", with the accent on the first word, providing a contrast with "derivative form". In the other usage of "differential form"—the one you're not equipped to understand, but that is used in the definition-box on p. 59—the pronunciation of "differential form" never has the accent on the first word; we either say "differential form", with the accent on the second word, or we accent both words equally. The paragraph directly below the "Exact Differential Form" box is not part of the current assignment. However, for future reference, this paragraph is potentially confusing or misleading, because while the first sentence uses "form" in the way it's used on the preceding page and in my notes, the third sentence uses it with the other, more advanced meaning. Said perhaps more accurately, this paragraph does not make sense unless the term "differential form" is used to mean a form of an equation (with the standard-English meaning of "form") on line 2, and with the meaning of a differential on line 4. Most likely, the authors didn't intentionally use "differential form" with two different meanings, but didn't notice that this paragraph (and several others) make no sense if "differential form" is assigned any consistent meaning.     Choosing to use the term "exact differential form" in the first equation of this paragraph is, itself, rather unusual. In any context involving the objects that, with the more advanced meaning, are called "differential forms", there is a standard meaning of the term "exact differential form" (and a synonymous term is "exact form"). In this meaning, an exact differential form is a differential form (with the more advanced meaning) with a certain property that makes us call it "exact". I.e., once we combine the word "exact" with "differential form", there are no longer two different things that "differential form" can mean, without departing from standard convention. In standard convention, "exact differential form" is never a type of equation. In the context of the current paragraph, there is only one standard meaning of "exact differential form", and it's exact differential—a type of differential, not a type of equation. The standard terminology for what the sentence in question calls "[differential equation] in exact differential form" is exact equation (or exact differential equation), just as you see in the definition-box on p. 59. The terminology "exact equation" has its own intrinsic problems, but is standard nonetheless. In Example 1 on p. 58, the sentence beginning "However" is not correct. In this sentence, "the first form" refers to the first equation written in the sentence beginnning "Some". An equation cannot be a total differential. An equation makes an assertion; a total differential (like any differential) is simply a mathematical expression; it is no more an equation than "\(x^3\)" is an equation. To correct this sentence, replace the word "it" with "its left-hand side". The following is just FYI; it's not a problem with the book: What the book calls the total differential of a function F is what my notes call simply the differential of F. Both are correct. The word "total" in "total differential" is superfluous, so I choose not to use it. Comments, part 3. You're going to find these sections more difficult to read than the book's Section 2.4 (and probably more difficult than the earlier sections of my notes). A large part of the reason for this is buried in the sentence containing the book's equation (3) on p. 58. You'll find the sentence plausible, but you should be troubled by the fact that since \(\frac{dy}{dx}\) is simply notation for an object that is not actually a real number "\(dy\)" divided by a real number "\(dx\)", just how is it that an equation of the form \(\frac{dy}{dx}=f(x,y)\) can be "rewritten" in the form of equation (3)? Are the two equations equivalent? Just what does an equation like (3) mean? In a derivative-form DE, there's an independent variable and a dependent variable. Do you see any such distinction between the variables in (3)? Just what does a solution of such an equation mean? Is it the same kind of animal as a solution of equation (1) or (2) on p. 6 of the book, even though no derivatives appear in equation (3) on p. 58? If so, why; if not, why not? Even if we knew what "solution of an equation in differential form" ought to mean, and knew how to find some solutions, would we have ways to tell whether we've found all the solutions? Even for an exact equation, how do know that all the solutions are given by an equation of the form \(F(x,y)=C\), as asserted on p. 58? The textbook is easier to read than my notes because these questions and their answers (which are subtler and deeper than you might think) aren't mentioned. The same is true of all the DE textbooks I've seen; overall, this book is better than any other I've seen on the market.
M 9/17/18	2.4/ 1–8. Strictly speaking, for differential-form DEs, there is no such thing as a linear equation. In these problems, you are meant to classify an equation in differential form as linear if at least one of the associated 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 several of these exercises. For example, #5 is linear as an equation for \(y(x)\), but not as an equation for \(x(y)\). In my notes, read from Remark 3.76 on p. 85 (part of Section 3.2.4) through the bottom of p. 99 (mid-way through Section 3.3). Again, the material in blue (which is quite a lot of this portion of the notes) is optional reading; you're required to read only the material in black. 2.2 (not 2.3 or 2.4)/ 5, 15, 16. (I did not assign these when we were covering Section 2.2 because we had not yet discussed "differential form".) An equation in differential form is called separable if, in some region of the \(xy\) plane (not necessarily the whole region on which the given DE is defined), the given DE is algebraically equivally equivalent to 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.
W 9/19/18	Read the remainder of Section 2.4 of the textbook. 2.4 / 9, 11–14, 16, 17, 19, 20. Read the online handout A terrible way to solve exact equations. The example in this version of the 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 \(\int \sin x \cos x\, dx\) 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.    Where the handout says "(we proved it!)", substitute "(we will prove it, time permitting)". 2.2 (not 2.3 or 2.4)/ 22. Note that although the differential equation doesn't specify independent and dependent variables, the initial condition does. Thus your goal in this exercise 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 "\(y(x_0)=y_0\)'' indicates), then the differential equation you're interested in at the start is one in derivative form (which in exercise 22 would be \(x^2 +2y \frac{dy}{dx}=0\), or an algebraically equivalent version), not one in differential form. Putting the DE into differential form is often 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 "\(y(x_0)=y_0\)'' or "\(x(y_0)=x_0\)''. What's logical to ask for is a solution whose graph passes through the point \((x_0,y_0)\), which in exercise 22 would be the point (0,2). 2.4 (resumed)/ 21, 22 (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 what I termed "schizophrenic" IVPs. In all of these, the goal is 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 algebraic equation ``\(F({\rm variable}_1, {\rm variable}_2)=0\)'' that you get via the exact-equation method (in these schizophrenic IVPs) is impossible to solve for the dependent variable in terms of the independent variable, you have to settle for an implicit solution. In my notes, read the remainder of Section 3.3, and start reading Section 3.4. (Completing Section 3.4 and reading Section 3.5 will be part of your next assignment; divide this reading between the two assignments in whatever way works best for you.)
F 9/21/18	2.4/ 32, 33ab Read the statement of non-book problem 10. I'm not requiring you to do the problem, but I want you to be aware of the fact stated in the problem. Read The Math Commandments. In my notes, finish reading Section 3.4 and read Section 3.5.
M 9/24/18	First midterm exam (assignment is to study for it). In case you'd like additional 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. 79 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. Even if you don't have time to work through the problems on p. 79, they're good practice for figuring out the appropriate methods are.     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, 6, and 9. 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 Chapter 3 of my notes (minus the portions I said were optional) as well as doing book and non-book exercises, and the textbook chapters/sections we've covered are 1.1, 1.2, 2.2, 2.3, and 2.4.
W 9/26/18	No new homework.
F 9/28/18	Read section 4.1 of the book. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)
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. I don't post solutions to my exams. Depending on the exam, I sometimes give a (paper) handout in class with comments on the exam and/or answers to some problems. I haven't decided yet whether I'll be doing that for this exam. Please read the rest of this message, read the information on the grade-scale page, and re-read the syllabus and course information page, so that you don't inadvertently ask me to spend time answering a general question that I've already answered, or ask by email a question that I never answer by email. Questions specific to your individual situation should be asked in office hours. If you'd like to see me in my office, but have a conflict with all my scheduled office hours, please let me know your complete schedule—all the days/times with which you have conflicts—and I'll try to find a day and time that works for both of us. (I.e. please don't just ask me "Could I see you [or would it be convenient for you to see me] Day X or Y during period Z or W?", the most convenient days and times for you. When you give me your full schedule, feel free to let me know what preferences you have within your non-conflicting time-slots; just don't send me your preferences instead of your complete schedule.) Exams will be returned at the end of class on Monday 10/1/18. 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 out any exam that has not been picked up, unless the student has made prior arrangements with me and has a valid excuse. (Usually I hold onto exams longer; just don't count on it.) Failure to read this notice on time will not count as a valid excuse, since you're supposed to be checking this homework page at least three times a week to get the homework assignment that's due by the next class.
M 10/1/18	Read Section 4.2. Unfortunately, hardly any of the exercises are doable until the whole section has been covered. Based on your reading, try to start on the exercises in the next assignment. There are a lot of them. 4.7 (yes, 4.7)/ 30. No reading is necessary for this one.
W 10/3/18	4.2/ 1, 3, 4, 7, 8, 10, 12, 13–16, 18, 26, 27–32, 46ab
F 10/5/18	The problems in this assignment are numerous but short. You may find them repetitive, but for basic skills, repetition builds retention. Nothing else does. You didn't learn to walk by taking five steps and saying to your toddling self, "Got it! Whew! Think I'll just sit in the stroller for the next month." (Okay, okay, learning to walk may have been a bit more fun and a trifle more useful than learning to solve DEs, but you get the point.) Repetition builds retention is a good mantra. 4.2/ 2, 5, 9, 11, 17, 19, 20, 35 4.3/ 1–18. In the instructions for 1–8, "complex roots" should be replaced by "non-real roots", "non-real complex roots", or "no real roots". Every real number is also a complex number (just like every square is a rectangle); thus "complex" does not imply "non-real". A real number is just a complex number whose imaginary part is 0. 4.7 (yes, 4.7)/ 1–8, 25, 26abc. First read from the bottom of p. 192 (Theorem 5, the \(n^{\rm th}\)-order version of which was stated in class a few days ago) through Example 1 on p. 193. 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]."     These problems from Section 4.7 should actually have been part of an earlier assignment. It's okay to defer them to the next assignment if you don't have time to do them on top of the above problems from Sections 4.2 and 4.3, but understand that they're most closely connected to material that was presented in class last week, not this week.
M 10/8/18	Read Section 4.3. In the title of this section, "complex roots" should be replaced by "non-real roots", "non-real complex roots", or "no real roots". 4.3/ 21–26, 28, 32, 33 (students in electrical engineering may do #34 instead of #33) Read Section 4.4 up through Example 3. Read Section 4.5 up through Example 2.
W 10/10/18	No new homework. If you are behind on your homework, use this opportunity to catch up.
F 10/12/18	Read the remainder of Sections 4.4 and 4.5. I was not able to get far enough in class on Wed. 10/10 to begin covering these sections in class, but to the extent that you're able to do problems based on your reading, I would recommend starting on the problems I'm posting that are due next week. This will help you avoid a work-crunch later. In the near future, I will be assigning almost all the exercises in sections 4.4 and 4.5. We will be covering Sections 4.4 and 4.5 simultaneously, more or less, rather than one after the other. What most mathematicians (including me) call "the Method of Undetermined Coefficients" is what the book calls "the Method of Undetermined Coefficients plus superposition." You should think of Section 4.5 as completing the (second-order case of) the Method of Undetermined Coefficients, whose presentation is begun in Section 4.4. Unfortunately, the way the Section 4.4 exercises are structured, you can't do more than a handful of the exercises before having completed the whole section, which is at least two full lectures worth of material. To help avoid giving you one massive assignment when we're done with 4.4 and 4.5, I've spread the problems out over several assignments, with the hope that the reading will enable you to do some before we cover all the material in class. The order in which I'm assigning problems from Sections 4.4 and 4.5 corresponds to the order in which I'll be covering the material, which is different from the order of presentation in the book.
M 10/15/18	4.5 (not 4.4)/ 1–8. We may not have yet discussed in class how one might find the \(y_p\)'s in these problems, but you don't need to know that for these problems, since (modulo having to use superposition in some cases) the \(y_p\)'s are handed to you on a silver platter. 4.4/ 1–8 4.5/ 9–16
W 10/17/18	4.4/ 9 (note that \(-9=-9e^{0t}\) ), 10, 11, 14, 18 Add parts (b) and (c) to 4.4/ 9–11, 14, 18 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 18: \(y(0)=1, y'(0)=2\). 4.5/ 25, 26, 28 4.4/ 13, 15–17, 19–26
F 10/19/18	4.4/ 27–32. In the instructions for 27–32, the word "form" should be replaced by "MUC form" ("MUC" is the abbreviation we'll use for "Method of Undetermined Coefficients). 4.5/ 17–22, 24–30, 31–36, 41, 42. In the instructions for 31–36, the word "form" should be replaced by "MUC form".     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 10/22/18	Do these non-book exercises on the Method of Undetermined Coefficients. The answers to these exercises are here. 4.4/ 12, 33–36. Note that #12 can also be done by Chapter 2 methods. The purpose of this exercise, in this chapter, is to see that it also can be done using the Method of Undetermined Coefficients. 4.5/ 23 (the same comment as for 4.4/12 applies), 38, 40.     Regarding #38 and #40: As stated briefly in class several lectures ago, in a constant-coefficient differential equation \(L[y]=g\), the functions \(g\) to which the MUC applies are the same regardless of the order of the DE, and, for a given \(g\), the MUC form of a particular solution is also the same regardless of this order. The degree of the characteristic polynomial is the same as the order of the DE (to get the characteristic polynomial, just replace each derivative appearing in \(L[y]\) by the corresponding power of \(r\), remembering that the "zeroeth" derivative—\(y\) itself—corresponds to \(r^0\), i.e. to 1, not to \(r\).) However, a polynomial of degree greater than 2 can potentially have have roots of multiplicity greater than 2. The possibilities for the exponent "\(s\)" in the general MUC formula (for functions of "MUC type" with a single associated "\(\alpha + i\beta\)") range from 0 up to the largest multiplicity in the factorization of \(p_L(r)\).     Thus the only real difficulty in applying the MUC when \(L\) has order greater than 2 is that you have to factor a polynomial of degree at least 3. Explicit factorizations are possible only for some such polynomials. Every cubic or higher-degree characteristic polynomial arising in this textbook is one of these special, explicitly factorable polynomials (and even among these special types of polynomials, the ones arising in the book are very simplest). For problem 38, note that if all terms in a polynomial \(p(r)\) have even degree, then effectively \(p(r)\) can be treated as a polynomial in the quantity \(r^2\). Hence, a polynomial of the form \(r^4+cr^2+d\) can be factored into the form \((r^2-a)(r^2-b)\), where \(a\) and \(b\) either are both real or are complex-conjugates of each other. You can then factor \(r^2-a\) and \(r^2-b\) to get a complete factorization of \(p(r)\). (If \(a\) and \(b\) are not real, you may not have learned yet how to compute their square roots, but in problem 38 you'll find that \(a\) and \(b\) are real.) For problem 40, you should be able to recognize that \(p_L(r)\) is \(r\) times a perfect cube. 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.
W 10/24/18	No new homework.
F 10/26/18	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. 231, except those in which the DE is not a constant-coefficient equation. However, be aware that the types of problems on this page, or elsewhere in the book, do not represent all the types of problems you could see on your exam. I have done some other types of problems in class, and your homework has included non-book problems.     To solve the order-3 constant-coefficient DEs on p. 231 you need to be able to factor the characteristic polynomial, so here's a hint: All the cubic characteristic polynomials arising in this textbook have at least one root that is an integer of small absolute value. If you are able to guess one root, you can factor a cubic polyomial \(p(r)\). (If the root you know is \(r_1\), divide \(p(r)\) by \(r-r_1\), yielding a quadratic polynomial \(q(r)\). Then \(p(r)=(r-r_1)q(r)\), so to complete the factorization of \(p(r)\) you just need to factor \(q(r)\).) If you know the Rational Root Theorem then for all the cubic characteristic polynomials arising in this textbook, you'll be able to guess an integer root quickly. If you do not know the Rational Root Theorem, you will still be able to guess an integer root quickly, but perhaps slightly less quickly. (From the book's examples and exercises, you might get the impression that plugging-in integers is the only tool for trying to guess a root of a polynomial of degree greater than 2. If you were a math-team person in high school, you should know that this is not the case.)
General info	The most likely date for the third midterm is now Monday, Nov. 19 (the Monday before the Thanksgiving break).
M 10/29/18	In Section 4.7, read from the middle of p. 193 (the box "Cauchy-Euler, or Equidimensional, Equations") through the end of Example 3 on p. 195. 4.7/ 9–20, 23, 24, 27 Reminder about some terminology. As I've said in class, "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. 194, "indicial equation", is what's used for Cauchy-Euler DEs in most textbooks I've seen. Part of what problem 23 shows is that the indicial equation for the Cauchy-Euler DE is the same as the characteristic equation for the associated constant-coefficient DE obtained by the Cauchy-Euler substitution \(t=e^x\). (That's if \(t\) is the independent variable in the given Cauchy-Euler equation; you then get a constant-coefficient equation with independent variable \(x\).) 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 you'll need to be aware that that's what the book does. I won't consider it a mistake for you to use the book's terminology for that equation, but you do need to know how to use that equation correctly (whatever you call it), and need to understand me when I say "indicial equation". Do non-book problem 12. Read Section 4.6.
W 10/31/18	4.7/ 23, 24cd Prepare for small humanoids to prowl the land in packs.
M 11/5/18	4.7/ 27
W 11/7/18	4.6/ 2, 5–8, 9, 10, 11, 12, 15, 17, 19 (first sentence only). 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. 188. (The unknowns to be solved for in this system are \(v_1',v_2'\).) 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. 4.7/ 37–40.     Note that it is possible to solve all the DEs 37–40 (as well as 24cd) either by the Cauchy-Euler substitution "\(t=e^x\)" applied to the inhomogeneous DE, or by the indicial equation just to find a FSS for the associated homogeneous DE 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/ 40 by starting with the substitution \(y(t)=t^{1/2}u(t)\) and seeing where that takes you. Read Section 6.1. (We will not be covering Chapter 5.)
F 11/9/18	6.1/ 1–6, 7–14. (Do these based on your reading; they're not very difficult.) Do 7–14 without using Wronskians. The sets of functions in these problems are so simple that, if you know your basic functions (see The Math Commandments), Wronskians will only increase the amount of work you have to do. Furthermore, in these problems, if you find that the Wronskian is zero then you can't conclude anything (from that alone) about linear dependence/independence. If you do not know your basic functions, then Wronskians will not be of much help. Read Section 6.2.
W 11/14/18	6.1/ 19, 20, 23. 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.
F 11/16/18	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 11/19/18	Third midterm exam (assignment is to study for it). Fair-game material for this exam includes everything we've covered since the last exam, up through Section 6.3 of the textbook (where "covered" includes classwork and homework, and "homework" includes reading the relevant portions of the textbook, with the exceptions noted below). In Section 6.1, you are not responsible for knowing what the Wronskian is except when \(n=2\). In Theorem 2, p. 322, replace the condition involving the Wronskian with "If the solutions \(y_1, \dots, y_n\) are linearly independent on \( (a,b) \)." (Linear dependence and independence are defined on p. 323.) 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.
M 11/26/18	Read sections 7.1 and 7.2. If you feel sufficiently prepared by your reading, start on the exercises in the next assignment. You may be able to do the non-calculational exercises that start with 7.2/21 before you can do some or all of the earlier exercises.
W 11/28/18	7.2/ 1–4, 10, 12 (note: "Use Definition 1" in the instructions for 1–12 means "Use Definition 1", NOT Table 7.1 or any other table of Laplace Transforms). 7.2/ 13–20 (for these, do use Table 7.1) 21–23, 26–28, 29a–d,f,g,j. Read Section 7.3. 7.3/ 1–10 7.4/ 11, 13, 14, 16, 20. You should be able to do these with or without reading Section 7.4 first (recall from the "Prerequisite" paragraph in the syllabus that the method of partial fractions is something you're expected to know), but there's additional review in Section 7.4 if you need it.
F 11/30/18	Read Section 7.4. 7.4/ 1–10, 21–24, 26, 27, 31. Normally, I would not assign these until after talking about the inverse Laplace transform in class, but time is short. If you are unable to do these based on your reading, it's okay to wait, but then you'll have a much longer assignment due on Monday. I'm trying to spread the homework problems out over enough days that you'll have time to do all the problems. 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 \(y(t)\). Read Section 7.6.
M 12/3/18	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". 7.6/ 5–10 , 11–18, 29–32. 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 19–33 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 23 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.
W 12/5/18	No new homework. In Wednesday's class (which is our last class), my top priority will be to work some problems similar to the homework problems from Section 7.6. Thus we won't have our usual pre-exam Q&A review (although if there's time left after I've worked what I think are enough problems of "Section 7.6", I can field questions on earlier material). If there is sufficient demand, I may be able to hold an attendance-optional review on Friday. My usual office-hour schedule does not apply after Dec. 5. I will have an office hour on Friday, but it may or may not be at 4:05 p.m. If I'm going to hold the office hour at a different time that day, I'll announce that by email, so make sure to check for emails from me. Between now and the 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.
12/10/18 Final Exam	The final exam will be given on Monday, December 10, starting at 10:00 a.m., in our usual classroom.

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