Homework Assignments
MAP 2302, Section 3875—Elementary Differential Equations
Spring 2017

Last updated Fri Apr 14 16:42 EDT 2017

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

F 1/6/17
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. In problems 1–12, you may (for now) ignore the instruction involving the words "linear" and "nonlinear"; that will be part of the next assignment.

Do non-book problem 1.

M 1/9/17
For the DEs in 1.1/1, 2, and 4–12, classify each equation as linear or nonlinear.

1.2/ 2–6, 19–22 See Notes on some book problems. "Explicit solution" is synonymous with "solution".
In these notes, read from the beginning of "Notes for Students" on p. 3 through the end of the exercise on p. 13 (and do the exercise).
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 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."

W 1/11/17
Read the remainder of Section 2.2 of these notes, but not all at once. Read through the end of Example 2.8 on p. 15 before doing the exercises below, then do all the exercises below, then return and do the rest of the reading.
    Note: I have added a table of contents for these notes, so most of the page numbers as of 1/9/17 differ by 1 from what they were 1/8/17 and earlier. The new p. 4 is the old p. 3, etc. for all pages beyond that point. I also changed the title of the notes and added something at the end, but nothing in between has changed (yet).

1.2/ 1, 9–12, 14–18, 30
Do non-book problem 2.

F 1/13/17
1.2/ 23–27, 30, 31bc
Read Section 1.3.
1.3/ 2, 3
2.2/ 1–5
In my notes, read the portion of Section 2.3 from the beginning through Example 2.25. You may skip the portion of this that the notes say you may skip. I've now set up the notes so that references to examples, equations, definitions, etc., are clickable links. On p. 26, if you click on the "2.23" in "Example 2.23", you'll get taken to Example 2.23 without having to hunt for it.

W 1/18/17
2.2/ 7–14, 17–19, 21, 24, 30. "Solve the equation" means "Find all solutions of the equation".
Finish reading Section 2.3 of these notes.
Reminder: reading the notes I've written for the class 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 in class; we would not get through all the topics we're supposed to cover.

Do non-book problems 3–6.
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 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 "$\int \ln x\, dx =\frac{1}{x} +C$", 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, even if this takes a lot of time.

F 1/20/17
No new exercises.
In the book, read "Formal Justification of Method" on p. 42. I will go over this in class; it is only part of the justification of why, under hypotheses not stated in Section 2.2, our method for solving separable DEs gives the set of all solutions.
Read the following parts of my notes:

Section 2.4.
Section 2.5, up through the first paragraph on p. 42.
Throughout my notes, I recommend that you skip anything that says "Note(s) to instructors". They're not secrets; you're just very unlikely understand them, and some of them are very long.

M 1/23/17
2.3/ 1–6
Read Section 2.3 of the textbook.
In Section 2.5 of the notes, read from the point you reached in the last assignment through statement (2.78) on p. 43.

W 1/25/17
2.3/ 7–9, 12–15 (note which variable is which in #13!), 17–20, 22, 28, 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 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". 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 problem #7.
Read the remainder of Section 2.5 of the notes.
Note: One of the things you'll see in exercise 2.3/33 is that (as indicated above) 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 1/27/17
2.3/ 27a, 30– 32, 35
In my notes, read from the beginning of Section 2.6 through the end of p. 51.
In the textbook, read Section 2.4 through the boxed definition "Exact Differential Form" on p. 57. Everywhere in this box, wherever you see the term "differential form", the word "form" should be deleted. This applies also to 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 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. In my notes I talk about "derivative form" and "differential form" of a differential equation. My use of the word "form" in the notes and in class is standard English, whereas when a differential itself (rather than an equation containing a differential) is called a "differential form", the word "form" means something entirely different. (With the latter meaning, "differential forms" are something that are not discussed in any undergraduate-level courses at UF, with the possible exception of the combined graduate/undergraduate course Modern Analysis 2, and occasional special-topics courses.)
    The paragraph directly below the "Exact Differential Form" box is not part of this 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 in my notes (as is also the case in each occurrence on pp. 55–56), the third sentence uses it with an entirely different meaning.
    A related correction: In Example 1 on p. 56, 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".

M 1/30/17
Read the remainder of Section 2.4 of the textbook.
2.4/ 1–8. In these problems, you are meant to 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. (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, 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 (not 2.4)/ 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 $x$ 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).
If you feel well-enough prepared from reading the book's section 2.4, I suggest getting a head-start on the exercises that are due Wednesday.

Read the following parts of the notes :

Section 2.6.1 (starts on p. 52), through the end of the first paragraph on p. 55.
    (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.)

Section 2.6.2 (pp. 56–59).
    (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. 61–64).

Section 2.6.5 through Definition 2.61 (pp. 64–66).
    (The remainder of Section 2.6.5 is optional reading.)

W 2/1/17
2.4/ 9, 11–14, 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 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 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.

Read the following parts of the notes :

Section 2.7, minus Example 2.63 (this example, which occupies most of pp. 68–71, is optional reading). Do read Example 2.64 on p. 71.
Section 2.8, from p. 71 through the middle of p. 72 (the end of the paragraph that has "Pretend" in boldface), then from the middle of p. 76 (paragraph beginning "To have a name ...") through the end on p. 77. The remainder of Section 2.8 is optional reading.
Section 2.9, through Example 2.67 on pp. 80–81. Also read Example 2.69 (pp. 88–89). The remainder of Section 2.9 is optional reading.

F 2/3/17
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 $\int \sin x \cos x\, dx$ 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.
Read The Math Commandments.
Optional: If you are interested in seeing a differential $M dx + N dy$ that passes the exactness test $M_y=N_x$ 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 2/6/17 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.

W 2/8/17 No new homework.

F 2/10/17
Read Sections 4.1 and 4.2. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.) 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.

M 2/13/17 4.2/ 2–5, 7, 8, 10–17, 26, 27–32, 35, 46ab

W 2/15/17
4.2/ 1, 6, 9, 18–20.
4.3/ 1–18
4.7/ 1–8, 25. 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]."

F 2/17/17
4.3/ 21–26, 28, 32, 33 (students in electrical engineering may do #34 instead of #33)

M 2/20/17 Read Section 4.4.

W 2/22/17
4.3/ 36
4.5 (not 4.4)/ 1–8. Use the "$y=y_p+y_h$" approach discussed in class, plus superposition (problem 4.7/30, previously assigned) where necessary, plus your knowledge (from Sections 4.2 and 4.3) of how to solve the associated homogeneous equations for all the DEs in these problems.
    We have not 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.

F 2/24/17
4.4/ 9 (note that $-9=-9e^{0t}$ ), 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
If you feel that, based on your reading of Section 4.4 (a previous homework assignment), you understand how to find more $y_p$'s than are represented in the problems above, 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 2/27/17
4.4/ 1–8, 13, 15, 16, 18–26, 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.
    You'll have to do many of these problems based on your reading of Section 4.4; what we've done in class so far doesn't cover most of these exercises. 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. In Monday's class I expect to finish presenting Section 4.4 and Section 4.5. But if I were to assign now only the problems you can do as of today's class (Friday), you'd have far too many problems to do next week. It's important that you get as many done over this weekend as possible. It's not critical that you get them all done this weekend, as long as you get this entire assignment plus the next one done before Wednesday's class.

4.5/ 17–22, 24, 27–30, 41.

W 3/1/17
4.4/ 33–36.
4.5/ 9–16 (read instructions carefully or you'll do a ton of extra work!), 17–22, 24–30, 31–36 (again, read instructions carefully!) , 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.

Do these non-book exercises on the Method of Undetermined Coefficients. The answers to these exercises are here.

F 3/3/17 Second midterm exam (assignment is to study for it).

M 3/13/17 No new homework.

W 3/15/17
4.7/ 9–20, 24ab
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. 195, "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 $t=e^x$. (That's if $t$ is the independent variable in the given Cauchy-Euler equation; you then get a constant-coeffiecient equation with independent variable $x$. In class on Monday 3/13 I used $x$ as the the independent variable in the given Cauchy-Euler equation, and therefore used the substitution $x=e^t$, yielding a constant-coeffiecient 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.
Do non-book problem 10.
F 3/17/17
4.7/ 24cd, 27. 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).
Read Sections 6.1 and 6.2. (We're not done with Chapter 4 yet; we're just covering certain Chapter 6 material first.)
6.1/ 1–6

M 3/20/17
6.2/ 1, 9, 11, 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.

W 3/22/17
6.2/ 13.
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.

F 3/24/17
4.7/ 37–40, 44.
Redo 4.7/ 24cd using Variation of Parameters.
    Note that it is possible to solve all the DEs in 24cd, 38, and 41–43 either by the Cauchy-Euler substitution ($t=e^x$ in the book's notation; $x=e^t$ in the notation presented in class) 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.

M 3/27/17
Read sections 7.1 and 7.2. If you feel sufficiently prepared by your reading, start on the exercises due Wed. 3/29/17.
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 3/29/17
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). If you feel sufficiently prepared by your reading, start on the exercises due Fri. 3/31/17.

General info New date for third midterm exam: Friday, Apr. 7.

F 3/31/17
7.2/ 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.

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 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.

Read Section 7.4.

M 4/3/17
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 $y(t)$.

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".

W 4/5/17
Read Section 7.6 through Example 5.
7.6/ 1–4.
F 4/7/17 Third midterm exam (assignment is to study for it).
M 4/10/17
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 Wednesday (there are a lot).
W 4/12/17
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.

F 4/14/17
Skim Section 8.1. Carefully read Section 8.2 up through p. 432. 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. The only material in Section 8.2 that should be new to you is the material on analytic functions, which starts on p. 433. I plan to start with this right away on Friday (Apr. 14), and move quickly into Section 8.3.
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.

M 4/17/17
Read Section 8.3.
8.3/ 1, 3, 4, 5–8.
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.

W 4/19/17
8.3/ 11–14, 18, 20–22, 24, 25.
8.4/ 15, 20, 21, 23, 25.

Date due	Section # / problem #'s
F 1/6/17	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. In problems 1–12, you may (for now) ignore the instruction involving the words "linear" and "nonlinear"; that will be part of the next assignment. Do non-book problem 1.
M 1/9/17	For the DEs in 1.1/1, 2, and 4–12, classify each equation as linear or nonlinear. 1.2/ 2–6, 19–22 See Notes on some book problems. "Explicit solution" is synonymous with "solution". In these notes, read from the beginning of "Notes for Students" on p. 3 through the end of the exercise on p. 13 (and do the exercise). 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 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."
W 1/11/17	Read the remainder of Section 2.2 of these notes, but not all at once. Read through the end of Example 2.8 on p. 15 before doing the exercises below, then do all the exercises below, then return and do the rest of the reading. Note: I have added a table of contents for these notes, so most of the page numbers as of 1/9/17 differ by 1 from what they were 1/8/17 and earlier. The new p. 4 is the old p. 3, etc. for all pages beyond that point. I also changed the title of the notes and added something at the end, but nothing in between has changed (yet). 1.2/ 1, 9–12, 14–18, 30 Do non-book problem 2.
F 1/13/17	1.2/ 23–27, 30, 31bc Read Section 1.3. 1.3/ 2, 3 2.2/ 1–5 In my notes, read the portion of Section 2.3 from the beginning through Example 2.25. You may skip the portion of this that the notes say you may skip. I've now set up the notes so that references to examples, equations, definitions, etc., are clickable links. On p. 26, if you click on the "2.23" in "Example 2.23", you'll get taken to Example 2.23 without having to hunt for it.
W 1/18/17	2.2/ 7–14, 17–19, 21, 24, 30. "Solve the equation" means "Find all solutions of the equation". Finish reading Section 2.3 of these notes. Reminder: reading the notes I've written for the class 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 in class; we would not get through all the topics we're supposed to cover. Do non-book problems 3–6. 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 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 "\(\int \ln x\, dx =\frac{1}{x} +C\)", 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, even if this takes a lot of time.
F 1/20/17	No new exercises. In the book, read "Formal Justification of Method" on p. 42. I will go over this in class; it is only part of the justification of why, under hypotheses not stated in Section 2.2, our method for solving separable DEs gives the set of all solutions. Read the following parts of my notes: Section 2.4. Section 2.5, up through the first paragraph on p. 42. Throughout my notes, I recommend that you skip anything that says "Note(s) to instructors". They're not secrets; you're just very unlikely understand them, and some of them are very long.
M 1/23/17	2.3/ 1–6 Read Section 2.3 of the textbook. In Section 2.5 of the notes, read from the point you reached in the last assignment through statement (2.78) on p. 43.
W 1/25/17	2.3/ 7–9, 12–15 (note which variable is which in #13!), 17–20, 22, 28, 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 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". 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 problem #7. Read the remainder of Section 2.5 of the notes. Note: One of the things you'll see in exercise 2.3/33 is that (as indicated above) 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 1/27/17	2.3/ 27a, 30– 32, 35 In my notes, read from the beginning of Section 2.6 through the end of p. 51. In the textbook, read Section 2.4 through the boxed definition "Exact Differential Form" on p. 57. Everywhere in this box, wherever you see the term "differential form", the word "form" should be deleted. This applies also to 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 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. In my notes I talk about "derivative form" and "differential form" of a differential equation. My use of the word "form" in the notes and in class is standard English, whereas when a differential itself (rather than an equation containing a differential) is called a "differential form", the word "form" means something entirely different. (With the latter meaning, "differential forms" are something that are not discussed in any undergraduate-level courses at UF, with the possible exception of the combined graduate/undergraduate course Modern Analysis 2, and occasional special-topics courses.) The paragraph directly below the "Exact Differential Form" box is not part of this 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 in my notes (as is also the case in each occurrence on pp. 55–56), the third sentence uses it with an entirely different meaning. A related correction: In Example 1 on p. 56, 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".
M 1/30/17	Read the remainder of Section 2.4 of the textbook. 2.4/ 1–8. In these problems, you are meant to 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. (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, 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 (not 2.4)/ 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 \(x\) 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). If you feel well-enough prepared from reading the book's section 2.4, I suggest getting a head-start on the exercises that are due Wednesday. Read the following parts of the notes : Section 2.6.1 (starts on p. 52), through the end of the first paragraph on p. 55. (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.) Section 2.6.2 (pp. 56–59). (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. 61–64). Section 2.6.5 through Definition 2.61 (pp. 64–66). (The remainder of Section 2.6.5 is optional reading.)
W 2/1/17	2.4/ 9, 11–14, 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 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 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. Read the following parts of the notes : Section 2.7, minus Example 2.63 (this example, which occupies most of pp. 68–71, is optional reading). Do read Example 2.64 on p. 71. Section 2.8, from p. 71 through the middle of p. 72 (the end of the paragraph that has "Pretend" in boldface), then from the middle of p. 76 (paragraph beginning "To have a name ...") through the end on p. 77. The remainder of Section 2.8 is optional reading. Section 2.9, through Example 2.67 on pp. 80–81. Also read Example 2.69 (pp. 88–89). The remainder of Section 2.9 is optional reading.
F 2/3/17	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 \(\int \sin x \cos x\, dx\) 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. Read The Math Commandments. Optional: If you are interested in seeing a differential \(M dx + N dy\) that passes the exactness test \(M_y=N_x\) 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 2/6/17	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.
W 2/8/17	No new homework.
F 2/10/17	Read Sections 4.1 and 4.2. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.) 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.
M 2/13/17	4.2/ 2–5, 7, 8, 10–17, 26, 27–32, 35, 46ab
W 2/15/17	4.2/ 1, 6, 9, 18–20. 4.3/ 1–18 4.7/ 1–8, 25. 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]."
F 2/17/17	4.3/ 21–26, 28, 32, 33 (students in electrical engineering may do #34 instead of #33)
M 2/20/17	Read Section 4.4.
W 2/22/17	4.3/ 36 4.5 (not 4.4)/ 1–8. Use the "\(y=y_p+y_h\)" approach discussed in class, plus superposition (problem 4.7/30, previously assigned) where necessary, plus your knowledge (from Sections 4.2 and 4.3) of how to solve the associated homogeneous equations for all the DEs in these problems. We have not 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.
F 2/24/17	4.4/ 9 (note that \(-9=-9e^{0t}\) ), 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 If you feel that, based on your reading of Section 4.4 (a previous homework assignment), you understand how to find more \(y_p\)'s than are represented in the problems above, 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 2/27/17	4.4/ 1–8, 13, 15, 16, 18–26, 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. You'll have to do many of these problems based on your reading of Section 4.4; what we've done in class so far doesn't cover most of these exercises. 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. In Monday's class I expect to finish presenting Section 4.4 and Section 4.5. But if I were to assign now only the problems you can do as of today's class (Friday), you'd have far too many problems to do next week. It's important that you get as many done over this weekend as possible. It's not critical that you get them all done this weekend, as long as you get this entire assignment plus the next one done before Wednesday's class. 4.5/ 17–22, 24, 27–30, 41.
W 3/1/17	4.4/ 33–36. 4.5/ 9–16 (read instructions carefully or you'll do a ton of extra work!), 17–22, 24–30, 31–36 (again, read instructions carefully!) , 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. Do these non-book exercises on the Method of Undetermined Coefficients. The answers to these exercises are here.
F 3/3/17	Second midterm exam (assignment is to study for it).
M 3/13/17	No new homework.
W 3/15/17	4.7/ 9–20, 24ab 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. 195, "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 \(t=e^x\). (That's if \(t\) is the independent variable in the given Cauchy-Euler equation; you then get a constant-coeffiecient equation with independent variable \(x\). In class on Monday 3/13 I used \(x\) as the the independent variable in the given Cauchy-Euler equation, and therefore used the substitution \(x=e^t\), yielding a constant-coeffiecient 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. Do non-book problem 10.
F 3/17/17	4.7/ 24cd, 27. 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). Read Sections 6.1 and 6.2. (We're not done with Chapter 4 yet; we're just covering certain Chapter 6 material first.) 6.1/ 1–6
M 3/20/17	6.2/ 1, 9, 11, 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.
W 3/22/17	6.2/ 13. 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.
F 3/24/17	4.7/ 37–40, 44. Redo 4.7/ 24cd using Variation of Parameters. Note that it is possible to solve all the DEs in 24cd, 38, and 41–43 either by the Cauchy-Euler substitution (\(t=e^x\) in the book's notation; \(x=e^t\) in the notation presented in class) 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.
M 3/27/17	Read sections 7.1 and 7.2. If you feel sufficiently prepared by your reading, start on the exercises due Wed. 3/29/17. 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 3/29/17	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). If you feel sufficiently prepared by your reading, start on the exercises due Fri. 3/31/17.
General info	New date for third midterm exam: Friday, Apr. 7.
F 3/31/17	7.2/ 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. 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 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. Read Section 7.4.
M 4/3/17	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 \(y(t)\). 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".
W 4/5/17	Read Section 7.6 through Example 5. 7.6/ 1–4.
F 4/7/17	Third midterm exam (assignment is to study for it).
M 4/10/17	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 Wednesday (there are a lot).
W 4/12/17	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.
F 4/14/17	Skim Section 8.1. Carefully read Section 8.2 up through p. 432. 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. The only material in Section 8.2 that should be new to you is the material on analytic functions, which starts on p. 433. I plan to start with this right away on Friday (Apr. 14), and move quickly into Section 8.3. 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.
M 4/17/17	Read Section 8.3. 8.3/ 1, 3, 4, 5–8. 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.
W 4/19/17	8.3/ 11–14, 18, 20–22, 24, 25. 8.4/ 15, 20, 21, 23, 25.

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Homework Assignments MAP 2302, Section 3875—Elementary Differential Equations Spring 2017

Homework Assignments
MAP 2302, Section 3875—Elementary Differential Equations
Spring 2017