Homework Assignments
MAP 2302, Section 3149—Honors Elementary Differential Equations
Spring 2018

Last updated Wed Apr 18 20:57 EDT 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

W 1/10/18
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. 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.
    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. 9. In this and future assignments from these notes, you should skip all items labeled "Note to instructors".

F 1/12/18
For the DEs in 1.1/ 1, 2, and 4–12, classify each equation as linear or nonlinear.

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. 9 through the end of Section 3.1.2, and do the exercise on p. 13. Also re-read Definition 3.4 on p. 7; I changed it a little in an update that I posted to the website around 1:40 a.m. 1/11/18.
    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. In case you've already started reading beyond the end of the reading that was due 1/10/18, using the version older than the update mentioned above: other than Definition 3.4, the only changes up through the end of Section 3.1.2 that have changed, other than those in notes to instructors, are on p. 13 (starting with Definition 3.6). The changes to notes to instructors have affected where some pages begin or end, though.
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 1/17/18
Read the remainder of Section 3.1.3 of my notes, but not all at once. Read through the end of Example 3.9 on p. 15 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, 14–18, 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 should is much weaker than the Implicit Function Theorem, and should not be called by that name.

F 1/19/18
1.2/ 23–28, 30, 31bc. In 23–27 "unique solution" means "unique maximal solution (in an appropriate region)", in the terminology used in class Wednesday and in my notes.
Read Section 1.3.
1.3/ 2, 3
2.2/ 1–4, 6.
In this updated version of my notes, read the portion of Section 3.1.4 up through Example 3.27 (re-numbered 3.28 on 1/20/18).
      Almost all of the changes since the previous version of my notes are after the end of the last section you were assigned to read. However, I've made some enhancements and a few minor changes to the earlier material:
Definition 3.1 (now on p. 4) has some new wording, but the wording in the previous version is still completely correct.
Section 3 now has roughly a page (p. 3) that precedes Section 3.1, which changes the page-numbers on which items appear (relative to the previous version).
A previously un-numbered equation has been given a number (equation (3.14) on p. 9), which changes the subsequent equation numbers.
I'm keeping the previous version posted on the website to spare you from searching for things you've already read, and so that the page-numbers in previous assignments are still correct. (The links to my notes in previous assignments will take you to the previous version of the notes.)

M 1/22/18
2.2/ 7–14, 17–19, 21, 24 . "Solve the equation" means "Find all (maximal) solutions of the equation".
In my notes, read Definition 3.19 at the beginning of Section 3.1.4; this definition had been accidentally omitted at the time of the previous assignment. Then read from Example 3.29 (numbered 3.28 at the time of the last assignment) through to the end of Section 3.1.4, and read Section 3.1.5.

Do non-book problems 3, 4, and 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.
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.

W 1/24/18
2.2/ 27abc

Do non-book problem 7. Answers to this and some other non-book problems are posted on the "Miscellaneous handouts" page.

In my notes, read Section 3.1.6 and the portion of Section 3.1.7 up through the paragraph on p. 50 that contains equation (3.87).
       Theorem 3.41 in my notes is 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: the book's argument does not quite establish what it purports to, and 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.)

F 1/26/18
2.3/ 1–6
Read Section 2.3 of the textbook.
Read the remainder of Section 3.1.7 of my notes.

M 1/29/18
2.3/ 7–9, 12–15 (note which variable is which in #13!), 17–20, 22, 28, 33.
   When you apply the method introduced 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". 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 #8.
Read Section 3.1.8 of my 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 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.

W 1/31/18
2.3/ 27a, 30– 32, 35
In my notes, read from the beginning of Section 3.2 up through, but not including, the beginning of Section 3.2.1 on p. 67.
In the textbook, read Section 2.4 through the boxed definition "Exact Differential Form" on p. 59. 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 p. 58), the third sentence uses it with an entirely different meaning.
    A related correction: 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".

F 2/2/18
Read Sections 3.2.1, 3.2.2, and 3.2.3 of my notes. A couple of paragraphs were added on p. 66 since the last assignment was posted (the paragraph after Definition 3.53 and the paragraph after Example 3.54), so Section 3.2.1 now starts on p. 68.

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. In the near future, we will define what a solution of a differential-form DE is, and show that equations of the form $h(y)dy=g(x)dx$ can be solved by integrating both sides. Just assume this for now.

M 2/5/18
Read the remainder of Section 2.4 of the textbook.
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 $x(y)$.
2.4 (continued)/ 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)" for now.
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 $x$ 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.
Read Sections 3.2.4 and 3.2.5 of my notes.

W 2/7/18
2.4/ 27a, 28a, 32, 33ab
Do non-book problems 9 and 10.
Read The Math Commandments.

Read the following parts of my notes :

Section 3.2.6. You may treat Example 3.80, and the material between the end of Example 3.80 and the paragraph before Example 3.81, as optional reading. The remainder of this section is required reading.
Sections 3.3 and 3.4. In Section 3.4, you may treat Example 3.85 as optional reading; the remainder is required.

F 2/9/18 First midterm exam (assignment is to study for it).

In case you'd like additional 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 8.
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 and the empty Section 3.5) 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.
If you want to see how the class that took your "practice exam" as a real exam did, go here and here.

M 2/12/18 Read section 4.1 of the book. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)

W 2/14/18
Re-read Example 3.17 in my notes. See also Example 3.13 and the paragraph before Example 3.16. The entirety of Section 3.1.3 (which houses these items) was required reading in earlier homework assignments. It is very unwise to treat any portion of required homework as optional. To make problem 6 on your exam easier, I gave you an algebraic equation and differential equation that were exactly the same as in an example in the notes, but all you needed to retain from your reading were the ideas (including the definition of "implicit solution") and how to apply them; you didn't need to memorize Example 3.17 in order to do the exam problem.
Re-read your class notes regarding constant solutions. These came up first when we discussed separable derivative-form equations (for which we spent a lot of time on constant solutions), and later in the setting of differential-form equations. We discussed constant solutions both as part of the general solution and as potential solutions to initial-value problems. We covered an example in which assuming that the solution of an initial-value problem took the form that we had derived for non-constant solutions led to a contradiction, when in fact the solution of that IVP was one of the constant solutions of the DE.
    It wouldn't hurt you to re-read the portions of my notes relating to constant solutions, either. (Most, but not all, of this was in the section on separable equations.)
Re-read the "Fundamental Theorem of Ordinary Differential Equations" (FTODE), a weak version of which is the book's Theorem 1 on p. 11, and a more useful version of which is Theorem 5.1 in my notes. The FTODE is one of several theorems you're taught whose proof is beyond the level of Calc 1-2-3 or an intro DE course. But you're not being taught false theorems. Theorems that are stated in the book, or in class, or in my notes, are true facts, and you're expected to be able to use those facts even though you won't have the tools learn know why they're true until (and unless) you take Advanced Calculus. The same is true of many theorems you were taught in Calculus 1—for example, the Intermediate Value Theorem, the Extreme Value Theorem, and part of the Fundamental Theorem of Calculus—but you are held responsible for the content of those theorems as well.
    One of the things the FTODE tells you (whichever version you use) is that, if certain hypotheses are satisfied, an initial-value problem does have some solution. Therefore if you're given an IVP for which these hypotheses are met, and you think you've figured out that there is no solution, you should know that you've made a mistake.

F 2/16/18
Read Section 4.2. Unfortunately, hardly any of the exercises are doable until the whole section has been covered. 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 2/19/18
4.2/ 1–20, 26, 27–32, 35, 46ab

W 2/21/18
4.7 (yes, 4.7)/ 1–8, 25. First read from the bottom of p. 192 (Theorem 5, which was stated in class) 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]."
Read Section 4.3.
4.3/ 1–18

F 2/23/18
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.

M 2/26/18
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.

W 2/28/18
Read the remainder of Section 4.4.

F 3/2/18
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/ 1–8. These require you to have finished reading Section 4.4, which was assigned in earlier homework, but do not require solving any equations.
4.5/ 9–16. The same comment applies, but see note below.
Note: We are 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 think of Section 4.5 simply as completing the (second-order case of) the method of undetermined coefficients.
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. Over the next few classes, I will be assigning almost all the exercises in sections 4.4 and 4.5. To help avoid giving you one massive assignment when we're done with 4.4 and 4.5, I've put 4.4/ 1–8 and 4.5/ 9–16 into the current assignment.

M 3/12/18 Spring break is meant to be a break. Enjoy it! It's okay if you don't get this assignment done by Mar. 12. I'm putting these problems into this assignment to help you pace yourself, since you will need to finish them, as well as the problems in the next assignment, shortly after spring break.

4.4/ 13, 15–17, 19–26, 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.

W 3/14/18
4.4/ 33–36.
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.

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

4.5/ 45. This is a nice problem that requires you to combine several things you've learned. The strategy is similar to the approach in Exercise 41. Because of the "piecewise-expressed" nature of the right-hand side of the DE, there is a sub-problem on each of three intervals: $I_{\rm left}= (-\infty, -\frac{L}{2V}\,] $, $I_{\rm mid} = [-\frac{L}{2V}, \frac{L}{2V}] $, $I_{\rm right}= [\frac{L}{2V}, \infty) $. The solution $y(t)$ defined on the whole real line restricts to solutions $y_{\rm left}, y_{\rm mid}, y_{\rm right}$ on these intervals.
    You are given that $y_{\rm left}$ is identically zero. Use the terminal values $y_{\rm left}(- \frac{L}{2V}), {y_{\rm left}}'(- \frac{L}{2V})$, as the initial values $y_{\rm mid}(- \frac{L}{2V}), {y_{\rm mid}}'(- \frac{L}{2V})$. You then have an IVP to solve on $I_{\rm mid}$. For this, first find a "particular" solution on this interval using the Method of Undetermined Coefficients (MUC). Then, use this to obtain the general solution of the DE on this interval; this will involve constants $ c_1, c_2$. Using the IC's at $t=- \frac{L}{2V}$, you obtain specific values for $c_1$ and $c_2$, and plugging these back into the general solution gives you the solution $y_{\rm mid}$ of the relevant IVP on $I_{\rm mid}$.
    Now compute the terminal values $y_{\rm mid}(\frac{L}{2V}), {y_{\rm mid}}'(\frac{L}{2V})$, and use them as the initial values $y_{\rm right}(\frac{L}{2V}), {y_{\rm right}}'(\frac{L}{2V})$. You then have a new IVP to solve on $I_{\rm right}$. The solution, $y_{\rm right}$, is what you're looking for in part (a) of the problem.
    If you do everything correctly (which may involve some trig identities, depending on how you do certain steps), under the book's simplifying assumptions $m=k=F_0=1$ and $L=\pi$, you will end up with just what the book says: $y_{\rm right}(t) = A\sin t$, where $A=A(V)$ is a $V$-dependent constant (i.e. constant as far as $t$ is concerned, but a function of the car's speed $V$). In part (b) of the problem you are interested in the function $|A(V)|$, which you may use a graphing calculator or computer to plot. The graph is very interesting.
    Note: When using MUC to find a particular solution on $I_{\rm mid}$, you have to handle the cases $V\neq 1$ and $V = 1$ separately. (If we were not making the simplifying assumptions $m = k = 1$ and $L=\pi$, these two cases would be $\frac{\pi V}{L}\neq \sqrt{\frac{k}{m}}$ and $\frac{\pi V}{L}= \sqrt{\frac{k}{m}}$, respectively.) In the notation used in the last couple of lectures, using $s$ for the multiplicity of a certain number as a root of the characteristic polynomial, $V\neq 1$ puts you in the $s= 0$ case, while $V = 1$ puts you in the $s= 1$ case.

F 3/16/18
4.4/ 12. This problem 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). 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.

M 3/19/18 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, and possibly those in which the DE is 3rd-order. 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.
    You can solve the order-3 constant-coefficient DEs on p. 231 if you succeed in factoring 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. 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.)

W 3/21/18
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.

F 3/23/18
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.
Do non-book problem 11.
Read Section 4.6.

M 3/26/18
4.6/ 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.

W 3/28/18
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 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/ 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.)
6.1/ 1–6, 7–14. 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.

F 3/30/18
6.1/ 19, 20, 23.

Read Section 6.2.
6.2/ 1, 9, 11, 13, 15–18. The characteristic polynomial for #9 is a perfect cube (i.e. $ (r-r_1)^3$ for some $r_1$); for #11 it's a perfect fourth power.
6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the annihilator method (which we are skipping for reasons of time); just use what we've done in class with MUC and superposition.

M 4/2/18 No new homework.

W 4/4/18
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.
There is a long Laplace Transform table that's located on or near the inside front cover or inside back cover of the book. Find where the entries of Table 7.1 (p. 356) are located in this longer table. 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 found in it. The table you'll be given on the remaining exams will be essentially this longer table with most or all of the entries beyond line 27 removed, and possibly a few of the earlier entries removed.

F 4/6/18
7.2/ 1–4, 6–8, 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; we'll derive most or all of this table in class), 21–23, 26–28, 29a–d,f,g,j.

M 4/9/18

Read Section 7.3. In class, I'll fill in a gap in the book's proof of Theorem 4 (the fact that integration by parts works for functions that are piecewise continuously differentiable, a fact you are unlikely ever to have seen before).
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.

W 4/11/18
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 $y(t)$. (This is why I've grouped them separately from the other exercises from Section 7.5 below.)

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".

Read Section 7.6.
7.6/ 1–4.
F 4/13/18 No new homework.

M 4/16/18 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 7.5 of the textbook (where "covered" includes classwork and homework, and "homework" includes reading the relevant portions of the textbook, with the following exceptions:
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) $."
In Section 6.3, you are not responsible for the terminology "annihilate" or "annihilator", or for the "annihilator method". You are responsible for begin able to use the Method of Undetermined Coefficients, as presented in class, to solve DEs such as the ones in the homework problems assigned from this section.
In Section 7.5, on this exam you are not responsible for the material involving non-constant-coefficient IVPs, such as Example 4.
You are responsible for the definition of "fundamental set of solutions" that I gave in class.

W 4/18/18
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.

F 4/20/18
Skim Section 8.1. Carefully read Section 8.2 up through p. 431. 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. 432. I plan to start with this right away on Friday (Apr. 20), 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:
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. 427, "open interval of convergence" means the set $ \{x: |x-x_0|<\rho\}$. Don't spend time working out whether these series converge at the endpoints of these intervals. For this class, 100% of the way we'll apply power-series ideas to solving DEs involves only the open interval of convergence.
The instructions for problems 23–26, as well as for Example 3 on p. 430, 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 call 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 and exercises 8.2/ 27, 28.

M 4/23/18
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. 445, 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. 432) 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.

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

Date due	Section # / problem #'s
W 1/10/18	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. 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. 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. 9. In this and future assignments from these notes, you should skip all items labeled "Note to instructors".
F 1/12/18	For the DEs in 1.1/ 1, 2, and 4–12, classify each equation as linear or nonlinear. 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. 9 through the end of Section 3.1.2, and do the exercise on p. 13. Also re-read Definition 3.4 on p. 7; I changed it a little in an update that I posted to the website around 1:40 a.m. 1/11/18. 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. In case you've already started reading beyond the end of the reading that was due 1/10/18, using the version older than the update mentioned above: other than Definition 3.4, the only changes up through the end of Section 3.1.2 that have changed, other than those in notes to instructors, are on p. 13 (starting with Definition 3.6). The changes to notes to instructors have affected where some pages begin or end, though. 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 1/17/18	Read the remainder of Section 3.1.3 of my notes, but not all at once. Read through the end of Example 3.9 on p. 15 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, 14–18, 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 should is much weaker than the Implicit Function Theorem, and should not be called by that name.
F 1/19/18	1.2/ 23–28, 30, 31bc. In 23–27 "unique solution" means "unique maximal solution (in an appropriate region)", in the terminology used in class Wednesday and in my notes. Read Section 1.3. 1.3/ 2, 3 2.2/ 1–4, 6. In this updated version of my notes, read the portion of Section 3.1.4 up through Example 3.27 (re-numbered 3.28 on 1/20/18). Almost all of the changes since the previous version of my notes are after the end of the last section you were assigned to read. However, I've made some enhancements and a few minor changes to the earlier material: Definition 3.1 (now on p. 4) has some new wording, but the wording in the previous version is still completely correct. Section 3 now has roughly a page (p. 3) that precedes Section 3.1, which changes the page-numbers on which items appear (relative to the previous version). A previously un-numbered equation has been given a number (equation (3.14) on p. 9), which changes the subsequent equation numbers. I'm keeping the previous version posted on the website to spare you from searching for things you've already read, and so that the page-numbers in previous assignments are still correct. (The links to my notes in previous assignments will take you to the previous version of the notes.)
M 1/22/18	2.2/ 7–14, 17–19, 21, 24 . "Solve the equation" means "Find all (maximal) solutions of the equation". In my notes, read Definition 3.19 at the beginning of Section 3.1.4; this definition had been accidentally omitted at the time of the previous assignment. Then read from Example 3.29 (numbered 3.28 at the time of the last assignment) through to the end of Section 3.1.4, and read Section 3.1.5. Do non-book problems 3, 4, and 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. 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.
W 1/24/18	2.2/ 27abc Do non-book problem 7. Answers to this and some other non-book problems are posted on the "Miscellaneous handouts" page. In my notes, read Section 3.1.6 and the portion of Section 3.1.7 up through the paragraph on p. 50 that contains equation (3.87). Theorem 3.41 in my notes is 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: the book's argument does not quite establish what it purports to, and 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.)
F 1/26/18	2.3/ 1–6 Read Section 2.3 of the textbook. Read the remainder of Section 3.1.7 of my notes.
M 1/29/18	2.3/ 7–9, 12–15 (note which variable is which in #13!), 17–20, 22, 28, 33. When you apply the method introduced 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". 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 #8. Read Section 3.1.8 of my 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 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.
W 1/31/18	2.3/ 27a, 30– 32, 35 In my notes, read from the beginning of Section 3.2 up through, but not including, the beginning of Section 3.2.1 on p. 67. In the textbook, read Section 2.4 through the boxed definition "Exact Differential Form" on p. 59. 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 p. 58), the third sentence uses it with an entirely different meaning. A related correction: 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".
F 2/2/18	Read Sections 3.2.1, 3.2.2, and 3.2.3 of my notes. A couple of paragraphs were added on p. 66 since the last assignment was posted (the paragraph after Definition 3.53 and the paragraph after Example 3.54), so Section 3.2.1 now starts on p. 68. 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. In the near future, we will define what a solution of a differential-form DE is, and show that equations of the form \(h(y)dy=g(x)dx\) can be solved by integrating both sides. Just assume this for now.
M 2/5/18	Read the remainder of Section 2.4 of the textbook. 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 \(x(y)\). 2.4 (continued)/ 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)" for now. 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 \(x\) 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. Read Sections 3.2.4 and 3.2.5 of my notes.
W 2/7/18	2.4/ 27a, 28a, 32, 33ab Do non-book problems 9 and 10. Read The Math Commandments. Read the following parts of my notes : Section 3.2.6. You may treat Example 3.80, and the material between the end of Example 3.80 and the paragraph before Example 3.81, as optional reading. The remainder of this section is required reading. Sections 3.3 and 3.4. In Section 3.4, you may treat Example 3.85 as optional reading; the remainder is required.
F 2/9/18	First midterm exam (assignment is to study for it). In case you'd like additional 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 8. 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 and the empty Section 3.5) 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. If you want to see how the class that took your "practice exam" as a real exam did, go here and here.
M 2/12/18	Read section 4.1 of the book. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)
W 2/14/18	Re-read Example 3.17 in my notes. See also Example 3.13 and the paragraph before Example 3.16. The entirety of Section 3.1.3 (which houses these items) was required reading in earlier homework assignments. It is very unwise to treat any portion of required homework as optional. To make problem 6 on your exam easier, I gave you an algebraic equation and differential equation that were exactly the same as in an example in the notes, but all you needed to retain from your reading were the ideas (including the definition of "implicit solution") and how to apply them; you didn't need to memorize Example 3.17 in order to do the exam problem. Re-read your class notes regarding constant solutions. These came up first when we discussed separable derivative-form equations (for which we spent a lot of time on constant solutions), and later in the setting of differential-form equations. We discussed constant solutions both as part of the general solution and as potential solutions to initial-value problems. We covered an example in which assuming that the solution of an initial-value problem took the form that we had derived for non-constant solutions led to a contradiction, when in fact the solution of that IVP was one of the constant solutions of the DE. It wouldn't hurt you to re-read the portions of my notes relating to constant solutions, either. (Most, but not all, of this was in the section on separable equations.) Re-read the "Fundamental Theorem of Ordinary Differential Equations" (FTODE), a weak version of which is the book's Theorem 1 on p. 11, and a more useful version of which is Theorem 5.1 in my notes. The FTODE is one of several theorems you're taught whose proof is beyond the level of Calc 1-2-3 or an intro DE course. But you're not being taught false theorems. Theorems that are stated in the book, or in class, or in my notes, are true facts, and you're expected to be able to use those facts even though you won't have the tools learn know why they're true until (and unless) you take Advanced Calculus. The same is true of many theorems you were taught in Calculus 1—for example, the Intermediate Value Theorem, the Extreme Value Theorem, and part of the Fundamental Theorem of Calculus—but you are held responsible for the content of those theorems as well. One of the things the FTODE tells you (whichever version you use) is that, if certain hypotheses are satisfied, an initial-value problem does have some solution. Therefore if you're given an IVP for which these hypotheses are met, and you think you've figured out that there is no solution, you should know* that you've made a mistake*.
F 2/16/18	Read Section 4.2. Unfortunately, hardly any of the exercises are doable until the whole section has been covered. 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 2/19/18	4.2/ 1–20, 26, 27–32, 35, 46ab
W 2/21/18	4.7 (yes, 4.7)/ 1–8, 25. First read from the bottom of p. 192 (Theorem 5, which was stated in class) 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]." Read Section 4.3. 4.3/ 1–18
F 2/23/18	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.
M 2/26/18	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.
W 2/28/18	Read the remainder of Section 4.4.
F 3/2/18	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/ 1–8. These require you to have finished reading Section 4.4, which was assigned in earlier homework, but do not require solving any equations. 4.5/ 9–16. The same comment applies, but see note below. Note: We are 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 think of Section 4.5 simply as completing the (second-order case of) the method of undetermined coefficients. 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. Over the next few classes, I will be assigning almost all the exercises in sections 4.4 and 4.5. To help avoid giving you one massive assignment when we're done with 4.4 and 4.5, I've put 4.4/ 1–8 and 4.5/ 9–16 into the current assignment.
M 3/12/18	Spring break is meant to be a break. Enjoy it! It's okay if you don't get this assignment done by Mar. 12. I'm putting these problems into this assignment to help you pace yourself, since you will need to finish them, as well as the problems in the next assignment, shortly after spring break. 4.4/ 13, 15–17, 19–26, 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.
W 3/14/18	4.4/ 33–36. 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. Do these non-book exercises on the Method of Undetermined Coefficients. The answers to these exercises are here. 4.5/ 45. This is a nice problem that requires you to combine several things you've learned. The strategy is similar to the approach in Exercise 41. Because of the "piecewise-expressed" nature of the right-hand side of the DE, there is a sub-problem on each of three intervals: \(I_{\rm left}= (-\infty, -\frac{L}{2V}\,] \), \(I_{\rm mid} = [-\frac{L}{2V}, \frac{L}{2V}] \), \(I_{\rm right}= [\frac{L}{2V}, \infty) \). The solution \(y(t)\) defined on the whole real line restricts to solutions \(y_{\rm left}, y_{\rm mid}, y_{\rm right}\) on these intervals. You are given that \(y_{\rm left}\) is identically zero. Use the terminal values \(y_{\rm left}(- \frac{L}{2V}), {y_{\rm left}}'(- \frac{L}{2V})\), as the initial values \(y_{\rm mid}(- \frac{L}{2V}), {y_{\rm mid}}'(- \frac{L}{2V})\). You then have an IVP to solve on \(I_{\rm mid}\). For this, first find a "particular" solution on this interval using the Method of Undetermined Coefficients (MUC). Then, use this to obtain the general solution of the DE on this interval; this will involve constants \( c_1, c_2\). Using the IC's at \(t=- \frac{L}{2V}\), you obtain specific values for \(c_1\) and \(c_2\), and plugging these back into the general solution gives you the solution \(y_{\rm mid}\) of the relevant IVP on \(I_{\rm mid}\). Now compute the terminal values \(y_{\rm mid}(\frac{L}{2V}), {y_{\rm mid}}'(\frac{L}{2V})\), and use them as the initial values \(y_{\rm right}(\frac{L}{2V}), {y_{\rm right}}'(\frac{L}{2V})\). You then have a new IVP to solve on \(I_{\rm right}\). The solution, \(y_{\rm right}\), is what you're looking for in part (a) of the problem. If you do everything correctly (which may involve some trig identities, depending on how you do certain steps), under the book's simplifying assumptions \(m=k=F_0=1\) and \(L=\pi\), you will end up with just what the book says: \(y_{\rm right}(t) = A\sin t\), where \(A=A(V)\) is a \(V\)-dependent constant (i.e. constant as far as \(t\) is concerned, but a function of the car's speed \(V\)). In part (b) of the problem you are interested in the function \(\|A(V)\|\), which you may use a graphing calculator or computer to plot. The graph is very interesting. Note: When using MUC to find a particular solution on \(I_{\rm mid}\), you have to handle the cases \(V\neq 1\) and \(V = 1\) separately. (If we were not making the simplifying assumptions \(m = k = 1\) and \(L=\pi\), these two cases would be \(\frac{\pi V}{L}\neq \sqrt{\frac{k}{m}}\) and \(\frac{\pi V}{L}= \sqrt{\frac{k}{m}}\), respectively.) In the notation used in the last couple of lectures, using \(s\) for the multiplicity of a certain number as a root of the characteristic polynomial, \(V\neq 1\) puts you in the \(s= 0\) case, while \(V = 1\) puts you in the \(s= 1\) case.
F 3/16/18	4.4/ 12. This problem 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). 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.
M 3/19/18	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, and possibly those in which the DE is 3rd-order. 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. You can solve the order-3 constant-coefficient DEs on p. 231 if you succeed in factoring 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. 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.)
W 3/21/18	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.
F 3/23/18	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. Do non-book problem 11. Read Section 4.6.
M 3/26/18	4.6/ 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.
W 3/28/18	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 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/ 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.) 6.1/ 1–6, 7–14. 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.
F 3/30/18	6.1/ 19, 20, 23. Read Section 6.2. 6.2/ 1, 9, 11, 13, 15–18. The characteristic polynomial for #9 is a perfect cube (i.e. \( (r-r_1)^3\) for some \(r_1\)); for #11 it's a perfect fourth power. 6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the annihilator method (which we are skipping for reasons of time); just use what we've done in class with MUC and superposition.
M 4/2/18	No new homework.
W 4/4/18	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. There is a long Laplace Transform table that's located on or near the inside front cover or inside back cover of the book. Find where the entries of Table 7.1 (p. 356) are located in this longer table. 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 found in it. The table you'll be given on the remaining exams will be essentially this longer table with most or all of the entries beyond line 27 removed, and possibly a few of the earlier entries removed.
F 4/6/18	7.2/ 1–4, 6–8, 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; we'll derive most or all of this table in class), 21–23, 26–28, 29a–d,f,g,j.
M 4/9/18	Read Section 7.3. In class, I'll fill in a gap in the book's proof of Theorem 4 (the fact that integration by parts works for functions that are piecewise continuously differentiable, a fact you are unlikely ever to have seen before). 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.
W 4/11/18	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 \(y(t)\). (This is why I've grouped them separately from the other exercises from Section 7.5 below.) 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". Read Section 7.6. 7.6/ 1–4.
F 4/13/18	No new homework.
M 4/16/18	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 7.5 of the textbook (where "covered" includes classwork and homework, and "homework" includes reading the relevant portions of the textbook, with the following exceptions: 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 \( (a,b) \)." In Section 6.3, you are not responsible for the terminology "annihilate" or "annihilator", or for the "annihilator method". You are responsible for begin able to use the Method of Undetermined Coefficients, as presented in class, to solve DEs such as the ones in the homework problems assigned from this section. In Section 7.5, on this exam you are not responsible for the material involving non-constant-coefficient IVPs, such as Example 4. You are responsible for the definition of "fundamental set of solutions" that I gave in class.
W 4/18/18	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.
F 4/20/18	Skim Section 8.1. Carefully read Section 8.2 up through p. 431. 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. 432. I plan to start with this right away on Friday (Apr. 20), 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: 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. 427, "open interval of convergence" means the set \( \{x: \|x-x_0\|<\rho\}\). Don't spend time working out whether these series converge at the endpoints of these intervals. For this class, 100% of the way we'll apply power-series ideas to solving DEs involves only the open interval of convergence. The instructions for problems 23–26, as well as for Example 3 on p. 430, 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 call 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 and exercises 8.2/ 27, 28.
M 4/23/18	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. 445, 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. 432) 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.
W 4/25/18	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 3149—Honors Elementary Differential Equations Spring 2018

Homework Assignments
MAP 2302, Section 3149—Honors Elementary Differential Equations
Spring 2018