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 a few hours after class or later that night.
Assignments with due-dates later than the next lecture
are estimates. In particular,
problems or reading not
currently listed for a future assignment may be added
by the time that assignment is finalized, and
due dates for
particular exercises, reading, or entire assignments,
may end up being
moved either forward or back
(but
not moved back to an assignment whose due-date
has already passed).
Note that on any given due-date there
may be problems due from more than one section of the book.
It is critical that you keep up with the homework daily. Far too
much homework will be assigned for you to catch up after a several-day
lapse, even if your past experience makes you think that you'll be
able to do this. I cannot stress this strongly enough. Students
who do not keep up with the homework frequently receive D's or worse
(or drop the class to avoid receiving such a grade). Every time I
teach this class, there are students who make the mistake of thinking
that this advice does not apply to them. No matter
how good a student you are, or what your past experiences
have been, this advice applies to YOU. Yes, YOU.
A great many students don't do as well as they'd
hoped, for reasons that can be chalked up to not following their
instructors' best advice from the start. Much of my advice (and
the book's) will require more time, and more consistent effort, than
you're used to putting into your classes. It's easy to dig yourself
into a hole by thinking, "I've never had to work after every single
class, or put in as many hours as following
advice like this would take, and
I've always done well. And the same goes for my friends. So I'll just
continue to approach my math classes the way I've always done." By
the time a student realizes that this plan isn't working, and asks his
or her professor "What can I do to improve?" it's usually too late to
make a big difference.
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. (Or use the opportunity to get ahead in
your other classes, so that you'll have more time available when I do
give you a longer assignment.)
Unless otherwise indicated, problems are from our textbook
(Nagle, Saff, & Snider) 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.
In the table below, "NSS" stands for our textbook. Exercises are
from NSS unless otherwise specified.
| Date due |
Section # / problem #'s |
| W 1/15/25 |
Read
the class home page and
syllabus webpages.
Go to the Miscellaneous Handouts page (linked to the
class home page) and read the web handouts
"Taking and Using
Notes in a College Math Class," "Sets and Functions,", and
"What is a solution?"
Never treat any reading portion
of any assignment as optional, or as something you're sure
you already know, or as something you can postpone
(unless I tell you otherwise)! I can pretty much guarantee that
every one of my handouts has something in it that you don't know,
no matter how low-level the handout may appear to be at
first.
Read Section 1.1 and do problems 1.1/ 1–16.
Since not everyone may have access to the
textbook yet, here is a
scan of the first 15 pages (Sections
1.1–1.2, including all the exercises).
Do non-book problem 1.
In my notes on first-order
ODEs (also linked to the Miscellaneous Handouts page), read
the first three paragraphs of the introduction, all of Section
3.1, and Section 3.2.1 through Definition 3.1. In
all readings I assign from these notes,
you should skip anything labeled "Note(s) to instructors".
Whenever I update these notes (whether
substantively or just to
fix typos), I update the
version-date line on p. 1. Each time
you're going to look at the notes,
re-load them to make sure that you're looking at
the latest version.
|
| F 1/17/25 |
1.2/ 1, 3–6, 17, 19–22.
Something I didn't have time to talk
about in Wednesday's class: whenever you
see the term "explicit solution" in the book, you should
(mentally) delete the word "explicit".
(Until the third author was added to later
editions of the textbook, what NSS now calls an explicit
solution is exactly what it had previously called, simply and
correctly,a solution.. The authors tried to "improve" the
completely standard meaning of "solution of a DE". They did not
succeed.
See Notes on some book
problems for additional corrections to the wording of several
of the Section 1.2 problems.
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.
However, since my posted notes are only
on first-order ODEs, the reading portions of the
assignments will become much lighter once we're finished with
first-order equations (which will take the first month or so of
the semester).
In
my notes, read from where you left off in the last assignment
through Example 3.11 (p. 15).
|
| W 1/22/25 |
In the textbook, read the
first page of Section 2.2, minus the last sentence.
(We will discuss how to solve separable
equations after we've finished discussing linear equations, the topic
of Section 2.3. The only reason I'm
having you read the first page of Section 2.2 now is so that you can
do the first few exercises of Section 2.3. But as a "bonus", you'll also
be able to do the exercises in Section 2.2 assigned
below.)
2.2/ 1–4, 6
2.3/ 1–6
In
my notes, read from where you left off in the last assignment
through the one-sentence paragraph after Definition 3.19. (Exception:
you may treat the blue Remark 3.16 as optional reading.)
|
| F 1/24/25 |
In Section 2.3, read up through p. 50, but mentally
makes some modifications:
- Replace the book's transition from
equation (6) to equation (7) by what I said in class: that
\(\mu'/\mu
=\frac{d}{dx} \ln |\mu|.\) (The book's reference to separable
equations is unnecessary, and does not lead directly to equation
(7); it leads to a similar equation but with \(|\mu(x)|\) on the
left-hand side, just as my argument on
Wednesday 1/22 did. On Friday 1/24, I'll go over why we can get rid of
the absolute-value symbols in this setting. Equation (7) itself
is fine, modulo the meaning of indefinite-integral notation;
it's only the book's derivation that has problems.)
-
Remember that whenever you see an indefinite
integral in the book, e.g. \(\int f(x)\, dx,\) the meaning is
my "\(\int_{\rm spec} f(x)\, dx\)." To review
what I said about notation for indefinite integrals, go to the
my Spring 2024 homework page,
locate the assignment that was due 1/19/24, and in the first
bullet-point, read from the beginning of the second sentence ("Remember
...") to the end of the green text.
In
my notes, read from where you left off in the last assignment
through the end of Section 3.2.4 (the middle of p. 27). For a while,
the readings I'm assigning will be beyond what we've gotten to yet in
class. On any of the topics in my notes, there's simply too much to
read for me to assign it all at once when we get to that topic in
class. There is also simply too much there to talk about all of it
in class. Much of it is material that could and should
have been taught in Calculus 1 (and that used to be
taught there).
In Wednesday's class I didn't get quite far enough to
write a clean summary of the method we had effectively
derived, but the box on p. 50 serves that purpose (modulo the
indefinite-integral notation). Armed with this, you should
be able to do most of the exercises I'll be assigning from
Section 2.3, but I'm putting these exercises in to
the next assignment since I didn't get quite far
enough. However, I recommend
that,
by Friday
1/24, you do as many of these exercises as you can, so that
the next assignment isn't extra-long.
If you want to read more examples
before starting on the exercises, it's
okay to look at Section 2.3's Examples 1–3, but be warned:
Examples 1 and 2 have some extremely poor writing that you
probably won't realize is poor, and that
reinforces certain bad habits that
most students have, but few are aware of. (Example 3 is
better written, but shouldn't be read before the other two.)
Specific
problems with Example 2 (one of which also occurs in Example
1) are discussed in the same
Spring 2024 assignment as above. The most pervasive of these
is the one in last small-font paragraph in that assignment.
|
| M 1/27/25 |
2.3/ 7–9,
12–15 (note which variable is which in #13!),
17–20
When you apply the integrating-factor
method
don't forget
the first step: writing the equation in "standard linear form",
equation (15) in the book. (If the original DE had an \(a_1(x)\)
multiplying \(\frac{dy}{dx}\) — even
a constant function other than 1—you have to divide
through by \(a_1(x)\) before you can use the formula for \(\mu(x)\) in
the box on p. 50; otherwise the method doesn't work.) 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 \(\frac{1}{x}\).
One thing I didn't get to say in Friday's
lecture:
The general solution of any
derivative-form
DE—the set of all maximal solutions—is the same
as the set of maximal solutions of all possible IVP's for that DE.
Hence, by figuring out all solutions to all IVPs for the DE, we are
simultaneously figuring out the general solution of the DE.
2.3 (continued)/ 22, 23, 25a, 27a, 28, 31, 33, 35
See my Spring 2024 homework
page, assignment due 1/22/24, for corrections to some of the
Section 2.3 exercises. Also, in that same assignment read the three
paragraphs at the bottom of the assignment.
(The reason I'm not simply recopying such items into this semester's
homework page is that some of my 2023-2024 students said that,
although my comments and corrections were intended to be helpful, they
made the look of those assignments
overwhelming.)
2.2/ 34. Although this exercise is in the section on
"Separable Equations" (which we haven't discussed yet), the DE
happens to be linear as well as separable, so you're equipped to
solve it. For solving this equation, the "linear equations
method" is actually simpler than—I would even say
better than—the (not yet discussed) "separable
equations method".
(The
same is true of Section 2.1's equation (1), which the book solves
by the "separable equations method"—and makes two
mistakes in the sentence containing equation (4). This is why
I did not assign you to read Section 2.1.)
Do non-book problem
2.
In
my notes, read from the
beginning of Section 3.2.5 (p. 27) through the end of
Definition 3.23 (p. 33), plus the paragraph after that definition.
(All of this is needed for a
proper understanding of the word "determines" in the book's Definition
2 in its Section 1.2! [I still haven't defined "implicit solution of a
DE" yet; the above reading is needed just to understand the single
word "determines" in that definition.] This is one of the biggest
reasons I didn't assign you to read Section 1.2.)
Then
do the exercise that's shortly after Definition 3.23 in my notes.
|
| W 1/29/25 |
In
my notes:
- Read the remainder of Section 3.2.5 (pp. 33–34).
- Read Section 5.5 (The Implicit Function
Theorem).
- In Section 3.2.6, read up through the end of Example 3.27
(pp. 37–38).
After you've done that reading,
do the following exercises from the
textbook: 1.2/ 2, 9–12, 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
is much weaker than the Implicit Function Theorem, and should
not be called by that name. In fact, problem 30 cannot even be done using
the book's theorem, because of the the words "near the point (0,1)"
at the end of the problem.
Note that the next assignment includes a more
challenging version of problems 9–12. Since the reading
portion of the current assignment is substantial (if you take the
time
needed to understand what you're reading), I didn't want to
make the exercise portion too time-consuming.
However, if you already feel up to tackling the more-challenging
versions of these problems, then, by all means, get started on them early!
|
| F 1/31/25 |
Re-do 1.2/ 9–12 without the book's instruction
to assume that "the relationship does define \(y\) as a function
of \(x\)."
In my notes, read
Sections 5.2 and 5.4.
(If you have any uncertainty
about what an interval is, read Section 5.1 as well.
If you need to review
anything about the Fundamental Theorem of Calculus, read Section 5.3.)
My notes' Theorem
5.8, the "FTODE", is what the textbook's Theorem 1 on
p. 11 should have said (modulo my having used
"open set"
in the FTODE instead of the book's "open rectangle").
1.2/ 18, 23–28, 31. Do not do these until
after you've read Section 5.4 in my notes.
Anywhere that the book asks you whether its Theorem 1 implies
something, replace that Theorem 1 with the FTODE stated in my notes.
See my Fall 2024 homework
page, assignment due 9/11/24, for corrections to some of these
exercises, and some other brief comments.
In my notes, read
Section 3.2.6 up through the
paragraph before Example 3.31 on p. 42.
Reminder: reading my notes is not optional
(except for portions that I [or the notes] say you may
skip, and the footnotes or parenthetic comments that say "Note to
instructor(s)").
You should do
your best to complete each reading assignment by the due date I
give you. If you let yourself fall
significantly behind, planning to catch up later, you will
have far too much to absorb in too little
time. What I've put in the notes are things that are
not adequately covered in our textbook (or any current textbook
that I know of). Unfortunately there isn't enough time to go
over most of these carefully in class; we would not get through
all the topics we're supposed to cover.
|
| M 2/3/25 |
Skim Section 2.2 in the textbook, up through Example 3.
I'm always uneasy about having my
students read this section. The book's explanations and
definitions in this section say many of the right
things, but don't hold up under scrutiny, and there's lot of
poor writing that I hate exposing you to. Furthermore, the
most prominent item in the section—the box on
p. 42—is misleading. The correct "method for solving
separable DEs" has two parts, one of which is
the (not quite finished) mechanical method in the box, the
"brain off" method that I illustrated in class. The correct
name for the method in the box on p. 42 is
separation of variables.
Furthermore, this PART of the method for solving
separable DEs has (potentially) one
more step: solving equation (3) explicitly for \(y\) in terms
of \(x\) when possible (as it was in the example I did
Friday's
in class, \(\frac{dy}{dx}= x(y-1)^2\)).
We still have a partial lecture's worth of
conceptual material that's absent from the book, before which
doing the exercises in Section 2.2 would amount to little more than
pushing the symbols around the page a certain way. However,
you do need to start getting some practice with the mechanical
separation-of-variables method; otherwise you'll have too much to
do in too short a time. So I've assigned some exercises from
Section 2.2 below, for you to attempt based on your reading, but
with special temporary instructions.
2.2/ 7–14. For now (with the Monday 2/3
due date), all I want you to do in these exercises is
to achieve
an answer of the form of equation (3) in the box on
p. 42—without worrying about intervals, regions, or exactly what
an equation of this form has to do with (properly
defined) solutions of a DE.
Save your work, so that when I re-assign these exercises later,
at which time your goal will be to get a complete answer that you fully
understand, you won't have to re-do this part of the work.
In my notes, finish
reading Section 3.2.6.
|
| W 2/5/25 |
In my notes:
- Read Section 3.2.10 up through the paragraph after the
statement of Theorem 3.45. (This
one-sentence paragraph explains the notation in equation (3.103)
in Theorem 3.45.) This theorem assures us that,
when its hypotheses are met, every solution of
\(\frac{dy}{dx}=g(x)p(y)\) in the indicated region \(R\) is either
a constant solution or can be found, at least in implicit form, by
separation of variables (the "brain-off" method in the box on
p. 42 of the textbook).
On my Spring 2024 homework
page, go to the assignment that was due 1/26/24, and read the
(whole) second bullet-point (which continues until the end of that
assignment). This details several of the items that are misleading or
just plain wrong in the book's Section 2.2. In the last
non-parenthetic sentence of that assignment, "the method we've
studied" is the method that we've just begun to study this
semester (the method summarized by Theorem 3.45 in my notes, and
justified by the proof of that theorem a few pages later).
Return to exercises 2.2/ 7–14 that I had you partially
do in the previous assignment. Using Theorem 3.45 in my notes,
this time find all the maximal solutions. Don't worry
about graphing the solution-curves for any of the exercises in
the current assignment; that's more than the exercises are asking
for, and would take more time than it's worth.
You are not yet expected to understand
yet why the two-part method given by my notes' Theorem
3.45 works, or to fully understand "implicit solutions". For now, you are just getting practice with
the two-part procedure for solving
separable DEs
(one part being separation
of variables [the box on NSS p. 42], the other being
finding any constant solutions the DE may have [it may not
have any]).
Do non-book problems
3–5 . Although I haven't finished discussing various
subtleties, or justified the separation-of-variables technique yet,
the two-part procedure mentioned above does
find all the solutions of the DEs in this
assignment.
Answers to these
non-book problems are posted on the
"Miscellaneous handouts" page.
General comment. In doing the
exercises from Section 2.2 or the non-book problems you may
find that, often, the hardest part of doing
such problems
is 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).
If you need to review the method of partial fractions,
you can undoubtedly find it online somewhere, but our textbook has
its own review on pp. 370–374. This
review is interspersed with examples related to the topic of
Chapter 7, Laplace Transforms, which we are a long way from
starting to cover. For purposes of simply reviewing
partial fractions, ignore everything in Examples 5, 6, and 7
on these pages except for the partial fractions
computations. (For example, ignore any equation that has a curly
"L" in it.)
2.2/ 17–19, 21, 24
The book's IVP exercises are not rich
enough, by a long shot, to illustrate the dangers
of keeping your brain turned off after you've separated
variables (putting all \(y\)'s on one side of the equation and all
\(x\)'s on the other, if these are the variable-names) and done
the relevant integrals. Non-book problems 7 and 8, which will be
in an upcoming assignment, were constructed to remedy this
poverty. Feel free to tackle these before they're assigned.
|
| F 2/7/25 |
2.2/ 27abc
Do non-book problems
6–8.
Re-do 2.2/ 18 with the initial condition \(y(5)=1.\)
In my notes:
- Read the remainder of Section 3.2.7.
- Read Sections 3.2.8 and 3.2.9. It's okay if you read
one of these sections as part of this assigment, and the
other as part of the next assignment.
- In Section 3.2.10, starting where you left off,
read up through at least the portion of the proof of Theorem
3.45 that ends with statement (3.109).
|
| M 2/10/25 |
In my notes:
- Read the remainder of Section 3.2.10.
- Read Section 3.3.1. With the exception of
the definition of the differential \(dF\) of a two-variable function
\(F\), the material in Section 3.3.1 of my notes
is basically not discussed in the book at all, even though
differential-form DEs appear in (not-yet-assigned) exercises for the
book's Section 2.2 and in all remaining sections of Chapter 2. (Except
for "Exact equations"—Section 3.3.6 of my notes—hardly
anything in Section 3.3 of my notes [First-order equations in
differential form] is discussed in the book at all.)
In the textbook, read Section 2.4 up through the boxed
definition "Exact Differential Form" on p. 59. Also, on
my Spring 2024 homework page,
go to the assignment that was due 2/7/24, and read
"Comments, part 1" and "Comments, part 2."
|
| W 2/12/25 |
In my notes:
- Read
Section 3.3.2 and 3.3.3. You may skip the portions labeled
"optional reading".
- In Section 3.3.5, read up through Example 3.71.
Section 3.3.5 essentially addresses: what
constitutes a possible answer to various questions, based the
type of DE (derivative-form or differential-form) you're being asked
to solve? A proper answer to this question requires taking into
account some important facts omitted from the textbook (e.g. the fact
that DEs in derivative form and DEs in differential form
are not "essentially the same thing").
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".)
Previously, we defined what "separable" means
only for a DE in derivative 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 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.
As for how to solve these equations: you will
probably be able to guess the correct mechanical procedure. A natural
question is: how can you be sure that these mechanical procedures give
you a completely correct answer? That question is, essentially, what Sections
3.4–3.6 of my notes
are devoted to.
Warning. For
questions answered in the back of the book: not all answers there are
correct
(that's a general statement; I haven't done a separate
check for the exercises in this assignment)
and some may be misleading. But most are either correct, or
pretty close.
|
| F 2/14/25 |
In the textbook, continue reading Section 2.4, up through Example
3. Then do the next set of exercises:
2.4/ 1–8.
Note: For differential-form DEs, there is no
such thing as a linear equation. In these problems, the book
means for you
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, the associated derivative-form
DE for \(y(x)\) is linear; the associated derivative-form DE for
\(x(y)\) is not.
In my notes,
read the remainder of Section 3.3.5, and read Section 3.3.6
up through Example 3.76. (The remainder of Section 3.3.6
is optional reading.)
When reading anything in Sections 3.3
(all of the "3.3.x" subsections) and Sections 3.4–3.6,
remember that Section 3.7 summarizes all the definitions
and results in those sections. To avoid getting lost in the weeds,
refer to this summary as often as you need; that's the
whole reason for Section 3.7's existence.
|
| M 2/17/25 |
If I have not yet gone through the "exact equation method" in class,
read the rest of NSS Section 2.4 to see the mechanics of solving an
exact DE. (Just don't trust any "justifications" or
terminology in this section.) This should be enough to enable you to do the
exercises below, though not necessarily with confidence if I
haven't gone through this in class yet.
Don't invent a different method for solving
exact equations (or use a different method you may have
seen before).
Read the handout "A terrible method for solving exact equations"
that's posted on the Miscellaneous Handouts page. I can almost guarantee
that if you've invented (or have ever been shown) an alternative
to the method shown in the book (and that I'll go over in
class), this "terrible method" is that alternative method.
2.4 (continued)/ 9, 11–14, 16, 17, 19,
20
possibly more TBA
|