| Date due |
Section # / problem #'s |
| W 1/8/20 |
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. Since not everyone has access to the
textbook yet, here is a
scan of the first 15 pages (Sections
1.1–1.2, including all the exercises).
Note: the sentence on p. 4 that contains
equation (7) is not quite correct as a definition of "linear".
An ODE in the indicated variables is linear if it has the
indicated format, or can be put in this format just by
adding/subtracting expressions from both sides of the equation
(as is the case with the next-to-last equation on the page).
Do non-book problem 1.
In my notes on
first-order ODEs, read the first three paragraphs of the
introduction, all of Section 3.1.1, and Section 3.1.2 through
the third paragraph on p. 13 (the paragraph beginning with
"Graphically"). In this and future assignments from these notes,
you should skip all items labeled "Note to instructors". (In
particular, you should skip the footnote that begins on p. 11
and occupies all but the first three lines of p. 12.)
|
| F 1/10/20 |
For the DEs in 1.1/ 1–3 and 5–12, classify each equation
as linear or nonlinear.
1.2/ 1, 3–6, 14, 15, 17, 19–22.
See Notes on some book problems.
Do non-book problem 2.
In
my notes, read from where you left off on p. 13
through the end of Section 3.1.2, and do the exercise on p. 17.
I update these notes from time to time
during the semester, and update the version-date line on p. 1 whenever
I make a revision. Each time you're going to look at the notes, make
sure that what you're looking at isn't an older version cached by
your browser.
Note: The exercise portions of many (probably most) of
your homework assignments will be a lot more time-consuming than in
the assignments to date; I want to give you fair warning of this
before the end of Drop/Add. Often, most of the book
problems in a section aren't doable until we've finished covering
practically the entire section, at which time I may give you a large
batch to do all at once. Heed the suggestion near the top of this
page: "If one day's assignment seems lighter than average,
it's a good idea to read ahead and start doing the next assignment,
which may be longer than average."
|
| M 1/13/20 |
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 discuss how to solve
linear equations, which is the topic of Section 2.3. The purpose
of 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. As a
"bonus", you'll also be able to do the early exercises in
Section 2.2 assigned below.)
2.2/ 1–4, 6
2.3/ 1–6
In Section 3.1.3 of
my notes, read up through
the end of the black text on
p. 24. Then 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.
|
| W 1/15/20 |
2.3/ 7–9,
12–15 (note which variable is which in #13!), 17–20, 22.
Do non-book problems
9ab.
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",
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 it 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 \(\frac{1}{x}\).
Do non-book problem
9 (all three parts). Before doing 9c, see the notational reminder
in non-book problem 4.
In Section 3.1.3 of
my notes, read the black text on
pp. 27–32. (Reading any of the blue text in my notes is
optional, in all assignments, unless I say otherwise.) After reading
Remark 3.19, do textbook problem 1.2/ 16.
In my notes, read Section
3.1.4 up through Example 3.27. You are allowed to skip
a portion of this material, as indicated in the bold-faced
sentence beginning "Non-honors students ..." on p. 32.
Also read the first paragraph of
Section 3.1.5 and then Definition 3.37.
|
| F 1/17/20 |
2.3/ 25a, 27a, 28, 31, 33, 35. Note about wording in #35:
The term "a brine"
in this problem
is not
proper English; it's similar to saying "a water" or
"a sand". One should either say "brine" (without the "a") or "a
brine solution". Another phrase that should not be used is the
redundant "a brine solution of salt" (literally "a concentrated
salt water
solution of salt").
One of the things you'll see in exercise 2.3/33 is that
what
you might think is only a minor difference between the DE's in parts
(a) and (b)—a sign-change in just one
term—drastically changes the nature of the
solutions. When solving differential equations, a tiny algebra slip
can make your answers utter garbage. For this reason, there is
usually no such thing as a "minor algebra error" in solving
differential equations. This is a fact of life you'll have to get
used to. The severity of a mistake is not determined by the number
of pencil-strokes it would take to correct it, or whether your work
was consistent after that mistake. If a mistake (even something as
simple as a sign-mistake) leads to an answer that's garbage, or that
in any other way is qualitatively very different from the correct
answer, it's a very bad mistake, for which you can expect a
significant penalty. A sign is the only difference between a rocket
going up and a rocket going down. In real life,
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 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'll 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.
In
my
notes, read
the remainder of Sections
3.1.4, and read Section 3.1.5. (In Section 3.1.5 you've already
read a tiny bit in the last assignment. )
Reminder: reading
my notes is not
optional (except for portions that I [or the notes] say you may
skip, and the footnotes or parenthetic comments that say "Note to
instructor(s)"). Each reading assignment should be completed by the due
date I give you. Otherwise you will have far too much to absorb at
once.
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).
There is not enough time to cover most of these carefully in class;
we would not get through all the topics we're supposed to cover.
|
| W 1/22/20 |
In my notes, read
Section 5.1. With the term
"open rectangle" instead of "open set", my notes' Theorem 5.1 is what
the textbook's Theorem 1
on p. 11 should have said; see the paragraph in my notes
after Theorem 5.1. (The theorem is nicer with "open set"
instead of "open rectangle", but most DE textbooks don't define
"open set".)
1.2/ 18, 23–28, 31.
(For all but #18, it may help you to
look at Examples 8 and 9 on p. 13.) For 23–28, the
instructions should end with "... has a unique solution on some
open interval." Similarly, in 31a, "unique solution" should be
"unique solution on some open interval".
Read Section 1.3 of the textbook and do exercises 1.3/ 2, 3.
In my notes, read
Sections
3.1.6 and 3.1.8.
I know I've given you a lot of reading so far, much of which isn't
easy reading. The bad news is that this will continue for another
couple of weeks or so. The good news is that once we're done studying
first-order DEs (roughly the first month of the course), we'll be
done with my notes.
|
| F 1/24/20 |
In my notes, read
Section 3.1.7 up to where the proof of Theorem 3.43
begins. (You don't have to read the proof until the next
assignment.)
2.2/ 7–14.
(Note: "Solve
the equation" means "Find all [maximal] solutions of the
equation".)
Do non-book problems
3, 4, and 6.
Answers to these
non-book problems and some others are posted on the "Miscellaneous
handouts" page.
General comment. In doing the exercises
from Section 2.2 or the non-book problems 3, 4, and 6, you may
find that, often, the hardest part
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). Remember my warning
in the syllabus:
You will need a good working knowledge of Calculus 1 and 2. In
particular, you will be expected to know integration techniques ...
. If you are weak in any of these areas, or it's been a while since
you took calculus, you will need to spend extra time reviewing or
relearning that material. Mistakes in prerequisite material will be
graded harshly on exams.
Whenever you do do these exercises, whether as part of this
assignment or the next, don't just go through the motions, either saying to yourself, "Yeah, I know
what to do from here" but not doing it, or doing the integrals incorrectly, or
stopping when you reach an integral you don't remember how to do.
(This applies to the exercises that will be assigned in the future
as well.)
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.
|
| M 1/27/20 |
In my notes, read
the remainder of Section 3.1.7. (Remember
that the blue portions of my notes are optional reading; you
are allowed to skip them even if they're within the sections I'm
assigning.)
Theorems 3.43 and 3.44
in my notes are
closely related to the "Formal Justification of Method" on p. 45
of the textbook. You will find the book's presentation simpler
than mine, but this simplification comes at a high price: (1) the book's
argument is fallacious
(because it puts no hypotheses on the functions \(p\) and \(g\),
without which several steps in the book's argument
cannot be justified),
and (2) the conclusion it purports to
establish neglects 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).
In my notes, read
from the beginning of Section 3.2 (p. 69) up to, but not including,
the beginning of Section 3.2.1 (p. 74).
With the exception of the notes' Definition 3.56, the Section 3.2
material in the reading above is basically not discussed in the book
at all, even though differential-form DEs appear in (not-yet-assigned)
exercises for the book's Section 2.2 and in all remaining sections of
Chapter 2. (Very little of what's in Sections 3.2.1–3.2.4,
3.2.6, or 3.3 of my notes is discussed in the book either.) I'm
assigning the early portion of Section 3.2 now since the rest of the
current assignment is pretty light.
This will (partially) spare you from having to do a lot of reading and
exercises at the same time once we get to the book's Section 2.4.
|
| General info |
The date for your first midterm will not be Monday Feb. 3.
As of now (Monday Jan. 27), the earliest possible date is
Wed. Feb. 5. Before I give the exam, we need to finish the topic of
"exact equations", which we're about to start. Once I've determined
the date of the exam for sure, I'll let you know.
For any exam, I'll
always give you essentially a week's notice ("essentially" meaning
that, for example, I may let
you know on a Wednesday evening—via your homework page or an email—
that the exam will be the following Wednesday). Two lectures before
the exam, I will give you a copy of last semester's corresponding
exam. I will give this out only in class or, for students
with an excusable absence, in my office hours. I will not post or
email old exams or solutions.
|
| W 1/29/20 |
Do non-book problems 7,
8. (Some of #7 was done in
class, but it won't hurt you to do
the problem from scratch.)
In the textbook, read Section 2.4 through the boxed
definition "Exact Differential Form" on p. 59. See Comments,
part 1, below.
In my notes, read
from the beginning of Section 3.2.2 (p. 78) through the end of Example
3.75 on p. 87. Remember, you're allowed to skip anything in blue.
The term "regular parametrization" is defined in Definition 3.63 on
p. 76 (part of Section 3.2.1). With the exception of Definitions 3.63
and 3.64, you should have seen the material in Section 3.2.1 in
Calculus 2, so I'm not requiring you to read Section 3.2.1.
However, in the sections I am requiring you to read, you
may occasionally come across
terminology you're not familiar with because it was defined in
Section 3.2.1 or
somewhere in blue text that you skipped. If that happens, it's
generally safe to skip over the sentence(s) containing such
terminology, but you may instead want to find
that definition, read it briefly to get the general idea, and then
go back to where you were.
See Comments, part 2, below.
Comments, part 1.
There are
terminological
problems in Section 2.4 of the book, most notably an inconsistent
usage of the term "differential form". Many students
may
not notice the inconsistency,
but some may—especially the students with a deep interest in
mathematics—and I don't want anyone to come out of my class with an
improper education. Here are the problems, and fixes for
them:
- In this chapter, every instance in which the term
"differential form" is used for anything that's not
an equation—a statement with an "=" sign in
it—the word "form" should be deleted. In particular, this
applies to all instances of "differential form" in the
definition-box on p. 59 (including the title).
- The definition-box's use of the term
"differential form" is
not incorrect, but at the level
of MAP 2302 it is a very confusing use of the word "form", and the
less-misinterpretable term "differential" (without the word
"form") is perfectly
correct.
- Except for the title, the usage of "differential form" within the
definition-box is inconsistent with the usage
outside the definition-box. The usage in the title is ambiguous;
it is impossible to tell whether the title is referring to an
exact differential, or to an equation with an exact
differential on one side and zero on the other.
In my notes I talk about "derivative
form" and "differential form" of a
differential equation. The meaning of the word "form" in
my notes is standard mathematical English, and is the same as in
each of the two occurences of "form" on on p. 58 of the book.
In this usage, "form of an equation" refers to the way an
equation is written, and/or to what sort of objects
appear in it.
But when a differential itself (as
opposed to an equation containing a differential) is
called a "differential form", the word "form" means
something entirely different, whose meaning cannot be
gleaned from what "form" usually means in English. In this
other, more advanced usage, "differential forms" are
more-general objects than are differentials. (Differentials are
also called 1-forms. There are things called 2-forms,
3-forms, etc., which cannot effectively be defined at the level
of MAP 2302.) You
won't see these more general objects in this course, or in any
undergraduate course at UF—with the possible exception of
the combined graduate/undergraduate course Modern Analysis 2,
and occasional special-topics courses.) With the advanced
meaning of "differential form", the only differential
forms that appear in an undergraduate DE textbook
are differentials, so there's no
good reason in a such a course, or in its textbook, to use the term
differential form for a differential.
There is also a pronunciation-difference in the
two usages of "differential form". The pronunciation of this
term in my notes is "differential form", with the
accent on the first word, providing a contrast with
"derivative form". In the other usage of "differential
form"—the one you're not equipped to understand, but
that is used in the book's definition-box on p. 59—the
pronunciation of "differential form"
never has the accent on the first word; we either say
"differential form", with the accent on the second
word, or we accent both words equally.
- The paragraph directly below the "Exact Differential
Form" box on p. 59 is not part of the current
assignment. However, for future reference, this paragraph is
potentially confusing or misleading, because while the first
sentence uses "form" in the way it's used on p.58 and in my
notes, the third sentence uses it with the other, more
advanced meaning. This
paragraph does not make sense unless the term "differential
form" is used to mean a form of an equation (with the
standard-English meaning of "form") on line 2, and with the
meaning of a differential on line 4.
Choosing to use the term "exact
differential form" in the first equation of this paragraph is,
itself, rather unusual. In any context
involving the advanced meaning of "differential
forms" (the meaning you haven't been given), there is a
standard meaning of "exact differential form".
In this meaning, an exact differential form is
a differential form (with the advanced meaning) with a certain
property that makes us call it "exact". I.e., once we combine
the word "exact" with "differential form", there are no
longer two different things that "differential form" can
mean, without departing from standard
definitions. In standard convention, "exact
differential form" is
never a type of equation. In the context of the
paragraph under discussion, there is only one standard meaning of
"exact differential form"—namely, exact differential—a
type of differential, not a type of equation.
The standard terminology for
what the offending sentence calls "[differential equation]
in exact differential form" is exact equation
(or exact differential equation), just
as you see in the definition-box on p. 59. (The terminology
"exact equation" in the box has its own intrinsic problems,
but
is standard nonetheless.)
- In Example 1 on
p. 58, the sentence beginning "However" is not correct. In
this sentence, "the first form" refers to the
first equation written in the sentence beginning
"Some". An equation cannot be a total differential.
An equation makes an assertion; a total differential
(like any differential) is simply a
mathematical expression; it is no more an equation than
"\(x^3\) " is an equation. To correct this sentence, replace
the word "it" with "its left-hand side".
- The following is just FYI; it's not a problem with the
book: What the book calls the total differential of
a function F is what my notes call simply the
differential of F. Both are
correct. The word "total" in "total differential" is
superfluous, so I choose not to use it.
Comments, part 2. In my notes, you're going to
find section 3.2 and its subsections more difficult to read than the
book's Section 2.4 (and probably more difficult than the earlier
sections of my notes). A
major reason for this is that a lot of
important issues are buried in a sentence on the book's p. 58 (the
sentence that begins with the words "After all"
and contains equation (3)). You'll find the sentence plausible, but
you should be troubled by the fact that since \(\frac{dy}{dx}\) is
simply notation for an object that is not actually a real number
"\(dy\)" divided by a real number "\(dx\)", just how is it that an
equation of the form \(\frac{dy}{dx}=f(x,y)\) can be "rewritten" in
the form of equation (3)? Are the two equations equivalent? Just what
does an equation like (3) mean? In a derivative-form DE,
there's an independent variable and a dependent variable. Do you see
any such distinction between the variables in (3)? Just what
does solution of such an equation mean? Is such a solution the
same kind of animal as a solution of equation (1) or (2) on p. 6 of
the book, even though no derivatives appear in equation (3) on
p. 58? If so, why; if not, why not? Even if we knew what "solution
of an equation in differential form" ought to mean, and knew how to
find some solutions, would we have ways to tell whether we've
found all the solutions? Even for an exact equation, how do
know that all the solutions are given by an equation of the form
\(F(x,y)=C\), as asserted on p. 58?
The textbook is easier to read
than my notes because these questions and their answers (which are
subtler and deeper than you might think) aren't mentioned. The same
is true of all the DE textbooks I've seen; even with the problems
I've mentioned, our
textbook is still better than any other I've seen on the current market. But if
you had a good Calculus 1 class, you had it drilled into
you that "\(\frac{dy}{dx}\)" is not a real number
\(dy\) divided by a real number \(dx\), and you should be
confused to see a math textbook implying with words like
"After all" that's it's `obviously' okay to treat
"\(\frac{dy}{dx}\)" as if it were a fraction with real numbers
in the numerator and denominator. The Leibniz notation
"\(\frac{dy}{dx}\)" for derivatives has the miraculous feature
that the outcomes of certain symbol-manipulations
suggested by the notation can be justified (usually using
higher-level mathematics), even though the manipulations
themselves are not valid algebraic operations, and even though
it is not remotely obvious that the outcomes can be justified.
|
| F 1/31/20 |
2.4/ 1–8. Note: (1) In class on Wednesday 1/29, we
did not get far enough to discuss how to tell whether a DE in
differential form is exact. For this, use the "Test for
Exactness" in the box on p. 60 of the book. We will discuss this
test in Friday's class. (2)
For differential-form
DEs, there is no such thing as a linear equation.
In these problems, you are meant to classify
an equation in differential form as linear if at least one of
the associated derivative-form equations (the ones you get
by formally dividing through by \(dx\) and \(dy\), as if
they were numbers) is linear. It is possible for one of these
derivative-form equations to be linear while the other is
nonlinear. This happens in several of these exercises.
For example, #5 is
linear as an equation for \(y(x)\), but not as an equation for
\(x(y)\).
In my notes, read from
Remark 3.76 on p. 87 (part of Section 3.2.4) through statement
(3.162)
on p. 101 (mid-way through Section 3.3). Again, the material in blue
(which is quite a lot of this portion of the notes) is optional
reading; you're required to read only the material in black.
2.2 (not 2.3 or 2.4)/ 5, 15, 16. (I did not assign these when we were
covering Section 2.2 because we had not yet discussed
"differential form".) An equation in differential form is called
separable if, in some region of the \(xy\) plane (not
necessarily
the whole region on which the given DE is defined), the given DE
is algebraically 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.
|
| M 2/3/20 |
Read the remainder of Section 2.4 of the textbook.
2.4/ 9, 11–14, 16, 17, 19, 20.
Read the online handout
A terrible way to
solve exact equations. The example in this
version of the
handout is rather
complicated; feel free to read the simpler 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 argument, 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 constructed the more complicated
example to make the failure of the terrible method more obvious.
2.2 (not 2.3 or 2.4)/ 22. Note that although the differential
equation doesn't specify independent and dependent variables, the
initial condition does. Thus your goal in this exercise is to
produce an explicit solution "\(y(x)= ...\)". But this exercise is an
example of what I call a "schizophrenic" IVP. In practice, if you are
interested in solutions with independent variable \(x\) and dependent
variable \(y\) (which is what an initial condition of the form
"\(y(x_0)=y_0\)'' indicates), then the differential equation you're
interested in at the start is one in derivative form
(which in exercise 22 would be \(x^2 +2y \frac{dy}{dx}=0\), or an
algebraically equivalent version), not one in differential
form. Putting the DE into differential form is often a useful
intermediate step for solving such a problem, but differential form is
not the natural starting point. On the other hand, if what you are
interested in from the start is a solution to a
differential-form DE, then it's illogical to express a preference for
one variable over the other by asking for a solution that satisfies a
condition of the form "\(y(x_0)=y_0\)'' or "\(x(y_0)=x_0\)''. What's
logical to ask for is a solution whose graph passes through the
point \((x_0,y_0)\), which in exercise 22 would be the point
(0,2).
2.4 (resumed)/ 21, 22
(note that #22 is the same DE as #16, so you don't have to solve a new
DE; you just have to incorporate the initial condition into your old
solution). Note that exercises 21–26 are what I termed
"schizophrenic" IVPs.
Your goal in these
problems is to find an
an explicit formula for a solution, one expressing the dependent
variable
explicitly as a function of the independent variable —if algebraically
possible—with the choice of independent/dependent variables
indicated by the initial condition. However, if in the algebraic equation
``\(F({\rm variable}_1, {\rm variable}_2)=0\)'' that you get via the
exact-equation method (in these schizophrenic IVPs), it is impossible to
solve for the dependent variable in terms of the independent variable,
you have to settle for an implicit solution.
In my notes,
read the remainder of Section 3.3, and start reading Section
3.4. (Completing Section 3.4 and reading Section 3.5 will be
part of your next assignment; divide this reading between the
two assignments in whatever way works best for you.)
|
| W 2/5/20 |
In my notes,
finish reading Section 3.4 and read Section 3.5.
Do non-book problem 11.
I suggest doing the above reading first.
Note: The list of non-book problem was
updated
2/3/20 at about 7:45 p.m.; the current problem 11 (which has parts
(a)–(f)) is new. A typo in 11b was later fixed at about 11:35 p.m.
Read the statement
of non-book problem 10.
I'm not requiring you to do the problem, but I want you to be aware
of the fact stated in the problem.
Read The Math
Commandments.
2.4/ 32, 33a.
In #32, the instructions omit important
hypotheses and do not deal correctly with points \((x_0,y_0)\) for
which \(\frac{\partial F}{\partial x}(x_0,y_0)=0\) and/or
\(\frac{\partial F}{\partial y}(x_0,y_0)=0.\) To fix
this:
- In the set-up paragraph
and in part (a), assume that the function \(F\)
is continuously differentiable
(i.e. \(\frac{\partial F}{\partial x}\) and \(\frac{\partial
F}{\partial y}\) are continuous) on a
region \(R\). Furthermore, we restrict attention to the
sub-region \(R^*\) consisting of those points of \(R\) that are not
critical points of \(F\) (points at which both
\(\frac{\partial F}{\partial x}(x_0,y_0)\) and
\(\frac{\partial F}{\partial y}(x_0,y_0)\) are zero). The
Implicit Function Theorem can be used to show that, for each
constant \(k\), the set of points in \(R^*\) satisfying
\(F(x,y)=k\) (if there are any such points) is a smooth
curve or a union of non-intersecting smooth curves. (The
"union" statement is needed since, for example, the hyperbola
with equation \(xy=1\) is not a single smooth curve, but a
union of two non-intersecting smooth curves). If we allow
critical points into the picture, then the set of points
satisfying \(F(x,y)=k\) may or may not be a union of
non-intersecting smooth curves. (For example, the origin \(
(0,0)\) is a critical point of the function
\(F(x,y)=x^2-y^2\). Since \(F(0,0)=0\), the \(k\) for which
the origin satisfies \(F(x,y)=k\) is \(0\). The set of points
satisfying \(x^2-y^2=0\) is the pair of lines \( y=x\) and
\(y=-x\), which cross each other with different slopes at the
origin. Thus, at the origin, there is no such thing as "the
slope of the curve described by \(x^2-y^2=0\)".)
Everywhere that "curve(s)"
appears in problem 32, it should be replaced by "curve(s) in
\(R^*\) ". (The notation "\(R^*\)" is just for this
problem; it's not something general.)
- In the set-up paragraph, replace the sentence beginning with
"Recall" with the following: "For each curve in
\(R^*\) in this family, and point \( (x_0,y_0)\) of that curve:"
-
If \(\frac{\partial F}{\partial
y}(x_0,y_0) \neq 0\), then the slope of that curve at \((x_0,y_0)\) is
given by \(\frac{dy}{dx} = -\frac{\partial F}{\partial x}/
\frac{\partial F}{\partial y}\) (evaluated at \( (x_0,y_0)
\)).
-
If, \(\frac{\partial F}{\partial
y}(x_0,y_0) =0\) then the slope of that curve at \((x_0,y_0)\) is
infinite (the tangent line is vertical)."
Note that since the only points we are considering are in \(R^*\), if
\(\frac{\partial F}{\partial
y}(x_0,y_0) =0\) then \(\frac{\partial F}{\partial
x}(x_0,y_0)\) is not zero.
- In part (a), replace the sentence beginning with "Recall" with the
following: "Recall that the slope of a curve that is
orthogonal to a smooth curve \({\mathcal C}\) at a point \(
(x_0,y_0)\) is the negative reciprocal of the slope of
\({\mathcal C}\) at \( (x_0,y_0)\), provided that the slope of
\({\mathcal C}\) at \( (x_0,y_0) \) is finite and nonzero. If
the slope of \({\mathcal C}\) at \((x_0,y_0)\) is infinite
(i.e. if \({\mathcal C}\) has a vertical tangent line at
\((x_0,y_0)\)) then the slope of a curve orthogonal to
\({\mathcal C}\) at \( (x_0,y_0) \) is zero. Similarly, if the
slope of \({\mathcal C}\) at \((x_0,y_0)\) is zero, then the
slope of a curve orthogonal to \({\mathcal C}\) at \(
(x_0,y_0) \) is infinite."
|
| F 2/7/20 |
First midterm exam (assignment is to study for it).
- In case you'd like additional
exercises to practice
with:
If you
have done
all your homework (and I don't mean "almost all"), you should
be able to do all the review problems on p. 79 except #s
8, 9, 11, 12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35,
37, and the last part of 41. A good feature of the book's "review
problems" is that, unlike the exercises after each section, the
location gives you no clue as to what method(s) is/are likely to work.
Your exam will have no such clues on exams either. Even if you don't
have time to work through the problems on p. 79, they're good
practice for figuring out the appropriate methods are.
A negative feature of the book's exercises
(including the review problems) is that they
don't give you enough practice with a few important integration
skills. This is why I assigned my non-book problems 3,
4, 6, and 9.
- Reminder: the syllabus says,
"[U]nless I say otherwise, you are responsible for knowing any material
I cover in class, any subject covered in homework, and all the
material in the textbook chapters we are studying." I have not "said
otherwise", the homework has included
reading Chapter 3 of my notes (minus the portions I said were
optional) as well as doing book and
non-book exercises,
and the textbook chapters/sections we've covered are 1.1, 1.2, 2.2,
2.3, and 2.4.
|
| M 2/10/20 |
No new homework.
|
| W 2/12/20 |
Read section 4.1 of the book. (We're skipping Sections 2.5 and
2.6, and all of Chapter 3.)
We will be covering the
material in Sections 4.1–4.7 in an order that's different from the
book's. We have already covered a small portion of
Section 4.7.
4.7 (yes, 4.7)/ 1–8, 30. Problem #30 does not
require you to have read anything in Sections
4.1–4.7. For problems 1–8, you may wish to first
read from the bottom of p. 192 (Theorem 5, the \(n^{\rm
th}\)-order version of which was stated in class
on Monday 2/10/20) 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]."
|
| F 2/14/20 |
Read Section 4.2 up through the bottom of p. 161. (Note: on
p. 158, the authors mention that the "auxiliary equation" is also
known as the "characteristic equation". In class, I'll be using the
term "characteristic equation".) On p. 157, between the next-to-last
line and the last line, insert the words "which we may rewrite as".
4.7 (yes, 4.7)/ 25
In class on Wednesday 2/12, two general properties "(i)" and
"(ii)" of linear differential operators were introduced, from which
we derived several consequences. Redo
problem 4.7/30 using only these properties and/or the consequences
we derived.
Unfortunately, hardly any of Section 4.2's exercises are
doable until the whole section has been covered, which takes more than
a single day (we have just started it in class). In order for you not
to have a single massive assignment when we're done covering Section
4.2, I recommend that, based on your reading, you try to start on the
exercises listed in the next assignment. Problems that you should be
able to do after doing the reading assigned above are
4.2/ 1, 3, 4, 7, 8, 10, 12, 13–16, 18, 27–32.
Some reminders for students who had trouble with the first
exam (and other students too!):
- In my classes, if reviewing
your notes isn't part of your exam-study (or if you didn't take
good enough notes to begin with), or if you didn't do all
your homework, it is very unlikely that you will do well on my
exams.
- Mathematical knowledge
and skills are cumulative. Courses have prerequisites because you
need to know and be able to use, quickly and
accurately and without prompting, the mathematics you learned in
the past.
The syllabus for this class says,
"If you are weak in [various prerequisite topics], 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."
Unfortunately, most students who need to review prerequisite
material before they risk losing points on an exam, don't
do that review in time (if they do it at all).
Of course, it would be great if math
skills you've learned stuck with you, so that you
wouldn't have to review. So how do you learn math
skills in a way that you won't forget them? The answer
is repetition. Repetition builds retention.
Virtually nothing else does. I've known many intelligent
students (even within my own family!) who thought that the
"smart" use of their time, when faced with a lot of
exercises of the same type, was to skip everything after the
first or second exercise that they could do correctly. No.
This might help you retain a skill for a week, but probably
not through the next exam, let alone through the final exam,
let alone through the future courses in which you'll be
expected to have that skill. Would you expect to be able to
sink foul shots in a basketball game if you'd stopped
practicing them after one or two went in?
- One area in which many MAP 2302 students, in every section of
the course every semester, need review is exponentials. For this
reason, one of the resources I've had on the Miscellaneous
Handouts page for my MAP 2302 classes for many years is an
Exponential Review Sheet.
|
| M 2/17/20 |
4.2/ 1–20, 26, 27–32, 35, 46ab.
In #46, the instructions should say that the
hyperbolic cosine and hyperbolic sine functions can be
defined as the solutions of the indicated IVPs, not that
they are defined this way. The customary definitions are
more direct: \(\cosh t=(e^t+e^{-t})/2\) (this is what you're
expected to use in 35(d))
and \( \sinh t= (e^t-e^{-t})/2\). Part of what you're doing in
46(a) is showing that the definitions in problem 46 are equivalent
to the customary ones. One reason that these functions have
"cosine" and "sine" as part of their names is that the ordinary
cosine and sine functions are the solutions of the DE \(y''+y=0\)
(note the plus sign) with the same initial conditions at \(t=0\)
that are satisfied by \(\cosh\) and \(sinh\) respectively. Note
what an enormous difference the sign-change makes for the
solutions of \(y''-y=0\) compared to the solutions of \(y''+y=0\).
For the latter, all the nontrivial solutions (i.e. those that are
not identically zero) are periodic and oscillatory; for the
former, none of them are periodic or oscillatory, and all of them
grow without bound either as \(t\to\infty\), as \(t\to -\infty\),
or in both directions.
FYI: "\(\cosh\)" is pronounced the way it's
spelled; "\(\sinh\)" is pronounced "cinch".
|
| W 2/19/20 |
4.7/ 26abc
4.3/ 1–18. In the instructions for 1–8, "complex
roots" should be replaced by "non-real roots", "non-real complex
roots", or "no real roots". Every real number is also a complex
number (just like every square is a rectangle); thus "complex"
does not imply "non-real". A real number is just a complex number
whose imaginary part is 0.
|
| F 2/21/20 |
4.3/ 1–18, 21–26, 28, 32, 33 (students in
electrical engineering may do #34 instead of #33). Before
doing problems 32 and 33/34, see Examples 3 and 4 in Section 4.3.
In the
instructions for 1–8, "complex roots" should be replaced by
"non-real roots", "non-real complex roots", or "no real roots".
Every real number is also a complex number (just like every square
is a rectangle); thus "complex" does not imply "non-real". A real
number is just a complex number whose imaginary part is 0.
Read Section 4.3. In the title of this section,
"complex roots" should be replaced by "non-real roots", "non-real
complex roots", or "no real roots". In the box on p. 168, after
"\(\alpha \pm i\beta\)", the parenthetic phrase "(with \(\beta\neq
0\))" should be inserted.
Note: The book uses the complex
exponential function (which we have not yet discussed in class; we
will discuss it later if time permits) to derive the fact that in the case of
non-real-roots \(\alpha\pm i\beta\), the functions \( t\mapsto
e^{\alpha t} \cos \beta t\) and \(t\mapsto e^{\alpha t} \sin \beta
t\) are solutions of the DE (2) on p. 166, rather than showing this
by direct computation using only real-valued functions (the approach
used in class on Monday 2/17/20, with some calculations to the
student). The complex-exponential approach is very elegant and
unifying. It is also useful for studying higher-order constant-coefficient
linear DEs, and for showing the validity of a certain technique we
haven't gotten to yet (the Method of Undetermined Coefficients). It
is definitely worth at least reading about, and for most
purposes, I prefer it to the approach I took in class. The
drawbacks are:
- Several new objects (complex-valued functions in
general, and the derivative of a complex-valued function
of a real variable)
must be defined.
- Quite a few facts must be established, among them and the
relations between real and complex solutions of equation (2),
and the differentiation formula at the bottom of p. 166.
(There is no such thing as "proof by notation". Choosing
to call \(e^{\alpha t}(\cos \beta t + i\sin\beta t)\) a
"complex exponential function", and choosing to use the
notation \(e^{(\alpha + i\beta)t}\), doesn't magically give
this function the same properties that real exponential
functions have (any more than choosing to use the notation
"\(\csc( (\alpha+i\beta)t)\)" for
\(e^{\alpha t}(\cos \beta t + i\sin\beta t)\) would have given this
function properties of the cosecant function).
Exponential notation is used because it turns
out that the above function has the properties that the
notation suggests; the notation helps us remember these
properties. But
all of those properties have to be
checked based on defining \(e^{a+ib}\) to be
\(e^a(\cos b + i \sin b)\) (for all real numbers \(a\) and
\(b\)). This is a very worthwhile exercise, but time-consuming.
- On exams in this class, all final answers must be expressed
entirely in terms of real numbers; complex numbers are
allowed to appear only in intermediate steps. (The instructions on all
your exams starting with the second midterm will say so.) Every
year, there are students who use the complex exponential function
without understanding it, leading them to express some final answers
in terms of complex exponentials. Such answers receive little if any
credit.
There are also some problems with the book's presentation:
- Equation (4) on p. 168 is presented in a sentence that
starts with "If we assume that the law of exponents applies to complex
numbers ...". Unfortunately, the book is very fuzzy about the
distinction between definition and assumption, and never
makes clear that equations (4), (5), and (6) on p. 168 are not things
that need to be assumed. Rather, all these equations result
from defining \(e^{z}\), where \(z= a+ bi\), to be \(e^a(\cos b
+ i \sin b)\), a formula not written down explicitly in the book.
- A non-obvious fact, beyond the level of this course, is that
the above definition of \(e^z\) is equivalent to defining
\(e^z\) to be \(\sum_{n=0}^\infty \frac{z^n}{n!}\). This is a
series that—in a course on functions of a complex
variable—we might call the Maclaurin series for \(e^z\).
However, the only prior instance in which MAP 2302 students have
seen "Maclaurin series" (or, more generally, Taylor series)
defined is for functions of a real variable. To define
these series for functions of a complex variable requires a
definition of "derivative of a complex-valued function of a
complex variable". That's more subtle than you'd think. It's
something you'd see in in a course on functions of a complex
variable, but is beyond the
level of MAP 2302. So the sentence on p. 166 that's two lines
below equation (4) is misleading; it implies that we
already know what "Maclaurin series" means for
complex-valued functions of a complex variable. A
non-misleading way to introduce the calculation of
\(e^{i\theta}\) that's on this page is the following: "To
motivate the definition of \(e^{i\beta t}\)—or, more
generally, \(e^{i\theta}\) for any real number
\(\theta\)—that we are going to give below, let us see
what happens if we replace the real number \(x\) by the
imaginary number \(i\theta\) in the Maclaurin series for
\(e^x\)." Instead of the word "identification" that's used in
the line above equation (5), we would then use the much clearer
word "definition".
|
| M 2/24/20 |
Read Section 4.4 up through at least Example 3.
Read Section 4.5 up through at least Example 2.
We
will be covering Sections 4.4 and 4.5 simultaneously, more
or less, rather than one after the other. What most mathematicians
(including me) call "the Method of Undetermined Coefficients" is what
the book calls "the Method of Undetermined Coefficients plus
superposition." You should think of Section 4.5
as completing the (second-order case of) the Method of
Undetermined Coefficients, whose presentation is begun in Section 4.4.
|
| W 2/26/20 |
Read the remainder of Sections 4.4 and 4.5. You will not be
responsible for all of this on Friday's exam, but the order in which
the book covers this material makes it difficult to isolate the
sentences that are important for you to read before the exam; it is
easier to clarify what material you'll be
responsible (from these sections, this exam) in terms of the box
labeled "Method of Undetermined Coefficients"
on p. 178.
For Friday's exam, the Section 4.4 material
you'll be responsible for is essentially the first half of what's
in this box: using the MUC to find a solution ("particular
solution") of a DE of the form \(ay''+by'+cy = {\rm constant}\times
t^m e^{rt}\), where \(m\) is a nonnegative integer and \(r\) is a
real number. Note that this includes the special cases \(m=0\)
(in which case the right-hand side of the DE is just \(C e^{rt}\))
and \(r=0\) (in which case the right-hand side of the DE is just
\(C t^m\)). The number \(s\) in this box is what I called
the multiplicity of \(r\) as a root of the characteristic
polynomial.
To state this "multiplicity" fact cleanly, it
is imperative not to use the identical notation \(r\) in "\(t^m
e^{rt}\)" as in the characteristic polynomial \(p_L(r)=ar^2+br+c\)
and the characteristic equation \(ar^2+br+c=0\). Replace
the \(r\) in the box on p. 178 by the letter
\(\alpha\),
so that the right-hand side
of the first equation in the box is written as \(Ct^m e^{\alpha
t}\). Recall that \(ar^2+br+c\) can be factored as
\(a(r-r_1)(r-r_2)\), where \(r_1\) and \(r_2\) are (potentially)
complex. The multiplicity \(s\) of \(\alpha\) (as a
root of \(p_L(r)\)) is the number of times \(r-\alpha\) appears in
this factorization. Thus:
- \(s=0\) if \(r-\alpha\) is not a factor of
\(p_L(r)\) (equivalently, if \(p_L(\alpha)\neq 0\));
- \(s=1\) if \(r-\alpha\) appears
exactly once in the factorization \(a(r-r_1)(r-r_2)\)
(equivalently, if the roots \(r_1,r_2\)
are distinct and \(\alpha\) is equal to one of them); and
- \(s=2\) if \(p_L(r)\) has a double root, and that
root is exactly \(\alpha\) (equivalently, if
\(p_L(r)=a(r-\alpha)^2\)).
(FYI: the terminology "multiplicity of a root of a polynomial" is
completely standard; it's just not mentioned in your textbook.)
Once you've done this reading and the exercises below, you should be
able to do all the problems from last semester's second
exam except #2. (I would expect you to be able to do questions
like these on your exam.)
Exercises:
4.4/ 9, 10, 11, 14,
15, 18, 19, 21, 22, 28,
29, 32.
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/ 1–8, 24, 25, 26, 28.
Use the "\(y=y_p+y_h\)" approach
discussed in class , plus superposition (problem 4.7/ 30, previously
assigned) where necessary, plus your knowledge (from Sections 4.2 and 4.3)
of how to solve the associated homogeneous equations for all the DEs
in these problems.
Note that the MUC is not
needed to do exercises 1–8, since (modulo having to use
superposition in some cases) the \(y_p\)'s are handed to you on a
silver platter, just as in the "\(y''+y=t^3\)" example done in class
on Monday 2/24/20.
|
| F 2/28/20 |
Second midterm exam (assignment is to study for it).
|
| M 3/9/20 |
Enjoy your Spring Break!
|
| W 3/11/20 |
4.4/ 12, 17. Problem 12
can also be done by Chapter 2 methods. The purpose of this exercise
in Chapter 4 is to see that it also can be done using the Method of
Undetermined Coefficients.
4.5/ 17, 18, 20, 22 , 23, 27, 29,
32, 34, 36,
41, 42. In #23,
the same comment as for 4.4/12 applies. 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.
|
| F 3/13/20 |
You'll need to do some of the problems based on your reading; as of
Wed. 3/11/18 we had not yet gone over all of this material in
class.
4.4/ 1–8, 16, 20, 24, 30, 31. In the
instructions for 27–32, the word "form" should be replaced by
"MUC form".
4.5/ 9–12, 14–16, 19, 21, 31, 35
4.5 (continued)/ 45. This is a nice problem that requires you
to combine several things you've learned. The strategy is similar to
the approach outlined in Exercise 41 (which is not an assigned
problem, but which you should read to understand the strategy
for #45). 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.
|
| M 3/16/20 |
Do these non-book exercises on the Method
of Undetermined Coefficients. The answers to these exercises
are here.
4.5 (continued)/ 37–40. In these, note that you are
not being asked for the general solution (for which you'd need
to be able to solve a third- or fourth-order homogeneous linear
DE, which we haven't discussed).
In a constant-coefficient differential equation \(L[y]=g\),
the
functions \(g\) to which the MUC applies are the same regardless of
the order of the DE, and, for a given \(g\), the MUC form of a
particular solution is also the same regardless of this order. The
degree of the characteristic polynomial is the same as the order of
the DE (to get the characteristic polynomial, just replace each
derivative appearing in \(L[y]\) by the corresponding power of
\(r\), remembering that the "zeroeth" derivative—\(y\)
itself—corresponds to \(r^0\), i.e. to 1, not to \(r\).)
However, a polynomial of degree greater than 2 can potentially have
have roots of multiplicity greater than 2. The possibilities for the
exponent "\(s\)" in the general MUC formula (for functions of "MUC
type" with a single associated "\(\alpha + i\beta\)") range from 0
up to the largest multiplicity in the factorization of \(p_L(r)\).
Thus the only real difficulty in applying the
MUC when \(L\) has order greater than 2 is that you have to factor a
polynomial of degree at least 3. Explicit factorizations are
possible only for some such polynomials. Every
cubic or higher-degree characteristic polynomial arising in this textbook is
one of these special, explicitly factorable polynomials (and even
among these special types of polynomials, the ones arising in the
book are very simplest):
- In all the problems in this textbook in which you have to
solve a homogeneous, constant-coefficient,
linear DE of order greater than two, the
corresponding characteristic polynomial has at
least one root that is an integer of small absolute value
(usually 0 or 1). For any cubic polynomial
\(p(r)\), if you are able to guess even one root, you can factor
the whole polynomial. (If the root you know is \(r_1\), divide
\(p(r)\) by \(r-r_1\), yielding a quadratic polynomial
\(q(r)\). Then \(p(r)=(r-r_1)q(r)\), so to complete the
factorization of \(p(r)\) you just need to factor \(q(r)\). You
already know how to factor any quadratic polynomial, whether or
not it has easy-to-guess roots.)
From the book's examples and
exercises, you might get the impression that plugging-in
integers, or perhaps just plugging-in \(0\), \(1\), and \(-1\),
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.)
- 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.)
You can also do problem 38 by extending the
method mentioned above for cubic polynomials. Start by guessing one
root \(r_1\) of the fourth-degree characteristic polynomial \(p(r)\).
(Again, the authors apparently want you to think that the way to find
roots of higher-degree polynomials is to plug in integers, starting
with those of smallest absolute value, until you find one that works.
In real life, this rarely works—but it does work in all the
higher-degree polynomials that you need to factor in this book.) Then
\(p(r)=(r-r_1)q_3(r)\), where \(q_3(r)\) is a cubic polynomial that you
can compute by dividing \(p(r)\) by \(r-r_1\). Because of the
authors' choices, this \(q_3(r)\) has a root \(r_2\) that you should be
able to guess easily. Then divide \(q_3(r)\) by \(r-r_2\) to get a
quadratic polynomial \(q_2(r)\)—and, as mentioned above, you
already know how to factor any quadratic polynomial.
- For
problem 40, you should be able to recognize that \(p_L(r)\) is \(r\)
times a cubic polynomial, and then factor the cubic polynomial by
the guess-method mentioned above (or, better still, recognize that
this cubic polyomial is actually a perfect cube).
|
| W 3/18/20 |
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. Exercises TBA.
4.7/ 9–20, 23ab. In #23, ignore the
first sentence ("To justify ...").
Reminder about some terminology. As I've
said in class, "characteristic equation" and "characteristic
polynomial" are things that exist only for constant-coefficient
DEs.
This terminology should be avoided in the setting of
Cauchy-Euler DEs (and
was avoided for these DEs in early editions of our
textbook). The term I used in class (online) for Equation (7) on
p. 194,
"indicial equation", is what's used in
most textbooks I've seen, and really is better
terminology—you invite confusion when you choose to
give two different meanings to the same terminology. 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; the substitution
leads to a
constant-coefficient equation with independent variable \(x\).)
In my experience
it's unusual to hybridize the terminology and call the book's
Equation (7) the characteristic equation for the Cauchy-Euler
DE, but you'll need to be aware that that's what the book
does. I won't consider it a mistake for you to use the
book's terminology for that equation, but you do need to
know how to use that equation correctly (whatever you call
it), and need to
understand me when I say "indicial equation".
In our textbook, p. 194's Equation (7) is actually introduced
twice for
Cauchy-Euler DEs, the second time as Equation (4) in Section
8.5. For some reason—perhaps an oversight—the authors
give the terminology "indicial equation" only in Section 8.5,
rather than when this equation first appears in the book's first
treatment of Cauchy-Euler DEs, i.e. in Section 4.7.
It's also rather unusual and ahistorical to
use the letter \(t\) as the independent variable in a Cauchy-Euler
DE, even though we're certainly allowed to use any letter we
want (that's not already being used for something else). The reason
we use `\(t\)' for constant-coefficient linear DEs (as well as some
others, especially certain first-order DEs), is that when these DEs
arise in physics, the independent variable represents time.
When a Cauchy-Euler DE arises in physics, almost always the
independent variable is a spatial variable, for which a
typical a letter is \(x\), representing the location of
something. In this case, the common substitution that reduces a
Cauchy-Euler DE to a constant-coefficient DE (for a different
function of a different variable) is the substitution
\(x=e^{<\mbox{new variable}>}\) rather than \(t=e^x\). Earlier
editions of our textbook used \(x\) as the independent variable in
Cauchy-Euler DEs, and made the substitution \(x=e^t\), exactly the
opposite of what is done in the current edition. (Again,
we're allowed to use whatever variable-names we want; the
letters we use don't change the mathematics. It's
just that in practical applications it's usually helpful mentally to
use variable-names that remind us of what the variables represent.)
Check directly that if the indicial equation for a
second-order homogeneous Cauchy-Euler DE
\(at^2y''+bty'+cy=0\) has complex roots \(\alpha \pm
i\beta\) (with \(\beta\neq 0\)), then the functions
\(y_1(t)=t^{\alpha}\cos(\beta \ln t)\) and
\(y_2(t)=t^{\alpha}\sin(\beta \ln t)\) are solutions of the DE.
Do non-book problem
13. (Parts (c) and (d) were done in the 3/16/20 online
lecture, but it won't hurt for you to do them again.)
Read Section 4.6.
|
| F 3/20/20 |
4.6/ 2, 5–8, 9, 10, 11, 12, 15, 17, 19 (first sentence only).
Remember that to apply Variation of
Parameters as presented in class (online), 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.
This is fine, but my personal preference is to put
the DE in standard form from the start, in which case the "\(a\)" in
the book's pair-of-equations (9) disappears.
4.7/ 37–40.
Redo 4.7/ 40 by starting with the substitution
\(y(t)=t^{1/2}u(t)\)
and seeing where
that takes you.
|
| M 3/23/20 |
Read Section 6.1. (We will not be covering Chapter 5.)
6.1/ 1–6, 7–14, 19, 20, 23.
Do
7–14 without using Wronskians.
The sets of
functions in these problems are so simple that, if you know
your basic functions
(see The Math
Commandments), Wronskians will only increase the
amount of work you have to do. Furthermore, in these
problems, if you find that
the Wronskian is zero then you can't conclude anything (from
that alone) about
linear dependence/independence. If you do not know your basic
functions, then Wronskians will not be of much help.
Read Section 6.2.
|
| W 3/25/20 |
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.
For some of these problems and the ones
below from Section 6.3, it may help you to first review my
instructions/hints for the assignment that was due 3/16/20.
6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the
annihilator method (which we are skipping for reasons of time); just
use what we've done in class with MUC and superposition.
|
| F 3/27/20 |
Read sections 7.1 and 7.2.
7.2/ 1–4, 6–8, 10, 12. (Note: "Use
Definition 1" in the instructions for
1–12 means "Use Definition 1", NOT Table
7.1 or any other table of Laplace Transforms.)
|
| M 3/30/20 |
7.2/ 13–20,
21–23, 26–28, 29a–d,f,g,j.
For 13–20,
do use Table 7.1 on p. 356, even though we haven't derived
all of the formulas there yet in class, or discussed
linearity of the Laplace Transform (Theorem 1 on p. 355) in class.
On your third midterm and final exam, you'll be given
this Laplace
Transform table. Familiarize yourself with where the entries
of Table 7.1 (p. 356) are located in this longer table. The
longer table comes from an older edition of your textbook, but is
very similar to one you can still find on the inside front cover
or inside back cover of hard-copies of the current edition, and
somewhere in the e-book (search there on "A Table of Laplace
Transforms").
Warning: On line 8 of this table, "\( (f*g)(t)\)"
is not \(f(t)g(t)\); the symbol "\(*\)" in this line denotes an
operation called convolution (defined in Section 7.8 of the
book, which we won't be covering), not simple multiplication.
For the ordinary product \(fg\) of functions \(f\)
and \(g\), there is no simple formula that expresses
\({\mathcal L}\{fg\}\) in terms of \({\mathcal L}\{f\}\) and
\({\mathcal L}\{g\}\).
|
| W 4/1/20 |
Read Section 7.3.
7.3/ 1–10, 12–14, 20, 31.
|
| F 4/3/20 |
7.4/ 11, 13, 14, 16, 20. You should be able to do these
with or without reading Section 7.4 first (see the
"Prerequisite" paragraph in the
syllabus, but there's
additional partial-fractions review in Section 7.4 if you need it.
Read Section 7.4 up through at least Example 4. The rest of the
section is the partial-fractions review of mentioned above.
7.4/ 1–10. (You should be able to do these based on your
reading and the 4/1/20 online lecture.)
7.5/ 15, 17, 18, 21, 22. Note that in these problems, you're being
asked only to find \(Y(s)\), not \(y(t)\).
The
"application/example of general importance" worked-through in
the online lecture 4/1/20 shows how to do problems like this.
|
| M 4/6/20 |
No new homework.
|
| W 4/8/20 |
Third midterm exam, assuming that all the logistical issues
brought up by the Honorlock-practice exam are solved early
enough. If they aren't, the exam will likely be delayed till
Wed. 4/8/20. Whichever day the exam is, your assignment due that
day is to study for the exam.
Fair-game material for this exam includes everything we've covered
since the last exam, up through the material in Sections 7.4 and 7.5
corresponding to the exercises due 4/3/20. (Remember that
"covered" includes classwork and homework, and "homework" includes
reading the relevant portions of the textbook,
with the exceptions noted below). As of
the date of the last exam, we had not yet finished with the Method
of Undetermined Coefficients for 2nd-order equations (Sections
4.4 and 4.5). The homework from these sections with due-dates in
March reflects the material from these sections that we covered
after Spring Break.
11/20
- In Section 6.1, you are not responsible for knowing what
the Wronskian is except when \(n=2\). In Theorem 2, p. 322,
replace the condition involving the Wronskian with "If the
solutions \(y_1, \dots, y_n\) are linearly independent on \(
(a,b) \)." (Linear dependence and independence are defined on
p. 323. The definition on p. 323 looks different from the one
I gave you, but is equivalent. For example, if \(f_1\) is a
linear combination of \(f_2, f_3, \dots, f_m\), then there are
constants \(c_2, c_3, \dots, c_m\) such that \(f_1 = c_2 f_2 +
c_3 f_3 + \dots + c_m f_m\). But then \((-1)f_1 + c_2 f_2
+c_3 f_3 + \dots + c_m f_m=0\), so if we set \(c_1=-1\), the
book's equation (23) is satisfied, so \(f_1, f_2, \dots, f_m\)
are linearly dependent according to the book's definition.
Conversely, if the functions \(f_1, f_2, \dots, f_m\) are
linearly dependent according to the book's definitio, then
equation (23) is satisfied for some constants \(c_1, c_2,
\dots, c_m\) that are not all zero; at least one must be
nonzero. If, for example, \( c_1\neq 0\), then we can divide
both sides of equation (23) by \(c_1\) and subtract the
resulting terms involving \(f_2, \dots, f_m\) from both sides
of the equation. This yields \(f_1=k_2 f_2 + k_3 f_3 + \dots
+ k_m f_m\), where \(k_i= - c_i/c_1\). Thus \(f_1\) is a
linear combination of \(f_2, \dots, f_m\), so the functions
\(f_1, f_2, \dots, f_m\) are linearly dependent according to
the definition given in class.)
- 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.
| F 4/10/20 |
- 7.4/ 21–24, 26, 27, 31.
- 7.5/1–8, 10, 29.
To learn some shortcuts for the partial-fractions work that's
typically needed to invert the Laplace Transform, you may want
first to read the web handout
"Partial fractions and
Laplace Transform problems".
|
| M 4/13/20 |
- Read Section 7.6 through p. 389 (the end of Example 5).
- 7.6/ 1–10, 11–18
|
|
W 4/15/20 |
- 7.6/ 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 exercises 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.
- 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 much, if 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.
- 8.2/1–6, 7, 8, 9, 10, 11–14, 17‐20,
23, 24, 27, 28, 37. Note:
- 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.
|
| F 4/17/20 |
- Read Section 8.3.
- 8.3/ 1, 3, 4, 5–8. (All you need to do these
problems is in the first page
of Section 8.3, ending with
Example 1.)
|
| M 4/20/20 |
- 8.3/ 11–14, 18, 20–22, 24, 25. These exercises will
become easier after I've done more examples in class than just the one
I did on Friday 4/17, but I think you'll have a better chance at
mastering this material before the final exam if you start now.
-
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 (and that I stated in the Wed. 4/15/20 lecture) 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 4/22/20 |
- 8.4/ 15, 20, 21, 23, 25.
|
|