| Date due |
Section # / problem #'s |
| F 8/23/19 |
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.
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.)
|
| M 8/26/19 |
1.2/ 1, 3–6, 19–22.
See Notes on some book problems.
Do non-book problem 2.
In
my notes, read from where you left off on p. 13
through the end of Section 3.1.2, and do the exercise on p. 17.
I update these notes from time to time
during the semester, 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."
|
| W 8/28/19 |
Read the remainder of Section 3.1.3 of
my notes, but not all at once.
Read through the end of Example 3.10 (the end is on p. 20) before
doing the exercises below, then do all the exercises below, then
return and do the rest of the reading. (As mentioned in an earlier
assignment, you may skip the material in blue, but don't overlook the
non-blue material on pp. 27–31.)
1.2/ 2, 9–12, 14–17, 30. In #30, ignore the
book's statement of the Implicit Function Theorem; use the
statement in my notes. The theorem stated in problem 30
is much weaker than the
Implicit Function Theorem, and should not be called by that
name.
|
| F 8/30/19 |
1.2/ 18, 23–28, 31. For 23–28, the instructions
should end with "... has a unique
solution on some open interval." Similarly, in 31a, "unique
solution" should be "unique solution on some open interval".
In the textbook, read Section 1.3, and read the first two
pages of Section 2.2 (through the end of Example 1).
1.3/ 2, 3
2.2/ 1–4, 6.
In my notes, read Section
3.1.4 up through Example 3.29 (the rest of this section will be in
the next assignment). Some of this has already been discussed in
class, so should go quickly. Also read Section 3.1.5 up through Definition
3.37. You are allowed to skip a portion of this material, as
indicated in the bold-faced sentence beginning
"Non-honors students ..." on p. 32.
In my notes, also read
Section 5.1 except for the
last paragraph. In class on Wednesday I stated this theorem
with "open rectangle" instead of "open set", since I'd not yet
given you the definition of "open set" (which is in the portion
of Section 3.1.5 assigned above). With the term
"open rectangle" instead of "open set", Theorem 5.1 is what
the textbook's Theorem 1
on p. 11 should have said; see the paragraph in my notes
after Theorem 5.1. (The theorem is nicer with "open set"
instead of "open rectangle", but most DE textbooks don't define
"open set" in the plane.)
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 9/6/19 |
In
my
notes, read
the remainder of Sections
3.1.4 and 3.1.5.
Reminder: reading
my notes is not
optional (except for portions that I [or the notes] say you may
skip, and the footnotes or parenthetic comments that say "Note to
instructor(s)"). Each reading assignment should be completed by the due
date I give you.
What I'm
putting in the notes are things that are not adequately covered in
our textbook (or any current textbook that I know of).
There is not enough time to cover most of these carefully in class;
we would not get through all the topics we're supposed to cover.
Do at least one of items 1 and 2 below:
- In my notes (version dated
8/30/19
or later), read
Sections 3.1.6 and 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.)
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 does not actually establish what it purports to
(because it puts no hypotheses on the functions \(p\) and \(g\),
without which several steps in the book's argument
cannot be justified),
and (2) the conclusion it purports to
establish skirts an important issue. (The question of whether
the method gives all the solutions, or even all the
non-constant solutions, is never mentioned, let alone answered.)
- In the textbook, read Section 2.2 and attempt to do problems
2.2/ 7–14 (note: "Solve the equation" means "Find
all [maximal] solutions of the equation"),
as well as non-book problems
3, 4, and 6.
(Download a fresh copy of the non-book problems page; I revised it a
few days ago.)
Answers to these
non-book problems and some others are posted on the "Miscellaneous
handouts" page.
Whichever of items 1 and 2 you don't do for this assignment will be
part of your next assignment. Advantages of doing item 2 now
are (i) you won't have as
many exercises to do later, (ii) the sooner you can start
putting methods into practice, the better, and (iii) you may
feel like doing something besides reading my notes. Disadvantages
are that I did not get quite far enough in Friday's lecture to
prepare you for doing problems—I think it would help you
to see me work one or two more problems first—and that the
preparation for these exercise the book gives you is really
not adequate.
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 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 9/9/19 |
In the previous assignment, you were given a choice of whether to
do item 1 or item 2 in the second bullet-point. Now do whichever of
these you did not already do.
2.2/ 17–19, 21, 24, 27abc
Do non-book problems 7,
8.
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.)
|
| W 9/11/19 |
2.3/ 1–6
Read Section 2.3 of the textbook.
In my notes, read
from the beginning of Section 3.2 (p. 68) up to, but not including,
the beginning of Section 3.2.1 (p. 73).
I will assign the
earlier Section 3.1.8 after we've spent another lecture on linear DEs.
With the exception of the notes' Definition 3.56, the Section 3.2
material in the reading above is basically not discussed in the book
at all, even though differential-form DEs appear in (not-yet-assigned)
exercises for the book's Section 2.2 and in all remaining sections of
Chapter 2. (Very little of what's in Sections 3.2.1–3.2.4,
3.2.6, or 3.3 of my notes is discussed in the book either.) I'm
assigning the early portion of Section 3.2 now since the rest of the
current assignment is pretty light.
This will (partially) spare you from having to do a lot of reading and
exercises at the same time once we get to the book's Section 2.4.
|
| F 9/13/19 |
2.3/ 7–9,
12–15 (note which variable is which in #13!), 17–20, 22.
When you apply the method
introduced in Wednesday'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 problems
9ab.
Read Section 3.1.8 of
my notes.
|
| M 9/16/19 |
2.3/
27a, 30–33, 35. See Comments, part 1, below.
Note about wording in #35: The term "a brine" in this
problem (and in the unassigned problem 2.2/ 33) is an instance
of improper English usage; 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"), which unfortunately appears in Example 1 of
Section 3.2.
In the textbook, read Section 2.4 through the boxed
definition "Exact Differential Form" on p. 59. See Comments,
part 2, below.
In my notes, read
from the beginning of Section 3.2.2 (p. 79) 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 3, below.
Comments, part 1.
One of the things you'll see in exercise 2.3/33 is that
(as indicated in the last assignment) what
you might think is only a minor difference between the DE's in parts
(a) and (b)—a sign-change in just one
term—drastically changes the nature of the
solutions. When solving differential equations, a tiny algebra slip
can make your answers utter garbage. For this reason, there is
usually no such thing as a "minor algebra error" in solving
differential equations. This is a fact of life you'll have to get
used to. The severity of a mistake is not determined by the number
of pencil-strokes it would take to correct it, or whether your work
was consistent after that mistake. If a mistake (even something as
simple as a sign-mistake) leads to an answer that's garbage, or that
in any other way is qualitatively very different from the correct
answer, it's a very bad mistake, for which you can expect a
significant penalty. A sign is the only difference between a rocket
going up and a rocket going down. In real life,
details like that
matter.
I urge you to develop (if you haven't already) the mindset of
"I really, really want to know whether my final answer is
correct, without having to look in the back of the book, or ask my
professor." Of course, you can find answers in the 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.
Comments, part 2.
There are some terminological
problems in Section 2.4 of the book, most notably an inconsistent
usage of the term "differential form". Most students will probably
not even notice the inconsistency,
but some may—especially the students with a deep interest in
mathematics—and I don't want anyone to come out 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 technically incorrect, but at the level
of MAP 2302 it is a very confusing use of the word "form", and the
less-misinterpretable term "differential" is perfectly
correct. The usage of "differential form" within the
definition-box, after the title, is inconsistent with the usage
outside the definition-box. The usage in the title is ambiguous;
it is impossible to tell whether the title is referring to an
exact differential, or to an equation with an exact
differential on one side and zero on the other.
In my notes I talk about "derivative
form" and "differential form" of a
differential equation. The meaning of the word "form" in
my notes is standard mathematical English, and is the same as in
each of the two occurences of "form" on on p. 58 of the book.
In this usage, "form of an equation" refers to the way an
equation is written, and/or 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 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 definition-box on p. 59—the
pronunciation of "differential form"
never has the accent on the first word; we either say
"differential form", with the accent on the second
word, or we accent both words equally.
-
The paragraph directly below the
"Exact Differential Form" box is not part of the current
assignment. However, for future reference, this paragraph is
potentially confusing or misleading, because while the first
sentence uses "form" in the way it's used on the preceding
book-page 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. Most likely, the authors didn't intentionally use
the term
"differential form" with two different meanings, but didn't notice
that this paragraph (and several others) make no sense if the term
"differential form" is assigned any consistent meaning.
Choosing to use the term "exact
differential form" in the first equation of this paragraph is,
itself, rather unusual. In any context
involving the advanced meaning of "differential
forms" (the meaning you haven't been given), there is a
standard meaning of "exact differential form"
(and a synonymous, simpler, and more common term is just
"exact 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", and it's 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 3. 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 large part of the reason for this is that a lot of
important issues are buried in the sentence on the book's p. 58
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; despite my criticisms, our
textbook is 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.
|
| W 9/18/19 |
2.4/ 1–8. Note (1) In class on Monday 9/16, 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 Wednesday'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. 86 (part of Section 3.2.4) through the bottom of
p. 100 (mid-way through Section 3.3). Again, the material in blue
(which is quite a lot of this portion of the notes) is optional
reading; you're required to read only the material in black.
2.2 (not 2.3 or 2.4)/ 5, 15, 16. (I did not assign these when we were
covering Section 2.2 because we had not yet discussed
"differential form".) An equation in differential form is called
separable if, in some region of the \(xy\) plane (not
necessarily
the whole region on which the given DE is defined), the given DE
is algebraically equivally equivalent to
an equation of the form
\(h(y)dy=g(x)dx\) (assuming the variables are \(x\) and \(y\)). This
is equivalent to the condition that the derivative-form equation
obtained by
formally dividing the original equation by
\(dx\) or \(dy\) is separable.
|
| F 9/20/19 |
Read the remainder of Section 2.4 of the textbook.
2.4/ 9, 11–14, 16, 17, 19, 20.
Read the online handout
A terrible way to
solve exact equations. The example in this
version of the
handout is rather
complicated; feel free to read the example in the
original version
instead. The only problem with the example in the original version is
that \(\int \sin x \cos x\, dx\) can be done three
ways (yielding three different antiderivatives, each differing from
the others by a constant), one of which happens to lead to the
correct final answer even with the "terrible method". Of course, if
the terrible method were valid, then it would work with any
valid choice of antiderivative. However, I've had a few students
who were unconvinced by this, and thought that because they saw a way to get
the terrible method to work in this example,
they'd be able to do it in any example. I've constructed the
more complicated example to make the failure of the terrible method
more obvious.
Where the handout says "(we proved it!)",
substitute
"(we will prove it if time permits)".
2.2 (not 2.3 or 2.4)/ 22. Note that although the differential
equation doesn't specify independent and dependent variables, the
initial condition does. Thus your goal in this exercise is to
produce an explicit solution "\(y(x)= ...\)". But this exercise is an
example of what I call a "schizophrenic" IVP. In practice, if you are
interested in solutions with independent variable \(x\) and dependent
variable \(y\) (which is what an initial condition of the form
"\(y(x_0)=y_0\)'' indicates), then the differential equation you're
interested in at the start is one in derivative form
(which in exercise 22 would be \(x^2 +2y \frac{dy}{dx}=0\), or an
algebraically equivalent version), not one in differential
form. Putting the DE into differential form is often a useful
intermediate step for solving such a problem, but differential form is
not the natural starting point. On the other hand, if what you are
interested in from the start is a solution to a
differential-form DE, then it's illogical to express a preference for
one variable over the other by asking for a solution that satisfies a
condition of the form "\(y(x_0)=y_0\)'' or "\(x(y_0)=x_0\)''. What's
logical to ask for is a solution whose graph passes through the
point \((x_0,y_0)\), which in exercise 22 would be the point
(0,2).
2.4 (resumed)/ 21, 22
(note that #22 is the same DE as #16, so you don't have to solve a new
DE; you just have to incorporate the initial condition into your old
solution). Note that exercises 21–26 are what I termed
"schizophrenic" IVPs. In all of these, the goal is to find an
explicit formula for a solution—if algebraically
possible—with the choice of independent/dependent variables
indicated by the initial condition. However, if 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.)
|
| M 9/23/19 |
2.4/ 32, 33ab
Read the statement
of non-book problem 10.
I'm not requiring you to do the problem, but I want you to be aware
of the fact stated in the problem.
Read The Math
Commandments.
In my notes,
finish reading Section 3.4 and read Section 3.5.
|
| W 9/25/19 |
First midterm exam (assignment is to study for it).
- In case you'd like additional
exercises to practice
with:
If you
have done
all your homework (and I don't mean "almost all"), you should
be able to do all the review problems on p. 79 except #s 9, 11,
12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35, 37, and the last part of
41. A good feature of the book's "review problems" is that, unlike
the exercises after each section, the
location gives you no clue as to what method(s) is/are likely to work.
Your exam will have no such clues on exams either. Even if you don't
have time to work through the problems on p. 79, they're good
practice for figuring out the appropriate methods are.
A negative feature of the book's exercises
(including the review problems) is that they
don't give you enough practice with a few important integration
skills. This is why I assigned my non-book problems 3,
4, 6, and 9. (Note: The answer to the
originally posted version of non-book problem 6 was wrong, and the correct
answer did not have a feature that I wanted it to have. I have
replaced the original problem with one that has the feature I wanted.)
- 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.
|
| F 9/27/19 |
Read section 4.1 of the book. (We're skipping Sections 2.5 and
2.6, and all of Chapter 3.)
|
| M 9/30/19 |
4.7 (yes, 4.7)/ 30. (You do not need any material from sections
4.1–4.7 to do this problem.)
|
| General info |
The grade scale for the first midterm is now posted on your
grade-scale
page, with a link to the list of scores so that you may see the
grade distribution. I don't post solutions to my exams.
Please read the rest of this message, read the information on the
grade-scale page, and re-read
the syllabus and course information
page,
so that you don't inadvertently ask me to spend time
answering a general question that I've already answered, or ask by
email a question that I never answer by email.
Questions specific to your
individual situation should be asked in office hours. If you'd
like to see me in my office, but have a conflict with all my
scheduled office hours, please let me know your complete
schedule—all the days/times with which you have
conflicts—and I'll try to find a day and time that works for
both of us. (I.e. please don't just ask me "Could I see you [or
would it be convenient for you to see me] Day X or Y during period Z
or W?", the most convenient days and times for you.
When you give me your full schedule, feel free to let me know
what preferences you have within your non-conflicting time-slots; just don't
send me your preferences instead of your complete
schedule.)
Exams will be
returned at the end of class on Monday 9/30/19. Students not in
class that day should pick up their exams during one of my office
hours as soon as possible.
After a week, I may toss out any exam that has not been picked up,
unless the student has made prior arrangements with
me and has a valid excuse. (Usually I hold onto exams longer;
just don't count on it.) Failure to
read this notice on time will not count as a valid excuse, since you're
supposed to be checking this homework page at least three times a week
to get the homework assignment that's due by the next class.
|
| W 10/2/19 |
4.7 (yes, 4.7)/ 1–8. 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]."
We will be covering
the Chapter 4 material in an order different from the book's
presentation. We haven't yet covered enough of the material in Section
4.2 for me to assign problems from that section.
Unfortunately, hardly any of that section's exercises are
doable until the whole section has been covered.
Regarding the now-returned exam: 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. Class performance on exam-problems 4 (58% average), 5
(53%), and especially problem 3 (36%) was disturbing (very
disturbing in the case of problem 3). Go over your returned
exam and do the following:
- Re-read your class notes regarding constant solutions.
These came up first when we discussed separable derivative-form
equations (for which we spent a lot of time on constant solutions),
and later in the setting of differential-form equations. We
discussed constant solutions both as part of the general solution
and as potential solutions to initial-value problems. We covered an
example in which assuming that the solution of an initial-value
problem took the form that we had derived for non-constant solutions
led to a contradiction, when in fact the solution of that IVP was
one of the constant solutions of the DE.
It wouldn't hurt you to re-read
the assigned portions of my notes relating to constant
solutions, either. (Most, but not all, of this was in the
section on separable equations.)
- Re-read your class notes on the example
"\(\frac{dy}{dx}=y^2-4\)" that I did in class, and compare this
example to the very similar DE in exam-problem 4. It's quite
frustrating to me that students' overall performance on this problem
was so poor despite my having given, in the classroom example, every
scrap of information and technique needed to do the exam problem (with
the exception that on your exam you needed also to be able to compute
\(\int x\,dx\), which was the one thing that everyone was able to do).
The way in which you'd solve for \(y(x)\) in the exam problem, after
doing the relevant integrals, is identical to the way I solved
for \(y(x)\) in the classroom problem. Additionally
disturbing is that the way that I solved for \(y(x)\) is high
school algebra that every student taking any college math class
should still be able to do. 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.
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 the first one or two that went in?
- Review non-book homework problem 8, and your class notes for
the day that I did this entire problem in class. The
IVP in
exam-problem 5a was nearly identical to this
problem in every important way (except that that on the exam, I
did not ask you for the domain of the solution).
- Re-read Theorem 5.1 in my notes, the "Fundamental Theorem of
Ordinary Differential Equations" (FTODE), which was assigned
reading and was covered in class, and was used several
times in the assigned readings from my notes, and that you
had practice applying in at least six homework problems (1.2/
23–28 in the assignment due 8/30/19), and for which
even the book's much weaker version (Theorem 1 on p. 11) was still
applicable to all the initial-value problems appearing on this
exam. The FTODE is one of several theorems we cover
whose proof is beyond the level of Calc 1-2-3 or an intro DE
course. But you're not being taught false theorems. Theorems that
are stated in the book, or in class, or in my notes, are true
facts, and you're expected to be able to use those facts
even though you won't have the tools to learn why they're
true until (and unless) you take Advanced Calculus. The same is
true of many theorems you were taught in Calculus 1—for
example, the Intermediate Value Theorem, the Extreme Value Theorem,
and part of the Fundamental Theorem of Calculus—but you are
held responsible for the content of those theorems as well.
One of the things the FTODE tells you
(whichever version you use) is that, if certain hypotheses are
satisfied, an initial-value problem does have some
solution. Therefore if you're given an IVP for which these
hypotheses are met on some open rectangle containing the
initial-condition point, and you think you've figured out that there
is no solution or that there's more than one solution, you
should know that you've made a mistake. Giving the
answer "This IVP has no solution" (or has more than one) for an IVP
that satisfies the hypotheses of the FTODE is contradicting
the FTODE, and shows that you don't understand what the theorem
says.
- Do you think that \(\frac{1}{1+1} = \frac{1}{1}
+\frac{1}{1}\), or that \(\frac{1}{5}=\frac{1}{2}+\frac{1}{3}\),
or that \(\frac{1}{1} +\frac{1}{-1} = \frac{1}{1-1}\)? Of course
not. So why would you think that the algebraic expression
\(\frac{1}{\sqrt{1+x^3}+C}\) is the same as
\(\frac{1}{\sqrt{1+x^3}} + \frac{1}{C}\)? When you make a mistake
like this, you are violating the second of
the Math
Commandments that I had you read for homework. Mistakes like
"\(\frac{1}{A+B}=\frac{1}{A}+\frac{1}{B}\)" and
"\(\sqrt{A+B}=\sqrt{A}+\sqrt{B}\) " should never be seen in a
college math class, no matter how complicated the expressions
\(A\) and \(B\) are.
- As you know, \(5=2+3\) and \(10=9+1\). Does this make you
think that \(10^5=10^2+10^3\) or that \(2^{10} =
2^9 + 2^1\)? Of course not. Do you think that \(e^5=e^2+e^3\) or
that \(e^{10} = e^9 + e^1\)? Hopefully not. So why would you
think that the algebraic expression \(e^{x^2+C}\) is the same as
\(e^{x^2} + e^C\)? Mistakes like this also violate the second
Math Commandment, and, again, should never be seen in a college math
class.
Although I shouldn't ever be seeing these errors,
I do see them every year; your class is not worse in this
respect than a typical MAP 2302 class. That's why one of the
resources that's been on the Miscellaneous Handouts page all
semester is an Exponential Review Sheet, and is also why the
syllabus says, in boldface, "If you are weak
in [exponentials and other topics areas listed in the previous
sentence], 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." (This was also stated again in the homework
assignment due 9/6/19.) Many students in this class need this
review. Unfortunately, most students who need to
review before they risk losing points on an exam, wait to
review until after they've lost points—if they even do their
review after that.
|
| M 10/7/19 |
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".)
4.7/ 25
4.2/ 1, 3, 4, 7, 8, 10, 12, 13–16, 18, 26, 27–32,
46ab
|
| W 10/9/19 |
4.7/ 26abc
Read pp. 162–163 through the end of Example 3.
4.2/ 2, 5, 9, 11, 17, 19, 20, 35
|
| F 10/11/19 |
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 (the rest of) 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, and
may not get to) 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 Wednesday 10/9/19, 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 10/14/19 |
No new homework. If you are behind on your homework, use this
opportunity to catch up.
|
| W 10/16/19 |
Read Section 4.4 up through at least Example 3. (See
below for exercises.)
Read Section 4.5 up through at least Example 2. (See
below for exercises.)
We
will be covering Sections 4.4 and 4.5 simultaneously, more
or less, rather than one after the other. What most mathematicians
(including me) call "the Method of Undetermined Coefficients" is what
the book calls "the Method of Undetermined Coefficients plus
superposition." You should think of Section 4.5
as completing the (second-order case of) the Method of
Undetermined Coefficients, whose presentation is begun in Section 4.4.
Unfortunately, the way the Section 4.4 exercises are
structured, you can't do more than a handful of the exercises before
having completed the whole section, which is at least two or three
full lectures worth of material.
Over the next few classes, I will be
assigning almost all the exercises in sections 4.4 and 4.5. To help
avoid giving you one massive assignment when we're done with 4.4 and
4.5,
I'll be spreading the problems out over several
assignments. The order in which
I'm assigning problems corresponds to the
order in which I'll be covering the material in class, which is
different from the order of presentation in the book.
Below, in the current assignment, are exercises that can be done based just on what we got through in class
on Monday 10/14/19 (plus earlier class work and homework).
If, based on your reading, you
are able to start doing problems from the next assignment as well, I
highly recommend that you do so. This will help you avoid a
work-crunch later.
4.4/ 9 (note that \(-9=-9e^{0t}\) ),
10, 11, 14 .
Add parts (b) and (c) to 4.4/ 9–11, 14, 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: \(y(0)=1, y'(0)=2\).
4.5 (not 4.4)/ 1–8.
Use the "\(y=y_p+y_h\)" approach
discussed in class , plus superposition (problem 4.7/ 30, previously
assigned) where necessary, plus your knowledge (from Sections 4.2 and 4.3)
of how to solve the associated homogeneous equations for all the DEs
in these problems.
We have not yet
discussed in class how one might
find all the \(y_p\)'s in these problems, but you don't need
to know that for these problems, since (modulo having to use
superposition in some cases) the \(y_p\)'s are handed to you on a
silver platter, just as in the "\(y''+y=t^3\)" example done in
class on Monday 10/14/19.
4.5 (continued)/ 25, 26, 28
|
| F 10/18/19 |
Read the remainder of Sections 4.4 and 4.5.
4.4/ 1–8,
13, 18. Add parts (b) and (c) to #18 as follows:
- (b) Find the general solution of the DE.
- (c) Find the solution of the initial-value problem for the DE
with the initial conditions \(y(0)=1, y'(0)=2\).
4.5/ 9–16
If, based on your reading, you
are able to start doing problems from the next assignment as well, I
highly recommend that you do so. This will help you avoid a
work-crunch later.
|
| M 10/21/19 |
4.4/ 15–17, 19–26, 27–32. In the
instructions for 27–32, the word "form" should be replaced by
"MUC form"
You'll have to do most of these problems
based on your reading of Section 4.4; what we've done in class so
far doesn't cover most of these exercises.
4.4 (continued)/ 12. This problem
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 / 23 (the same comment as for 4.4/12 applies)
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.
|
| W 10/23/19 |
4.5/ 17–22, 24–30, 31–36,
41, 42. In the
instructions for 31–36, the word "form" should be replaced by
"MUC form".
Problem 42b (if done
correctly) shows that the particular solution of the DE in part (a)
produced by the Method of Undetermined Coefficients actually has
physical significance.
Do these non-book exercises on the Method
of Undetermined Coefficients. The answers to these exercises
are here.
4.5 (continued)/ 38, 40.
Regarding #38 and #40:
In a constant-coefficient differential equation \(L[y]=g\),
the
functions \(g\) to which the MUC applies are the same regardless of
the order of the DE, and, for a given \(g\), the MUC form of a
particular solution is also the same regardless of this order. The
degree of the characteristic polynomial is the same as the order of
the DE (to get the characteristic polynomial, just replace each
derivative appearing in \(L[y]\) by the corresponding power of
\(r\), remembering that the "zeroeth" derivative—\(y\)
itself—corresponds to \(r^0\), i.e. to 1, not to \(r\).)
However, a polynomial of degree greater than 2 can potentially have
have roots of multiplicity greater than 2. The possibilities for the
exponent "\(s\)" in the general MUC formula (for functions of "MUC
type" with a single associated "\(\alpha + i\beta\)") range from 0
up to the largest multiplicity in the factorization of \(p_L(r)\).
Thus the only real difficulty in applying the
MUC when \(L\) has order greater than 2 is that you have to factor a
polynomial of degree at least 3. Explicit factorizations are
possible only for some such polynomials. Every
cubic or higher-degree characteristic polynomial arising in this textbook is
one of these special, explicitly factorable polynomials (and even
among these special types of polynomials, the ones arising in the
book are very simplest). For problem
38, note that if all terms in a polynomial \(p(r)\)
have even degree, then effectively \(p(r)\) can be treated as a
polynomial in the quantity \(r^2\). Hence, a polynomial of the form
\(r^4+cr^2+d\) can be factored into the form \((r^2-a)(r^2-b)\),
where \(a\) and \(b\) either are both real or are complex-conjugates
of each other. You can then factor \(r^2-a\) and \(r^2-b\) to get a
complete factorization of \(p(r)\). (If \(a\) and \(b\) are not real,
you may not have learned yet how to compute their square roots, but
in problem 38 you'll find that \(a\) and \(b\) are real.) For
problem 40, you should be able to recognize that \(p_L(r)\) is \(r\)
times a perfect cube.
|
| F 10/25/19 |
No new homework. Catch up on any homework you haven't done.
For students who want an extra supply of
exercises to practice
with: you should be able to do the review problems 1–36 on
p. 231, except those in which the DE is not a
constant-coefficient equation. However, be aware
that the types of problems on this page, or elsewhere in the
book, do not represent all the types of problems you could see
on your exam. I have done some other types of problems in
class, and your homework has included non-book problems.
To
solve the order-3 constant-coefficient
DEs on p. 231
you need to be able to factor the characteristic
polynomial, so here's a hint: All the cubic characteristic
polynomials arising in this textbook have at least one root that
is an integer of small absolute value. If you are able to guess
one root, you can factor a cubic polyomial \(p(r)\). (If the root you
know is \(r_1\), divide \(p(r)\) by \(r-r_1\), yielding a
quadratic polynomial \(q(r)\). Then \(p(r)=(r-r_1)q(r)\), so to
complete the factorization of \(p(r)\) you just need to factor
\(q(r)\).) If you know the
Rational Root Theorem then for all the cubic characteristic
polynomials arising in this textbook, you'll be able to guess
an integer root quickly. If you do not know the Rational Root
Theorem, you will still be able to guess an integer root
quickly, but perhaps slightly less quickly. (From the book's
examples and exercises, you might get the impression that
plugging-in integers is the only tool for trying to guess a root
of a polynomial of degree greater than 2. If you were a
math-team person in high school, you should know that this is
not the case.)
|
| M 10/28/19 |
Second midterm exam (assignment is to study for it).
|
| General info |
The most likely date for the third midterm is now Monday,
Nov. 25 (the Monday before the Thanksgiving break).
|
| W 10/30/19 |
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. Based on your reading, try to get a head-start
on the next assignment's exercises from Section 4.7.
Read Section 4.6.
|
| F 11/1/19 |
4.7/ 9–20, 23, 24, 27
Reminder about some terminology. As I've
said in class, "characteristic equation" and "characteristic
polynomial" are things that exist only for constant-coefficient
DEs. 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 will be using in class 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. In our
textbook, p. 194's Equation (7) is introduced twice as an
algebraic equation associated to a Cauchy-Euler DE, 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 Section
4.7's treatment of Cauchy-Euler DEs.
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 using the substitution
\(t=e^x\) in the Cauchy-Euler DE. (That's if \(t\) is the
independent variable in the given Cauchy-Euler equation; you then
get a constant-coefficient equation with independent variable
\(x\).)
In my experience
it's unusual to hybridize the terminology and call the book's
Equation (7) the characteristic equation for the Cauchy-Euler
DE, but you'll need to be aware that that's what the book
does. I won't consider it a mistake for you to use the
book's terminology for that equation, but you do need to
know how to use that equation correctly (whatever you call
it), and need to understand me when I say "indicial
equation".
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.)
Do non-book problem
12.
|
| M 11/4/19 |
No new homework. If you are behind on your homework, use this
opportunity to catch up.
|
| W 11/6/19 |
4.6/ 2, 5–8, 9, 10, 11, 12, 15, 17, 19 (first sentence only).
Remember that to apply Variation of
Parameters as presented in class, you must first put the DE in
"standard linear form", with the coefficient of the second-derivative
term being 1 (so divide by the coefficient of this term, if the
coefficient isn't 1 to begin with). The book's approach to remembering
this is to cast the two-equations-in-two-unknowns system as (9) on
p. 188.
4.7/ 37–40.
Note that it is possible to solve all the DEs
37–40 (as well as 24cd) either by the Cauchy-Euler substitution
"\(t=e^x\)" applied to the inhomogeneous DE, or by
the indicial equation
just to find a FSS for the associated homogeneous DE and then
using Variation of Parameters for the inhomogeneous DE. Both methods
work. I've deliberately assigned exercises that have you solving some
of these equations by one method and some by the other, so that you
get used to both approaches.
Redo 4.7/ 40 by starting with the substitution
\(y(t)=t^{1/2}u(t)\)
and seeing where
that takes you.
|
| F 11/8/19 |
Read Section 6.1. (We will not be covering Chapter 5.)
6.1/ 1–6, 7–14. (Do these based on your reading;
they're not very difficult.) Do
7–14 without using Wronskians.
The sets of
functions in these problems are so simple that, if you know
your basic functions
(see The Math
Commandments), Wronskians will only increase the
amount of work you have to do. Furthermore, in these
problems, if you find that
the Wronskian is zero then you can't conclude anything (from
that alone) about
linear dependence/independence. If you do not know your basic
functions, then Wronskians will not be of much help.
Read Section 6.2.
|
| W 11/13/19 |
6.1/ 19, 20, 23.
6.2/ 1, 9, 11, 13, 15–18. The characteristic polynomial for #9
is a perfect cube (i.e. \( (r-r_1)^3\) for some \(r_1\)); for #11 it's
a perfect fourth power.
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 assignments that had due-dates 10/23/19
(just the Section 4.5 problems) and 10/25/19.
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 11/15/19 |
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.)
If you feel sufficiently prepared by
your reading, start on the exercises in the next assignment.
|
| General info |
The third midterm will be given Monday,
Nov. 25 (the Monday before the Thanksgiving break).
|
| M 11/18/19 |
7.2/ 13–20 (for these,
do use Table 7.1 on p. 356, even though we haven't derived
all of the formulas there yet),
21–23, 26–28.
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 book, 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 \(f(t)g(t)\); "\(*\)" 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.
Unfortunately, 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
\({\mathcal L}\{g\}\).
Read Section 7.3.
|
| W 11/20/19 |
7.2/ 29a–d,f,g,j.
7.3/ 1–10, 12–14, 20, 31.
7.4/ 11, 13, 14, 16, 20. You should be able to do these
with or without reading Section 7.4 first (recall from the
"Prerequisite" paragraph in the
syllabus that
the method of
partial fractions is something you're expected to know), but there's
additional review in Section 7.4 if you need it.
|
| F 11/22/19 |
7.5/ 15–22. . Note that in these problems, you're being
asked only to find \(Y(s)\), not \(y(t)\).
)
Read Section 7.4 up through at least Example 4 (the rest of the
section is a review of partial fractions).
7.4/ 1–10. (You should be able to do these based on
your reading.)
|
| M 11/25/19 |
Third midterm exam (assignment is to study for it).
|
| M 12/2/19 |
Sorry to give you homework over the Thanksgiving break, but with
only two classes remaining after the break, it's an unfortunate
necessity.
7.4/ 21–24, 26, 27, 31.
Read Section 7.5 through Example 3.
7.5/1–8, 10, 29. (You should be able to do these based
on your reading. I'll do some examples like these in class on
Monday Dec. 2, but if you wait until after that to start on these
problems, you won't have enough time to learn new material and
practice what you'll need to be able to do on the Dec. 10 final
exam.) To learn some shortcuts for the partial-fractions work that's
typically needed to invert the Laplace Transform, you may want
first to read the web handout
"Partial fractions and
Laplace Transform problems".
|
| W 12/4/19 |
Read Section 7.6 through p. 389 (the end of Example 5).
7.6/ 1–10 , 11–18, 29–32.
For all of the above problems in which you
solve an IVP, write your final answer in "tabular form", by which
I mean an expression like the one given for \(f(t)\) in Example 1,
equation (4), p. 385. Do not leave your final answer in the
form of equation (5) in that example. On an exam, I would treat
the book's answer to problems 19–33 as incomplete, and would
deduct several points. The unit step-functions and "window
functions" (or "gate functions", as I call them) should be viewed
as convenient gadgets to use in intermediate steps, or
in writing down certain differential equations (the DEs
themselves, not their solutions). The purpose of these special
functions is to help us solve certain IVPs efficiently;
they do not promote understanding of solutions. In fact, when
writing a formula for a solution of a DE, the use of unit
step-functions and window-functions often obscures
understanding of how the solution behaves (e.g. what its graph
looks like).
For example, with the least
amount of simplification I would consider acceptable, the
answer to problem 23 can be written as
$$ y(t)=\left\{\begin{array}{ll} t, & 0\leq t\leq 2, \\
4+ \sin(t-2)-2\cos(t-2), & t\geq 2.\end{array}\right.$$
The book's way of writing the answer obscures the fact that the
"\(t\)" on the first line disappears on the second
line—i.e. that for \(t\geq 2\), the solution is purely
oscillatory (oscillating around the value 4); its magnitude does
not grow forever.
In this example, using trig identities the
formula for \(t\geq 2\) can be further simplified to several
different expressions, one of which is \(4+
\sqrt{5}\sin(t-2-t_0)\), where \(t_0=\cos^{-1}\frac{1}{\sqrt{5}} =
\sin^{-1}\frac{2}{\sqrt{5}}\). (Thus, for \(t\geq 2\), \(y(t)\)
oscillates between a minimum value of \(4-\sqrt{5}\) and a maximum
value of \(4+\sqrt{5}\).) This latter type of simplification is important
in physics and electrical engineering (especially for electrical
circuits). However, I would not expect you to do this further
simplification on an exam in MAP 2302.
In Wednesday's class (which is our last class), my
top priority will be to work some problems similar to the homework
problems from Section 7.6. Thus we won't have our usual pre-exam
Q&A review (although if there's time left after I've worked what I
think are enough problems of "Section 7.6", I can field questions
on earlier material).
|
| General info |
My usual office-hour schedule does not apply after
Wed. Dec. 4. I will have an office hour on Friday, but it may
or may not be at 3:00 p.m. If I'm going to hold the office hour at
a different time that day, I'll announce that by email, so make sure
to check for emails from me. I will probably have an office hour on
Mon. Dec. 9 (time TBA).
Between now and the final exam:
- Eat differential equations for breakfast, lunch, and dinner. Have
a differential equations snack before you go to bed, and dream about
differential equations.
- If you talk to anyone, talk only about differential equations.
- If you watch TV, watch only the Differential Equations Channel.
- Decide what differential equation you want to be when you grow
up.
|
12/10/19
Final Exam |
The final exam will be given on Tuesday, December 10, starting at
12:30 p.m., in our usual classroom.
|