In the table below, "NSS" stands for our textbook. Exercises are
from NSS unless otherwise specified.
| Date due |
Section # / problem #'s |
| M 8/26/24 |
Read
the class home page and
syllabus webpages.
Go to the Miscellaneous Handouts page (linked to the
class home page) and read the web handouts
"Taking and Using
Notes in a College Math Class," "Sets and Functions,", and
"What is a solution?"
Never treat any "reading" portion of any assignment
as optional, or as something you're sure you already know,
or as something you can postpone (unless I tell you
otherwise). I can pretty much guarantee that every one of my
handouts has something in it that you don't know, no matter how
low-level the handout may appear to be at first.
Read Section 1.1 and do problems 1.1/ 1–16.
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).
Do non-book problem 1.
In my notes on
first-order ODEs (also linked to the Miscellaneous Handouts page),
read the first three paragraphs of the
introduction, all of Section 3.1, and Section 3.2.1 through
Definition 3.1.
In this and future assignments from these notes,
you should skip anything labeled "Note(s) to instructors".
|
| W 8/28/24 |
It's OK if you don't get this assignment done
before the Wed. 8/28 class; it's a little ahead of where we ended
Monday's class (but is not all of what I'd planned to assign). I'm
posting it anyway so that you're not surprised with an extra-long
assignment in the near future.
1.2/ 1, 3–6
Something I didn't have time to talk
about in Monday's class: Whenever you
see the term "explicit solution" in the book, you should
(mentally) delete the word "explicit".
See Notes on some book
problems for additional corrections to the wording of several
of the Section 1.2 problems.
Note: The exercise portions of many
(probably most) of your homework assignments will be a lot more
time-consuming than in the assignments to date; I want to give you
fair warning of this before the end of Drop/Add.
However, since my posted notes are only
on first-order ODEs, the reading portions of the
assignments will become much lighter once we're finished with
first-order equations (which will take the first month or so of
the semester).
In
my notes, read from where you left off in the last assignment
through Example 3.11 (p. 15).
|
| F 8/30/24 |
1.2/ 14, 15, 17, 19–22
In the textbook, read the first page
of Section 2.2, minus the last sentence. (We will discuss how
to solve separable equations after we've finished discussing
linear equations, the topic of Section 2.3. The 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
my notes, read from where you left off in the last assignment
through the one-sentence paragraph after Definition 3.18 (currently
the top two lines on p. 22).
|
| W 9/4/24 |
In Section 2.3, read up through p. 50.
If you missed class on Friday 8/30, or to review
what I said about notation for indefinite integrals, go to the
first bullet-point in the assignment due 1/19/24 on
my Spring 2024 homework page,
and read from the beginning of the second sentence ("Remember
...") to the end of the green text.
On Wed. 9/4 we'll cover the method you're reading about on
pp. 48–50. If you want to read some examples to get a
feel for the method before we get to examples in class, it's
okay to look at Examples 1–3, but be warned:
Examples 1 and 2 have some extremely poor writing that you
probably won't realize is poor, and that
serves to reinforce certain bad habits—ones that
most students have, but few are aware of. (Example 3 is
better written, but shouldn't be read before the other two.)
Specific
problems with Example 2 (one of which also occurs in Example
1) are discussed in the same
Spring 2024 assignment as above. The most pervasive of these
is the one in last small-font paragraph.
I had hoped to get through most of Section 2.3 on Friday 8/30,
giving you the preparation needed for exercises I should be
assigning by now. We didn't get far enough for that, so I've
pushed those exercises into the next assignment, as part of
that assignment. I still encourage you to try to do as
many of those as you can before Wednesday based on your reading;
otherwise you're likely to end up with a double-length assignment
due Friday. If you don't feel prepared to get a big head-start on
the exercises I'm posting there tonight (Fri. 8/30) already posted
there, I encourage you to use the (estimated) time
you would have spent on them to get ahead in
your other classes as much as you can. That way you'll have
less pressure from those classes once you're ready to start on
your (not yet complete) DE assignment for Friday 9/6.
In
my notes, read from where you left off in the last assignment
through the end of Section 3.2.4 (the bottom of p. 26).
|
| F 9/6/24 |
2.3/ 7–9,
12–15 (note which variable is which in #13!),
17–20
When you apply the
procedure we derived for solving first-order linear DEs
(which is in
the box on p. 50, except that the book's "\(\int
P(x)\,dx\)" 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 \(a_1(x)\) before you can use the formula for \(\mu(x)\) in
the box on p. 50; otherwise the method doesn't work). Be
especially careful to identify the function \(P\) correctly; its
sign is
very important. For example, in 2.3/17,
\(P(x)= -\frac{1}{x}\), not just \(\frac{1}{x}\).
In
my notes, read from the
beginning of Section 3.2.5 (p. 27) through the end of
Definition 3.22 (p. 33), plus the paragraph after that definition.
(All of this is needed for a
proper understanding of the word "determines" in the book's Definition
2 in its Section 1.2! [I still haven't defined "implicit solution of a
DE" yet; the above reading is needed just to understand the single
word "determines" in that definition] This is one of the biggest
reasons I didn't assign you to read Section 1.2; would you have known
what "determines" is supposed to mean on p. 8?) Then
do the exercise on p. 33 of my notes.
2.2/ 34. Although this exercise is in the section on
"Separable Equations" (which we haven't discussed yet), the DE
happens to be linear as well as separable, so you're equipped to
solve it. For solving this equation, the "linear equations
method" is actually simpler than—I would even say
better than—the (not yet discussed) "separable
equations method".
(The
same is true of Section 2.1's equation (1) in , which the book solves
by the "separable equations method"—and makes two
mistakes in the sentence containing equation (4). This is why
I have not assigned you to read Section 2.1.)
|
| M 9/9/24 |
2.3/ 22, 23, 25a, 27a, 28, 31, 33, 35
See my Spring 2024 homework
page
for corrections to some of the Section 2.3 exercises. Also,
in that same assignment (the one that was due 1/22/24), read the
three paragraphs at the bottom of the assignment.
Do non-book problem
2.
In
my notes:
- Read the remainder of Section 3.2.5 (pp. 33-34).
- In Section 5.5, read the statement of the Implicit Function
Theorem (p. 150). [Note added 9/12/24
In this assignment, I should have had you read all of Section
5.5. I'm adding that to a later assignment.]
- In Section 3.2.6, read up through the end of Example 3.26
(p. 38).
- After you've done that reading,
do the following exercises from the
textbook: 1.2/ 2, 9–12, 30. In #30, ignore the
book's statement of the Implicit Function Theorem; use the
statement in my notes. The theorem stated in problem 30
is much weaker than the Implicit Function Theorem, and should
not be called by that name.
Note: Whenever I update my notes, including to correct typos that might
be confusing, I update the
version-date line on p. 1. 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. (For a while,
whenever I've clicked on the link to my notes in the homework page,
my
browser has been smart enough to re-load the document from the
source, thereby getting
the most up-to-date version, instead of
reloading a cached version. Browsers weren't always smart enough to
do that, and I don't know whether all of them are now.)
|
| W 9/11/24 |
In my notes, read
Section 5.2 and 5.4.
(If you have any uncertainty
about what an interval is, read Section 5.1 as well.
If you need to review
anything about the Fundamental Theorem of Calculus, read Section 5.3.)
My notes' Theorem
5.8, the "FTODE", is what the textbook's Theorem 1 on
p. 11 should have said (modulo my having used
"open set"
in the FTODE instead of the book's "open rectangle").
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.
1.2/ 18, 23–28, 31. Do not do these until
after you've read Section 5.4 in my notes.
Anywhere that the book asks you whether its Theorem 1 implies
something, replace that Theorem 1 with the FTODE stated in my notes.
For 23–28, given the book's reference
to (its) Theorem 1, for internal consistency the instructions
should have ended with "... has a unique solution on some open
interval." Similarly, in 31a, "unique solution" should have
been "unique solution on some open interval". However,
since I'm having you use the FTODE as stated in my notes, rather
than Theorem 1, what you should insert instead of "on some open
interval" is "on every sufficiently small
interval containing
[the relevant number]." The `relevant number' is \(x_0\) in 31ab;
in 23–28 and 31c it's whatever number is given for the value
of the independent variable at the initial-condition point.
For all these exercises except
#18, it may help you to
look at Examples 8 and 9 on p. 13. (In these examples, make same
replacements and/or insertions that I said to make for the
exercises.)
In my notes, read
Section 3.2.6 up through the
next-to-last paragraph on p. 42.
|
| F 9/13/24 |
Skim Section 2.2 in the textbook, up through Example 3.
I have very mixed feelings about having
you read this section. The book's explanations and
definitions in this section say many of the right
things, but don't hold up under scrutiny, and there'a lot of
poor writing that I hate exposing you to.
We still have at least one
lecture's worth of
conceptual material that's absent from the book, before which
doing the exercises in Section 2.2 would amount to little more than
pushing the symbols around the page a certain way. However,
you do need to start getting some practice with the mechanical
part of the
separation-of-variables method; otherwise you'll have too much to
do in too short a time. So I've assigned some exercises from
Section 2.2 below, for you to attempt based on your reading, but
with special temporary instructions.
2.2/ 7–14. For now (with the Friday 9/13
due date), all I want you to do in these exercises is
to achieve
an answer of the form of equation (3) in the box on
p. 42—without worrying about intervals, regions, or exactly what
an equation of this form has to do with (properly
defined) solutions of a DE.
Save your work, so that when I re-assign these exercises later,
when your goal will be to get a complete answer that you fully
understand, you won't have to re-do this part of the work.
In my notes:
- Read the
remainder of Section 5.5. (In an earlier assignment, I had you
read the statement of the theorem at the beginning of this section,
the Implicit Function Theorem.)
- Finish reading Section 3.2.6.
|
| M 9/16/24 |
In my notes:
- Read Section 3.2.7 up through the paragraph after
Definition 3.34.
- Read Section 3.2.10 up through the paragraph after the
statement of Theorem 3.44. (Currently,
this paragraph occupies the first two lines of p. 63, and explains
notation that you may not understand in Theorem 3.44, equation
(3.103).) This theorem assures us that, when its
hypotheses are met, every solution of \(\frac{dy}{dx}=g(x)p(y)\)
in the indicated region \(R\) is either a constant solution or can
be found, at least in implicit form, by separation of variables (the
"brain-off" method in the box on p. 42 of the textbook).
On my Spring 2024 homework
page, go to the assignment that was due 1/26/24, and read the
(whole) second bullet-point (which continues until the end of that
assignment). This details several of the items that are misleading or
just plain wrong in the book's Section 2.2. In the last
non-parenthetic sentence of that assignment, "the method we've
studied" is the method that we've just begun to study this
semester (the method summarized by Theorem 3.44 in my notes, and
justified by the proof of that theorem a few pages later).
Return to exercises 2.2/ 7–14 that I had you partially
do in the previous assignment. Using Theorem 3.44 in my notes,
this time find all the solutions. Don't worry about
graphing the solution-curves for any of the exercises in
the current assignment; that's more than the exercises
are asking for, and would take more time than it's
worth. In the example I did in class on Friday, I graphed the
solution curves just to emphasize that even a
simple-looking family of algebraic equations ( \( \big\{y=
1-\frac{2}{x^2+C}\big\}\) in that example) can represent a
surprisingly intricate collection of solution curves. Knowing that
that was something I wanted to illustrate, I specially chose
an example in which the
solution curves were relatively easy for me to
graph (which still didn't keep me from making a
couple of mistakes at the board ...).
You are not yet expected to understand
yet why the two-part method given by my notes' Theorem
3.44 works, or to fully understand "implicit solutions". For now, you are just getting practice with
the two-part procedure for solving
separable DEs
(one part being separation
of variables [the box on NSS p. 42], the other being
finding any constant solutions the DE may have [it may not
have any]).
Do non-book problems
3–6. Although I haven't finished discussing various
subtleties, or justified the separation-of-variables technique yet,
the two-part procedure mentioned above does
find all the solutions of the DEs in this
assignment.
Answers to these
non-book problems are posted on the
"Miscellaneous handouts" page.
General comment. In doing the exercises
from Section 2.2 or the non-book problems 3, 4, and 5, you may
find that, often, the hardest part of doing
such problems
is doing the integrals. I
intentionally assign problems that require you to refresh most of your
basic integration techniques (not all of which are adequately
refreshed by the book's problems).
2.2/ 17–19, 21, 24
The book's IVP exercises are not rich
enough, by a long shot, to illustrate the dangers
of keeping your brain turned off after you've separated
variables (putting all \(y\)'s on one side of the equation and all
\(x\)'s on the other, if these are the variable-names) and done
the relevant integrals. Non-book problems 7 and 8, which will be
in an upcoming assignment, were constructed to remedy this
poverty. Feel free to tackle these before they're assigned.
|
| W 9/18/24 |
Note: Some of the integrals in the
previous assignment(s) and the current one are best done using the
method of partial fractions. If you need to review the
method, you can undoubtedly find it online somewhere, but our
textbook has its own review on pp. 370–374, in Section 7.4.
This review is interspersed with examples related to the topic of
Chapter 7, Laplace Transforms, which we are a long way from
starting to cover. (It will be one of the last topics of the
semester, time permitting.) For purposes of simply reviewing
partial fractions, ignore everything in Esamples 5, 6, and
7 except for the partial fractions computations. (For
example, ignore any equation that has the a curly "L" in
it.) In the portion of the review focused on quadratic expressions
with no linear, real factors, you can used either of the
expansions the book gives on p. 373: the one in black with
coefficients that the book calls \(C_i\) and \(D_i\), or the one
in blue a few lines below, involving coefficients that the book
calls \(A_i\) and \(B_i\).
2.2/ 27abc
Do non-book problems
7 and 8.
Re-do 2.2/ 18 with the initial condition \(y(5)=1.\)
In my notes:
- Read Section 3.2.7 up through the paragraph after
Definition 3.34.
- Read Remark 3.35 at the end of Section 3.2.7.
(The two paragraphs of Section 3.2.7 that I have not assigned are
optional reading.)
- Read Section 3.2.8.
- In Section 3.2.10, starting where you left off,
read up through at least the portion of the proof of Theorem
3.44 that ends with statement (3.109).
|
| F 9/20/24 |
In my notes:
- In Section 3.2.10, read from where you left off up
through the end of Example 3.47 on p. 70.
- Read Section 3.3.1. With the exception of
the definition of the differential \(dF\) of a two-variable function
\(F\), the material in Section 3.3.1 of my notes
is basically not discussed in the book at all, even though
differential-form DEs appear in (not-yet-assigned) exercises for the
book's Section 2.2 and in all remaining sections of Chapter 2. (Except
for "Exact equations"—Section 3.3.6 of my notes—hardly
anything in Section 3.3 of my notes [First-order equations in
differential form] is discussed in the book at all.)
- Read Section 3.2.9. (This can be deferred to the
next assignment if you don't have time.)
In the textbook, read Section 2.4 up through the boxed
definition "Exact Differential Form" on p. 59. Also, on
my Spring 2024 homework page,
go to the assignment that was due 2/7/24, and read
"Comments, part 1" and "Comments, part 2."
|
| M 9/23/24 |
In my notes:
- Read Section 3.3.2 and 3.3.3.
- Read the remainder of Section 3.2.10.
- Read Section 3.2.9, if you haven't already.
|
| W 9/25/24 |
In Section 3.3.5 of my
notes, read up through Example 3.69.
Section 3.3.5 essentially addresses: what
constitutes a possible answer to various questions, based the
type of DE (derivative-form or differential-form) you're being asked
to solve? A proper answer to this question requires taking into
account some important facts omitted from the textbook (e.g. the fact
that DEs in derivative form and DEs in differential form
are not "essentially the same thing").
2.2 (not 2.3 or 2.4)/ 5, 15, 16.
(I did not assign these when we were
covering Section 2.2 because we had not yet discussed
"differential form".)
Previously, we defined what "separable" means
only for a DE in derivative form. An equation in differential
form is called separable if, in some region of the
\(xy\) plane (not necessarily the whole region on which the given DE
is defined), the given DE is algebraically equivalent to an equation
of the form \(h(y)dy=g(x)dx\) (assuming the variables are \(x\) and
\(y\)). This is equivalent to the condition that the derivative-form
equation obtained by
formally dividing the original equation by
\(dx\) or \(dy\) is separable.
As for how to solve these equations: you will
probably be able to guess the correct mechanical procedure. A natural
question is: how can you be sure that these mechanical procedures give
you a completely correct answer? That question is, essentially, what Sections
3.4–3.6 of my notes
are devoted to.
Warning. For
questions answered in the back of the book: not all answers there are
correct
(that's a general statement; I haven't done a separate
check for the exercises in this assignment)
and some may be misleading. But most are either correct, or
pretty close.
In the textbook, continue reading Section 2.4, up through Example
3. Then do the next set of exercises:
2.4/ 1–8.
Note: For differential-form DEs, there is no
such thing as a linear equation. In these problems, the book
means for you
to classify an equation in differential form as linear if
at least one of the associated derivative-form equations (the ones
you get by formally dividing through by \(dx\) and \(dy\),
as if they were numbers) is linear. It is possible for one of
these derivative-form equations to be linear while the other is
nonlinear. This happens in several of these exercises.
For example, the associated derivative-form
DE for \(y(x)\) is linear,the associated derivative-form DE for
\(x(y)\) is not.
In the textbook, read the rest of Section 2.4 to see the
mechanics of solving an exact DE. This should be enough to enable
you to do the exercises below, though not necessarily with
confidence yet. In class, I'll soon do some examples that
should help get you more confident in the method.
Don't invent a different method for solving
exact equations (or use a different method you may have
seen before). On the Miscellaneous Handouts page, there's
a handout called "A terrible method for solving exact equations"
that will be part of the next assignment. I can almost guarantee
that if you've invented (or have ever been shown) an alternative
to the method shown in the book (and that I'll go over in
class), this "terrible method" is that alternative method.
2.4 (continued)/ 9, 11–14, 16, 17, 19,
20
|
| F 9/27/24 |
UF closed, thanks to Hurricane Helene. What would have been this
assignment has been folded to the next assignment. Consequently that
assignment has more reading than usual, so get started on it as
soon as you can.
| |
| M 9/30/24 |
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.
At the time I'm posting this, the "(we proved
it!)" in the handout isn't yet true. Hopefully I'll have time to go
through the argument in class, or to post it.
For additional comments on this handout and the terrible method, see
my Spring 2024 homework page,
assignment due 2/12/24.
In my notes, read
Section 3.3.4, the remainder of Section 3.3.5,
and Section 3.3.6. (You may skip the portions
labeled as optional.)
My notes don't
present the basic method for (trying) to solve exact
equations. I plan to present that in class, but until I do, use what
you see in the book's Section 2.4.
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 a solution "\(y(x)= ...\)". This exercise, as written, 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). (That's how the exercise should have been written.)
2.4/ 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
from the previous assignment). 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.
2.4/ 29, modified as below.
- In part (b), after the word "exact", insert "on some regions
in \({\bf R}^2\)." What regions are these?
- In part (c), the answer in the back of the book is missing a solution
other than the one in part (d). What is this extra missing
solution?
- In part (c), the exact-equation method gives an answer of the
form \(F(x,y)=C\). The book's answer is what you get if you try
to solve for \(y\) in terms of \(x\). Because the equation you
were asked to solve was in differential form, there
is no reason to solve for \(y\) in terms of \(x\), any more
than there is a reason to solve for \(x\) in terms of \(y\).
As my notes say (currently on p. 78),
For any differential-form DE, if
you reverse the variable names you should get the same set of
solutions, just with the variables reversed in all your
equations. This will not be the case if you do what the book did
to get its answer to 29(c), treating your new \(x\) (old
\(y\)) as an independent variable.
In addition, in my notes:
- Skim
Section 3.3.7 up through the boldfaced statement (3.151). Read
statement (3.151) itself.
-
Read Example 3.77.
|
| W 10/2/24 |
Read Sections 3.4, 3.5, and 3.6 of my notes. In
these sections, the most important conclusions are displayed in
boldface, with equation numbers alongside for the sake of referencing
the statements. What you may want to do, for a first reading, is
scroll through and just read definitions and these highlights. Then
do a more careful reading when you have more time.
Do non-book problem
10. You may not get completely correct answers to parts of
problem 10 if you haven't read Sections 3.4–3.6 of my
notes.
|
| F 10/4/24 |
2.4/ 10, 15, 23, 26 (these last two are
"schizophrenic IVPs")
Read The Math
Commandments.
|
| M 10/7/24 |
No new homework
|
| F 10/11/24 |
Read Section 4.1 of the textbook.
(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.
|
M 10/14/24
(postponed from W 10/9) |
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,
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. You 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 several of my non-book problems.
Some reminders
(which I could also call warnings):
- The syllabus, the
reading of which was part of the very first assignment, 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
readings from my notes (Chapter 3, minus the portions labeled
as
optional, and parts of Chapter 5), as well as doing book and
non-book exercises.
The textbook chapters/sections we'll have covered before the exam
are 1.1, 1.2, 2.2,
2.3, and 2.4.
- The first homework assignment said this:
"Never treat any `reading' portion of
any assignment as optional ... (unless I tell you
otherwise)."
- The assignment due 9/11/24 had an even more emphatic
reminder:
"[R]eading
my notes is not
optional [.]"
But I've been asked some questions that have made
it clear that some students, probably many,
have not heeded my reminders/warnings. That's really not a
good idea.
One of the resources on the Miscellaneous Handouts page is
an Exponential Review Sheet. Many MAP 2302 students, in every
section of the course every semester, need review in this area.
|
| W 10/16/24 |
Do non-book problems 9 and
11.
4.7 (yes, 4.7) / 30.
(This exercise does not
require you to have read anything in Sections 4.1–4.7.)
Read Section 4.2 up through the bottom of p. 161. Some
corrections and comments:
- On p. 157, between the next-to-last line and the last line,
insert the words "which we may rewrite as".
(The book's " ... we obtain [equation 1], [equation 2]"
is a run-on sentence, the last part of which (equation 2) is a
non-sequitur, since there are no words saying how this
equation is related to what came before. As I've mentioned in
class,
this bad habit—writing [equation] [equation]
... [equation],
on successive lines, with no words or logical connectors
in between—is very commons among students,
and is tolerable from students at the level of MAP2302; they
haven't had much opportunity to learn better yet.
However, tolerating a bad habit until students can be trained
out of it is one thing; reinforcing that bad habit is
another. In older math textbooks, you would rarely if ever see this
writing mistake; in our edition of NSS, it's all over the
place.)
- On p. 158, the authors say that equation (3) is called the
auxiliary equation and say, parenthetically, that it is also known
as the characteristic equation.
While this is literally true, a more accurate depiction of reality would
be to say that equation (3) is called the
characteristic equation and to say, parenthetically, that it
is also known as the auxiliary equation.
"Characteristic equation" is more common, and that's the term
I'll be using.
- The second paragraph on p. 160 should say: "The proof of the
uniqueness statement in Theorem 1 is beyond the scope of a first
course in differential equations; in this text we defer that proof
to chapter 13.\(^\dagger\) However, in the present section and the
next, we will construct explicit solutions to (10) for all
constants \(a\, (\neq 0),\ b,\) and \(c,\) and all
initial values \(Y_0, Y_1\), thereby proving directly
the existence of at least solution to (10). For purposes of
an introductory course, we will simply take it on faith that the
uniqueness statement in Theorem 1 is true as well."
|
| M 10/21/24 |
No new homework.
|
| W 10/23/24 |
This assignment is being posted too late
for you to have enough time to finish it before the Wed. 10/23
class. Treat it as part of the assignment due
Fri. 10/25.
4.7 (yes, 4.7) / 1–8, 25 .
These exercises do not require anything from
Section 4.7 that we haven't covered in class already. "Theorem 5" (p.
192), referred to in the instructions for exercises 1–8, is
simply the 2nd-order case of the "Fundamental Theorem of Linear
ODEs" that I stated in class.
4.2/ 1, 3, 4, 7, 8, 10, 12, 13–16, 18,
27–32,
35,
46ab.
If you haven't gotten through all of these before
the Wed. 10/23 lecture, skip ahead to the next assignment and use
the full list of Section 4.2 exercises you'll see there.
(But still read the paragraphs below on #46.)
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.
Note: "\(\cosh\)" is
pronounced the way it's spelled; "\(\sinh\)" is pronounced "cinch".
--------------------------------------------------------------------------
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 unlikely that you will do well on my exams. The
homework problem/component you skip
(e.g. reading even just the first two
paragraphs of Example 3.47 in my notes) is always
the one that ends up on the exam. The universe is out to get you.
- Mathematical knowledge
and skills are cumulative. Math 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.
As 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).
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?
|
| F 10/25/24 |
In Section 4.2, read from the top of p. 162 through the middle
of p. 164 (just before Example 4 starts.)
4.2/ 2, 5, 9, 11, 17, 19, 20.
When combined with what was
in the previous assignment, the list of exercises assigned from this
section is:
4.2/ 1–20, 26, 27–32, 35, 46ab.
Read Section 4.3. For several comments and corrections,
see my
my Spring 2024 homework page,
assignment due 3/4/24.
|
| M 10/28/24 |
In the previous assignment, I had you read Section 4.3 and my
Spring 2024 comments and corrections for that section. I meant for
this reading to include the "Note" after the exercise-assignment,
4.3/1–18. If you
didn't get the latter part (or any part!) of the reading done
before the 10/25 class, do it now!
The one part of those comments I'll recopy here is this warning:
An
instruction you'll be seeing on the remaining exams is, "All
final answers must be in terms of real numbers (but complex
numbers may be used in intermediate steps)." 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.
4.7 (yes, 4.7)/ 26abc. NOTE: In
at least one e-reader, the formula for \(y_2(t)\) displays
incorrectly. The correct formula is \(y_2(t)=|t|^3\). If a
student hadn't shown this to me (thank you,
Grace!), I'd have never known about the mistake!
I don't know whether the formula was
transcribed incorrectly into the e-book, or if the
e-reader the publisher is selling UF students wasn't
programmed to display absolute-value symbols correctly (or
consistently). Without the absolute-value symbols, exercise 26
is pointless (and confusing), since \(y_1\) and \(y_2\) become
exactly the same function. I will contact the publisher about
this.
4.3/ 1–18, 21–26, 28, 32, 33 (students in
electrical engineering may do #34 instead of #33). These
exercises are numerous, but you should find 1–18 very short. Before
doing problems 32 and 33/34, see Examples 3 and 4 in Section 4.3.
Note: The book's solution of Example 4 starts with
"Equation (14) is a minor alteration of equation (12) in Example
3." This is true in somewhat the
same sense that the word "spit" is a minor alteration of the word
"suit". Changing one letter can radically alter the meaning of a
word. Any of the other words obtainable from "suit" by changing
the second letter has its own meaning, all very different from the
others.
It's true that the only difference between
the DEs in Examples 3 and 4 is the sign of the \(y'\)
coefficient, and that the only difference between equation (15) (the
general solution in Example 4) and equation (13) (the general
solution in Example 3) is that equation (15) has an \(e^{t/6}\)
where equation (13) has an \(e^{-t/6}\). But for modeling a
physical system, these differences are enormous; the
solutions are drastically different. Example 4 models a
system that does not exist, naturally, in our universe.
(More precisely: there could be
real-life physical (for example) system that could be
modeled approximately by equation (14) for a short enough
period of time. But the physical conditions that were used as
assumptions when modeling the system would break down after a while,
after which the system could no longer be modeled by the same
DE.) In this system, the amplitude of the
oscillations grow exponentially, without bound. This is
displayed in Figure 4.7 (except for the "without bound" part).
Example 3, by contrast,
models a realistic mass/spring system, one that could
actually exist in our universe. All the solutions exhibit
damped oscillation. Every solution \(y\) in Example 3 has
the property that \(\lim_{t\to\infty} y(t)=0\); the oscillations
die out. For a picture of this—which the
book should have provided either in place of the less-important
Figure 4.7 or alongside it—draw a companion diagram that corresponds to
replacing Figure 4.7's \(e^{t/6}\) with \(e^{-t/6}\). If you take away the
dotted lines, your companion diagram should look something like
Figure 4.3(a) on p. 154, modulo how many wiggles you draw.
When working with any linear,
constant-coefficient DE, it is crucial that you make NO
mistake in identifying the characteristic polynomial and its
roots. The most common result of misidentifying the characteristic
roots is to completely change the nature of the solutions.
|
| W 10/30/24 |
Read Section 4.4 up through Example 3.
Read Section 4.5 up through 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.
|
| F 11/1/24 |
Finish reading Sections 4.4 and 4.5.
The exercises below require much more of the
Method of Undetermined Coefficients (MUC) than the bare beginnings we
got through in the Wed. 10/30 class; you'll need to use what I've
assigned you to read in Sections 4.4 and 4.5. Do these exercises
as best you can before the Friday 11/1 class, and use whatever we
get through on Friday to review, continue, and/or finish your work
on these exercises later.
Over the next few days, I'll be
assigning almost all the exercises in these two sections, plus
some non-book exercises of my own.
If you wait even to get started, you won't be able to finish them
before your next exam (details still TBA).
4.4/ 9, 10, 11, 14,
15, 18, 19, 21–23, 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, 24–26, 28.
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.
Note: Anywhere that the book says
"form of a
particular solution," such as in exercises 4.4/ 27–32, it
should be "MUC form of a particular solution." The terms
"a solution" (as defined in the first lecture of this
course), "one solution", and "particular solution",
are synonymous. Each of these terms stands in contrast
to general solution, which means the set of all
solutions (of a given DE). Said another way, the general
solution is the set of all particular solutions (for a given
DE). Every solution of an initial-value problem for a DE is
also a particular solution of that DE.
The Method of Undetermined Coefficients, when applicable,
simply produces a particular solution
of a very specific form, "MUC form". (There is
an underlying theorem that guarantees that when the MUC
is applicable, there is a unique solution of that form.
Time permitting, later in the course I'll show you why the
theorem is true.)
|
| M 11/4/24 |
Some notes:
- In class I used (or will soon have used) the
term multiplicity of a root of the characteristic polynomial.
This is the integer \(s\) in the box on
p. 178. (The book eventually uses the term
"multiplicity", but not till Chapter 6; see the box on p. 337. On
p. 337, the linear constant-coefficient operators are allowed to have
any order, so multiplicities greater than 2 can occur—but not in
Chapter 4, where we are now.) In the the box on p. 178,
in order to restate cleanly what I said (or will be saying soon) in
class about multiplicity, it is imperative not to use the identical
letter \(r\) in "\(t^me^{rt}\)" 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 \(Ct^m e^{\alpha t}\).
-
In class, for the sake of simplicity and
time-savings, for second-order equations
I've consistently been using the letter \(t\) for the
independent variable and the letter \(y\) for the independent
variable in linear DE's. The book generally does this in Chapter 4
discussion as well, but not always in
the exercises—as I'm sure you've noticed. For each DE
in the book's exercises, you can still easily tell which variable is
which: the variable being differentiated (usually indicated with
"prime" notation) is the dependent variable, so by process of
elimination, the only other variable that appears must be the
independent variable.
While you're learning methods, it's
perfectly fine as an intermediate step to replace
variable-names with the letters you're most used to, as long as,
when writing your final answer, you remember to switch your
variable-names them back to the what they were in the problem you
were given. On exams, some past students have simply written a note
telling me how to interpret their new
variable-names. No. [Not if you want 100% credit
for an otherwise correct answer to
that problem. That translation is your job, not mine. Writing
your answer in terms of the given variables accounts for
part of the point-value and time I've budgeted
for.])
- It's important to remember that the MUC works only for
constant-coefficient linear differential operators \(L\)
(as well as only for certain functions \(g\) in "\(L[y]=g\)").
That can be easy to forget when doing Chapter 4 exercises, since
virtually all the DEs in these exercises are constant-coefficient.
(A linear DE \(L[y]=g\) is called a constant-coefficient equation
if \(L\) is a constant-coefficient operator; the function \(g\) is
irrelevant to the constant/non-constant-coefficient
classification.)
Exercises:
Non-book problem 12.
4.4/ 1–8, 12, 16, 17, 20, 24, 30, 31
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,
so make sure you do it the latter way.
4.5/ 9–12, 14–23, 27, 29, 31,
32, 34–36.
In #23,
the same comment as for 4.4/12 applies.
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.
Why so many exercises? As I said in an earlier assignment,
the "secret" to learning math skills
in a way that you won't forget them
is repetition. Repetition builds retention.
Virtually nothing else does (at least not for basic skills).
Do these non-book exercises on the
Method of Undetermined Coefficients. The answers to these
exercises are here. (These links
are also on the Miscellaneous Handouts page.)
4.5/ 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 yet discussed explicitly— although you would
likely be able to guess correctly how to do it for
the DEs in exercises 37–40). Some tips for 38 and 40 are
given below.
As mentioned in class (or will be mentioned soon), 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 the order of the DE. 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 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 may have to
factor a polynomial of degree at least 3, in order to correctly
identify root-multiplicities. Explicit factorizations are possible
only for some such
polynomials. (However, depending on the
function \(g\), you may not have to factor \(p_L(r)\) at all. For an
"MUC type" function \(g\) whose corresponding complex number is
\(\alpha +i \beta\), if \(p_L(\alpha +i \beta)\neq 0\), then
\(\alpha +i \beta\) is not a characteristic root, so the
corresponding "\(s\)" is zero.) 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 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, using the quadratic formula.)
- 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; they're misleadingly fine-tuned.)
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).
4.5/ 41, 42, 45. Exercise 45 is a nice (but
long)
problem that requires you to combine several things
you've learned. The strategy is similar to the approach
outlined in Exercise 41. Because of the "piecewise-expressed" nature of the
right-hand side of the DE, there is a sub-problem on
each of three intervals: \(I_{\rm left}= (-\infty,
-\frac{L}{2V}\,] \), \(I_{\rm mid} = [-\frac{L}{2V},
\frac{L}{2V}] \), \(I_{\rm right}= [\frac{L}{2V},
\infty) \). The solution \(y(t)\) defined on the whole
real line restricts to solutions \(y_{\rm left}, y_{\rm
mid}, y_{\rm right}\) on these intervals.
You are given that \(y_{\rm left}\)
is identically zero. Use the
terminal values \(y_{\rm left}(- \frac{L}{2V}), {y_{\rm
left}}'(- \frac{L}{2V})\), as the initial values \(y_{\rm
mid}(- \frac{L}{2V}), {y_{\rm mid}}'(- \frac{L}{2V})\). You then have
an IVP to solve on \(I_{\rm mid}\). For this, first find a
"particular" solution on this interval using the Method of
Undetermined Coefficients (MUC). Then, use this to obtain the general
solution of the DE on this interval; this will involve constants \(
c_1, c_2\). Using the IC's at \(t=- \frac{L}{2V}\), you obtain specific
values for \(c_1\) and \(c_2\), and plugging these back into the general
solution gives you the solution \(y_{\rm mid}\) of the relevant IVP on
\(I_{\rm mid}\).
Now compute the terminal values
\(y_{\rm mid}(\frac{L}{2V}), {y_{\rm
mid}}'(\frac{L}{2V})\), and use them as the initial
values
\(y_{\rm right}(\frac{L}{2V}), {y_{\rm
right}}'(\frac{L}{2V})\). You then have a new IVP to
solve on \(I_{\rm right}\). The solution,
\(y_{\rm right}\), is what you're looking for in part (a) of the
problem.
If you do everything correctly (which may
involve some trig identities, depending on how you do certain steps),
under the book's simplifying assumptions \(m=k=F_0=1\) and \(L=\pi\),
you will end up with just what the book says: \(y_{\rm right}(t) =
A\sin t\), where \(A=A(V)\) is a \(V\)-dependent constant
(i.e. constant as far as \(t\) is concerned, but a function
of the car's speed \(V\)). In part (b) of the problem you are interested in the
function \(|A(V)|\), which you may use a graphing calculator or
computer to plot. The graph is very interesting.
Note: When using MUC to find a
particular solution on \(I_{\rm mid}\), you have to handle the cases
\(V\neq 1\) and \(V = 1\) separately. (If we were not making the
simplifying assumptions \(m = k = 1\) and \(L=\pi\), these two cases
would be \(\frac{\pi V}{L}\neq \sqrt{\frac{k}{m}}\) and \(\frac{\pi
V}{L}= \sqrt{\frac{k}{m}}\), respectively.) 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 11/6/24 |
No new exercises.
On the Miscellaneous Handouts page, there's a section with
several MUC-related handouts. Look at the "granddaddy" file and
read the accompanying "Read Me" file, which is essentially a long
caption for the diagram in the "granddaddy file".
|
| F 11/8/24 |
No new homework.
|
W 11/13/24
|
Second midterm exam (assignment is to study for it).
|
| F 11/15/24 |
Read Section 4.7 up to, but not including, Theorem 7
(Variation
of Parameters).
Note on some terminology. "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 (meaning the book's authors) invite
confusion when you choose to give two different meanings to the
same terminology.
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
on the interval \( (0,\infty) \).
|
| M 11/18/24 |
4.7/ 9–14, 19, 20
Do non-book problem
13. (You'll need this before trying the exercises below.)
4.7/ 15–18, 23ab.
Problem 23b, with \(f=0\), 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".
|
W 11/20/24
and
F 11/22/24
|
On Monday, I did not finish presenting the
method for doing the exercises below from Sections 4.6 and 4.7.
For this reason I'm combining the assignment due Wednesday with
the one due Friday. Based on the classroom presentation on
Monday, and reading as much of Section 4.6 as you find helpful,
get as many of the exercises done as you can before Wednesday's
class, and the rest done before Friday's class. I may still add more
homework for you to do by Friday.)
One good piece of advice in the book is the sentence after
the box on p. 189: "Of course, in step (b) one could use the
formulas in (10), but [in examples] \(v_1(t)\) and \(v_2(t)\) are so
easy to derive that you are advised not to memorize them." (This
advice applies even if you've put the DE into standard linear form,
so that the coefficient-function \(a\) in equation (10) is 1.)
Incorrectly memorized formulas are worthless. If you attempt
to memorize a formula instead of learning the underlying method, and
your formula is wrong in any way (e.g. a sign is wrong), or
you misuse the correct formula in any way, do not expect
to get much partial credit on an exam problem.
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.
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/ 24cd, 37–40. Some comments on these exercises:
- Note that on the interval it is possible to solve the DEs
in all these exercises either by the using the Cauchy-Euler
substitution "\(t=e^x\)" (only for
the \(t\)-interval \((0,\infty)\); on the negative
\(t\)-interval the corresponding substitution is
\(t=-e^x\)) applied to the non-homogeneous DE,
or (without changing variables) by
first using the indicial equation
just to find a FSS for the associated homogeneous DE and then
using Variation of Parameters for the non-homogeneous 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.
- Note that in #37 and #39, the presence
of the expression \(\ln t\) in the given equation means that,
automatically, we're restricted to considering only the
domain-interval
\( (0,\infty) \). In #40, the instructions explicitly say to restrict
attention
to that interval.
But in #38, there is no need to restrict attention
to \( (0,\infty) \); you should solve on the negative-\(t\) interval
as well as the positive-\(t\) interval. However, observe that in
contrast
to the situation for homogeneous Cauchy-Euler DEs,
if a
function
\(y\) is a solution to #38's non-homogeneous
DE on \( (0, \infty) \), then the function
\(\tilde{y}\) on \( (-\infty,0) \) defined by \(\tilde{y}(t)
=y(-t)\) is not a solution of the same non-homogeneous
DE. You'll need to do something a little different to
get a solution to the non-homogeneous equation on \(
(-\infty,0) \).
In #40, to apply Variation of Parameters as I
presented it in class, don't forget to put the DE into standard form
first! But after you've done the problem
correctly, I recommend going back and seeing what happens if you
forget to divide by the coefficient of \(y''\). Go as far as seeing
what integrals you'd need to do to get \(v_1'\) and \(v_2'\). You
should see that if you were to do these (wrong) integrals, you'd be
putting in a lot of extra work (compared to doing the right
integrals), all to get the wrong answer in the end. I have made
this mistake before, myself!
Redo 4.7/40 by starting with the substitution
\(y(t)=t^{-1/2}u(t)\)
and seeing where
that takes you.
(This should
answer the question, "How did anyone ever figure out, or guess,
a FSS for the homogeneous DE in this problem?" Most, if not all,
of the homogeneous linear DEs for which anyone has ever figured
out a completely explicit FSS, are DEs that can be
"turned into" constant-coefficient DEs by some clever
substitution! Some substitutions change the independent variable
[e.g. the Cauchy-Euler substitution in 4.7/23]; some change the
dependent variable [e.g. the one I just gave you for
4.7/40].)
|
M 12/2/24
|
There is a substantial
homework assignment for you to do over the Thanksgiving
break.
Sorry, but this semester is already two whole weeks
shorter than Spring 2024 (37 MWFs vs. 43), and I'm not willing to
take any more out of the curriculum than this loss of days is
forcing me to. When you get back from Thanksgiving, we'll be
using our two remaining lectures to cover as much new material as
possible—which will be from Chapter 6, just as the
assignment below is—and you will NOT be in a position to
keep up unless that new material is a review of something
you've already read and done some exercises with. I won't slow
down those last few lectures to answer questions that you wouldn't
be asking if you'd done the assignment. Also, if you take a
nine-day break from differential equations that ends less than two
weeks before the final exam, you will not be able to learn the new
material and review the rest of the semester's material before the
final exam. Giving students this whole week off, especially at
this point of the semester, is doing them no favors.
Skim Section 6.1, a
lot of which is review of material we've covered already.
I'm
not fond of the way the section is organized or the material is
presented. Among other things:
- There is too much emphasis on the Wronskian,
especially since most students in
their first DE course haven't yet learned how to compute (or define) a
determinant that isn't \(2\times 2\) or \(3\times 3\). "Fundamental set of solutions" (or "fundamental
solution set") should not be defined using the
Wronskian.
- Linear dependence/independence of functions should
be introduced sooner, definitely before the Wronskian.
Here is how the material in Section 6.1 should be organized
(I suggest using this outline to guide your
thinking about the material in this section):
- Immediately after the "As
a consequence ..." sentence near the bottom of p. 320, before
anything else is said (or the book's "Is
it true ...?" question is asked), the term fundamental set
of solutions (FSS) should be defined. Specifically,
for
a homogeneous linear DE
&nbdsp; \(L[y]=0\) on an interval \(I\), a fundamental set
of solutions (FSS) should be defined in one of the
following equivalent ways.
(i) A set of
functions \( \{y_1, \dots, y_m\} \) on \(I\)
for which the general solution of \( L[y]=0\) on \(I\)
is the set of linear combinations \( \{c_1y_1+ \dots
+c_m y_m\} \), and for which \(m\) is as small as
possible among all such sets of
functions.
(ii) A set of solutions \( \{y_1, \dots,
y_m\} \) of \( L[y]=0\) on \(I\) for which the general
solution
is the set of linear combinations \( \{c_1y_1+ \dots
+c_m y_m\} \).
(As discussed in class several weeks ago, one consequence of "\(m\) is
as small as possible" is that \( \{y_1, \dots, y_m\}
\) is linearly independent.)
- The question should then be asked whether such a DE
always has a FSS, and if it does, whether the number of
functions (the \(m\) above) is always the same as the order of the
operator.
-
A theorem should then be stated that asserts that, for an
\(n^{\rm th}\)-order homogeneous linear DE \(L[y]=0\)
in standard form, with continuous
coefficient-functions, then
(1) a
FSS of \(L(y)=0\) on \(I\) exists (in
fact, infinitely many FSS's of this DE on \(I\)
exist);
(2) any such FSS has exactly \(n\) functions; and
(3) a set of solutions \( \{y_1, \dots, y_n\} \)
of \( L[y]=0\) on \(I\) is a FSS if and only if this set of
functions is linearly independent on \(I\).
(This is what the book's
Theorems 2 and 3, combined, should have said.)
-
The Wronskian should
then be introduced (and a reference for the definition and
properties of \(n\times n\) determinants for general \(n\) should be
given), and used as a tool for proving this theorem
and for checking whether a set of solutions of \(L[y]=0\) is
linearly independent. (Again:
a tool, not part of a definition of anything thing
important. Introducing the Wronskian any other way distracts from
concepts that are actually important.)
- Notation such as "\(y_h\)" should be introduced for the
general solution of the associated homogeneous equation. The
general solution is best treated as the set of all
solutions, not as a typical element of this
set. (The book does the opposite after
Theorem 2, as do many other books—generally, the same ones
that use indefinite-integral notation for an arbitrary
but specific antiderivative, rather than as the set
of all antiderivatives. Such a definition is defensible, but
misguided [in my opinion, of course], and should have
been retired by the 1960s if not earlier.)
- Theorem 4 should be stated and proved. But after equation (28),
before the next sentence, something like the following should be
inserted: "Then the general solution of (27) on \((a,b)\) is
\(y=y_p+y_h.\)" Then the book's next sentence (the one concluding
with
equation (29)) should be given, with "Then" replaced by "Thus".
- 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.
- 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 ones later in Section 6.3, it may help you
to first review my
comments about factoring
in the assignment due 11/4/24.
- Read Section 6.3.
- 6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the
annihilator method;
just
use
MUC and superposition.
|
| W 12/4/24 |
6.3/ 5–10, 11–20
In 5–10, you'll have to factor a cubic
(i.e. third-degree) characteristic
polynomial. Again, what I said previously (HW due 11/4) about factoring
polynomials of degree greater than two should be helpful.
Note: One of the things I said on this topic was:
"In all the problems in this textbook in which you
have to solve a 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)." One of the exercises in the 5–10
group is among the few that caused me to say
"usually 0 or 1," not "always 0 or 1."
FYI: The Rational Root Theorem says the
following: Let \(P(r)\) be an \(n^{\rm th}\) degree poynomial
\(a_nr^n + a_{n-1}r^{n-1}+\dots+ a_1 r + a_0\) with
integer coefficients. Then if P has a rational
root \(p/q\) in lowest terms, then the numerator \(p\) must be a
divisor of the constant term \(a_0\), and the denominator \(q\)
must be a divisor of \(a_n\). (Note: divisors can be
negative as well as positive.)
In particular, if the leading coefficient \(a_n\) is \(\pm 1\),
then the only potential rational roots (there may be no
rational roots at all) are the divisors of \(a_0\). (For
example, in 6.3/ 10, the only potential rational roots are
\(\pm 1, \pm 2, \pm 13,\) and \(\pm 26.\)) But the book's
exercises make this fact almost irrelevant, since all its
cubic-and-higher-degree polynomials were designed to have not just at
least one root that's a rational number, but at least one root
that's an integer of such small absolute value that you'd guess
the root even without knowing the Rational Root Theorem.
|