In the table below, "NSS" stands for our textbook. Exercises are
from NSS unless otherwise specified.
| Date due |
Section # / problem #'s |
| W 1/10/24 |
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.
Note: the sentence on p. 4 that contains
equation (7) is not quite correct as a definition of "linear
differential equation".
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 (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".
|
| F 1/12/24 |
1.2/ 1, 3–6, 14, 15, 17, 19–22.
Remember that 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 these
problems.
Note: The exercise portions of many
(probably most) of your homework assignments will be a lot more
time-consuming than in the assignments to date; I want to give you
fair warning of this before the end of Drop/Add.
However, since my posted notes are only
on first-order ODEs, the reading portions of the
assignments will become much lighter once we're finished with
first-order equations (which will take the first month or so of
the semester).
In
my notes, read from where you left off in the last assignment
through Example 3.11 (p. 15).
|
| W 1/17/24 |
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).
Whenever I update these notes, 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.
|
| F 1/19/24 |
As a summary of what we did in class,
read "Method for Solving Linear Equations" on p. 50. Remember
that whenever you see notation of the form "\(\int f(x) dx\)" in
this book, it means what I'm calling "\(\int_{\rm spec} f(x)\,dx\)",
(any) one specific antiderivative of \(f\) on the interval in
question.
On an interval in which \(f\) is continuous, $$ \int f(x)\, dx =
\left\{\ \int_{\rm spec} f(x)\, dx +C : C\in {\bf R} \
\right\},\ \ \ \ \ \ \ (*) $$ and thus \(\int f(x)\, dx = \int
f(x)\, dx +C\) (the collection of functions on the left-hand
side of the equation is the same as the collection of
functions on the right-hand side).
Note:
- I've written equation (*) in precise
"set-builder" notation, but in calculus textbooks and tables of
integrals, you'll usually see this written in the less precise
form $$\int f(x)\, dx = \int_{\rm spec} f(x)\, dx +C. \ \ \ \ \
\ \ (**); $$ e.g. "\(\int x\, dx = \frac{x^2}{2}+C.\)"
In this class I use the convention
that (**) is short-hand for (*); students are not required
to use the curly-brace notation I've used in (*).
In Section 2.3, read Example 2 to see one way of approaching
this IVP (essentially, finding the general solution of the
DE—a collection of functions, one for each value of the
arbitrary constant \(C\)—and then figuring out which value
of \(C\) is needed to get the one solution that also satisfies
the initial condition.
Comments and corrections for
Example 2:
- In "\(\,50 e^{-10 t}\,\)", the book neglects to mention
what units \(t\) is measured in, but from the solution in the
book, we can infer that the authors meant for \(t\) to be
measured in seconds.
The need to say explicitly what units a quantity is measured in
can be avoided by incorporating appropriate inverse units
(e.g. \(m^{-1}\) or \sec^{-1}\)), into formulas, equations, etc.
In the present example, we would replace
"\(10t\)" with "\(10t/sec\)".
(Then, for example, if \(t=1
\,\min\), then \(10t/\sec =10 \times (1\,\ \min)/\sec =
10\times (60 \,\sec)/\sec = 10\times 60 =600,\) and
\(e^{-10t/\sec} = e^{-600}.\) Units of time, length, mass,
etc., can't be exponentiated. Only "pure
numbers"—dimensionless quantities—can be
exponentiated.)
Throughout the problem, the usage of physical units is
schizophrenic. (However,
most other calculus and DE textbooks are no better
than NSS in this
regard.) For example, \(k\) is stated to be
\(2/\sec\), which is not the same as the dimensionless
number 2— but later the book says "we have substituted
\(k=2\)." Similarly, "40 kg" has units of mass, and
is not the same animal as the dimensionless number "40". But
the quantity \(y(t)\) is stated to be the mass of
\(RA_2\) present at time \(t\), not
the number of kilograms of \(RA_2\) present at time
\(t\). Equation (13), written correctly, should say
$$
\frac{dy}{dt}+ \frac{2}{\sec}y= 50\frac{{\rm kg}}{\sec} e^{-10t/\sec}.
\ \ \ \ \ \ \ (*)$$
(Note that "\(e^{-10t/\sec}\)" has the
same meaning no matter what units of time are used for
\(t\). For example, if is given in minutes—
say, \(t=\) 5 min, then
$$10t/\sec = 10(5 min)/\sec =
10(5 \times 60 \sec)/\sec = 10 \times 5 \times 60 =3000,
$$ and \(e^{-10t/\sec} = e^{-3000}\).) The "/sec" in
\(10t/\sec\)
guaranteed that the quantity being exponentiated
was dimensionless— a "pure number"— and
that it would have the same value whether \(t\) was given to
us in seconds, minutes, hours, megadays, or nanofortnights.
The DE (*) is in standard linear form, with coefficient
function
\(P(t)\) being a constant function with value
\(2/\sec\), so one specific antiderivative is
\(\int_{\rm spec} P(t)\, dt =(2/\sec) t\) (a dimensionless
quantity),
yielding \(\mu(t)= e^{2t/sec}\). Multiplying
both sides of equation (*) by \(\mu(t)\) then yields
\(\frac{d}{dt} [e^{2t/\sec}y(t)] = 50\frac{\rm kg}{\sec}
e^{-8t/\sec},\) which we can then integrate to find
\(e^{2t/\sec}y(t)=-\
\frac{50\,{\rm kg}/\sec}{-8/\sec}e^{-8t/\sec} +C
= -\,\frac{25}{4}{\rm kg}\,e^{-8t/sec} +C
.\)
Plugging in the initial condition "\(y(0\,\sec)=40\,{\rm kg}\)"
leads to \(1\times 40\, {\rm kg} = -\,\frac{25}{4}\,{\rm
kg}+C\),
implying \(C=\frac{185}{4}{\rm kg}\) (not the dimensionless
number "\(\frac{185}{4}\)"). Finally, plugging in this value
for \(C\) and solving for \(y(t)\) yields
\(y(t)=
(\frac{185}{4}e^{-2 t/\sec} - \frac{25}{4}e^{-10 t/\sec}){\rm
kg},\) an actual mass (as it should be),
not the dimensionless right-hand side of the book's equation
(14).
In intermediate steps of problems, I don't
require students to explicitly include all the physical
units as I did above. I just wanted to show you that
it can be done without extraordinary difficulty,
and how to do it. But in students' work, the
relevant arithmetic still needs to be done (including
any conversions, if needed), and the final answer should
have appropriate units (e.g. "70 kg" or "70,000 grams"
rather than just "70", if the answer is a mass).
- Except when you're listing independent
equations (e.g. the two equations that comprise a
first-order initial-value problem: the differential equation
and the initial condition), any writing of the form
"equation, equation" (with no words between the equations)
is terrible writing on several grounds:
grammatically, logically, and pedagogically. Unfortunately,
in our textbook, you'll see such writing countless times,
three of them in
this example alone. For
a student to write this way on a
timed exam in a calculus or DE class
is forgivable.
(Most of you have probably never been
told that there's anything wrong with "equation,
equation;" you may even have been encouraged,
implicitly or explicitly, to write mathematics in this
devoid-of-understanding way.) And, of course, when
writing notes for yourself you can do whatever you
want; just make sure, for your own sake, that what you
write is something you'll be able to understand later.
What I can't forgive is a textbook author setting
such a poor example for students.
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,
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 \(\frac{1}{x}\).
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).
|
| M 1/22/24 |
2.3/ 22, 23, 25a, 27a, 28, 31, 33, 35.
Do non-book problem
2.
In
my notes, read Section 3.2.5.
Corrections to some of the Section 2.3 exercises:
- #33: "Singular point" is not defined correctly. For
example, the point \(x=-5\) is not considered a singular
point of the DE \(y'+\sqrt{x}\, y=0.\) It is true
that \(0\) is a singular point of the DE
\(xy'+2y=3x,\) but the reason is that the coefficient
of \(y'\) is 0 when \(x=0\). (That's the actual
definition of "singular point" of a linear DE
\(a_1(x)\frac{dy}{dx} + a_0(x)
y=b(x)\).
[If \(x_0\) is a singular point of this
DE—i.e. if \(a_1(x_0)=0\)—then,
obviously, the expression \(\frac{a_1(x)}{a_0(x)} =:
P(x)\) is not defined when \(x=x_0\). But
\(x_0\) is not called a singular point of this
DE if \(\frac{a_1(x_0)}{a_0(x_0)}\) is
undefined simply because \(a_0(x_0)\) or
\(a_1(x_0)\) is undefined.])
- #35: The term "a brine" in this problem is not
English; it's
similar to saying "a sand". One should
either say "brine" (without the "a") or "a brine solution".
Another term that should not be used is the redundant "a
brine solution of salt" (literally "a concentrated
salt-water solution of salt"), which appears elsewhere in
the book.
One of the things illustrated by
2.3/33 is that
what you
might think is only a minor difference between the DE's in parts (a)
and (b)—a sign-change in just one term—drastically
changes the nature of the solutions. When solving differential equations, a tiny
algebra slip can make your answers utter garbage. For this
reason, there is usually no such thing as a "minor algebra error"
in solving differential equations.
This is a fact of life you'll have to get used to. The severity
of a mistake is not determined by the number of pencil-strokes it
would take to correct it, or whether your work was consistent after
that mistake. If a mistake (even something as simple as a
sign-mistake) leads to an answer that's garbage, or that in any other
way is qualitatively very different from the correct answer, it's
a very bad mistake (for which you can expect a significant
penalty on an exam). A sign is the only difference between a rocket
going up and a rocket going down. In real life, details like that
matter!
If you haven't already, I urge you to develop 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, for many exercises, you can find answers in
the back of the book, and you're 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 built-in 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.
(If you don't know what these built-in checks are, you haven't yet
understood what the term "solution of a differential equation"
or "solution of an initial-value problem" means. If that's the case,
do not wait another day to fix that.)
|
| W 1/24/24 |
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 first paragraph on p. 43.
|
| F 1/26/24 |
In my notes, read Sections
3.2.9 and 3.2.10. (I'll have you return later to where you left off
in Section 3.2.6.)
In NSS (our textbook), read from the beginning of Section 2.2
(p. 41) through Example 1, but ignore (for now) the last sentence in
the "Method for Solving Separable Equations" box (p. 42). In these
pages:
- Turn your brain off when reading the second sentence on
p. 41. Otherwise you risk brain damage.
- The title of the box on p. 42 should be "The Method of
Separation of Variables", which is part of the general method
for solving separable DEs. The other part is alluded to, with
vastly understated importance, in the "Caution" just below the box.
We'll cover the complete method more carefully in the next one or
two lectures, and in my notes.
- The part of the box on p. 42 that I said to ignore contains
the term implicit solution,
a proper definition of which is in my notes
(Definition 3.25). The book's "definition" of implicit solution in
Section 1.2 is ambiguous and misleading, and relies on terminology not
defined in the book. (I intentionally did not have you read Section
1.2, specifically because the terminology and definitions there are
very poor.)
Theorems 3.44 and 3.46
in my notes are closely related to the "Formal Justification of
Method" on p. 45 of the textbook. The book's presentation may
look simpler than mine, but unfortunately:
-
Contrary to what the title advertises, the book's
argument does not justify the method.
The
argument puts no hypotheses whatsoever on the functions \(p\)
and \(g\)—not even continuity—without which several
steps in the argument cannot be justified.
- The conclusion the book purports to establish neglects an
important issue. The question of whether the method
gives all the solutions, or even all
the non-constant solutions, is never even mentioned, let
alone answered. An example in my notes
(currently numbered as Example 3.47, involving the DE
\(\frac{dy}{dx}=6x(y-2)^{2/3}\)) illustrates how badly
the method fails to produce all the solutions if we don't
assume considerably more than the the minimal hypotheses needed
for the book's argument even to make
sense (the continuity of the functions \(p\)
and \(g\)). The example shows that if \(p\) is
not differentiable, the method we've studied for solving
separable equations can fail spectacularly to produce all
the solutions. (To guarantee that separation of variables will
give us all the non-constant solutions, we actually need to
assume even more, namely that \(p'\) is continuous [at least at
the points \(r\) for which \(p(r)=0\)].)
|
| M 1/29/24 |
In
my notes, finish reading Section 3.2.6.
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. (The book's explanations
and definitions say some of the right things, but don't hold up
under scrutiny.)
However, you do need
to start getting some practice with the mechanical ("brain
off") part of the 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, but with special temporary
instructions for them.
2.2/ 7–14. For now (with the Monday 1/29
due date), all I want you to do in these exercises is (a) 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— and (b) to find all the
constant
solutions,
if there are any.
Save your work, so that when I re-assign them later, with your
goal being to get a complete answer that you fully understand, you won't have
to re-do this part of the work.
When you do these exercises,
don't
just go through the motions, either saying to yourself, "Yeah, I know what to do
from here" but not doing it, or doing the integrals incorrectly, or
stopping when you reach an integral you don't remember how to do.
(This applies to the exercises that will be assigned in the future
as well.) Your integration skills need to good enough that you
can get the right answers to problems such as the ones
assigned 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. (In each of these mistakes, the
relation of \(\frac{1}{x}\) to \(\ln x\)
or \(\ln |x|\) is
reversed.)
This does not mean you should study integration techniques to
the exclusion of material you otherwise would have studied to do
your homework or prepare for exams. You need to both review the old
(if it's not fresh in your mind) and learn the new.
|
| W 1/31/24 |
No new homework
|
| F 2/2/24 |
No new homework
|
| M 2/5/24 |
Complete exercises 2.2/ 7–14, finding all solutions. (For
an exercise in which you could only get an implicit form of
the general solution, "completing the work" may amount to
just understanding your answer.)
2.2/ 17–19, 21, 24, 27abc. Also
(re)do #18 with the initial condition \(y(5)=1.\)
As always, "Solve the equation" means "Find all
(maximal) solutions of the equation or IVP"—explicitly if
possible; in implicit form otherwise. For an IVP, if the conditions
of the FTODE are met, there will be only one maximal solution, so there
should be no arbitrary constants in your answer, whether your answer
is in explicit or implicit form. (If you introduced an arbitary
constant along the way, use the initial condition to eliminate it.)
Do non-book problems
3–7.
Answers to
most of the non-book problems are posted on the
"Miscellaneous handouts" page.
General comment. In doing the exercises
from Section 2.2 or some of my
non-book problems, you may
find that, often, the hardest part of doing
such problems
is doing the integrals. I
intentionally assign problems that require you to refresh most of your
basic integration techniques (not all of which are adequately
refreshed by the book's problems). You
Read Section 3.2.7 of
my notes.
In the assignment due 1/26/24, I've inserted some blue and
green comments at the end. (In inserted them there, rather than
here, because they relate to Section 3.2.10, whose reading was
part of that assignment.) Read these.
|
| W 2/7/24 |
Read Section 3.3.1 of
my notes. 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.)
Do non-book problem
8.
In the textbook, read Section 2.4 through the boxed
definition "Exact Differential Form" on p. 59. See Comments,
part 1, below.
See Comments, part 2, below.
Comments, part 1. There are terminological
problems in Section 2.4 of the book, most notably an inconsistent
usage of the term "differential form". Many students may not notice
the inconsistency, but some may—especially
in an honors
class—and I don't want anyone to come out of my class with an
improper education. Here are the problems, and fixes for them:
- In this chapter, every instance in which the term
"differential form" is used for anything that's not
an equation—a statement with an "=" sign in
it—the word "form" should be deleted. In particular, this
applies to all instances of "differential form" in the
definition-box on p. 59 (including the title).
- The definition-box's use of the term
"differential form" is
not incorrect, but at the level
of MAP 2302 it is a very confusing use of the word "form", and the
less-misinterpretable term "differential" (without the word
"form") is perfectly
correct.
- Except for the title, the usage of "differential form" within the
definition-box is inconsistent with the usage
outside the definition-box. The usage in the title is ambiguous;
it is impossible to tell whether the title is referring to an
exact differential, or to an equation with an exact
differential on one side and zero on the other.
In my notes I talk about "derivative
form" and "differential form" of a
differential equation. The meaning of the word "form" in
my notes is standard mathematical English, and is the same as in
each of the two occurences of "form" on on p. 58 of the book.
In this usage, "form of an equation" refers to the way an
equation is written, and/or to what sort of objects
appear in it.
But when a differential itself (as
opposed to an equation containing a differential) is
called a "differential form", the word "form" means
something entirely different, whose meaning cannot be
gleaned from what "form" usually means in English. In this
other, more advanced usage, "differential forms" are
more-general objects than are differentials. (Differentials are
also called 1-forms. There are things called 2-forms,
3-forms, etc., which cannot effectively be defined at the level
of MAP 2302. (You
won't see these more general objects in this course, or in any
undergraduate course at UF—with the possible exception of
occasional special-topics courses.) With the advanced
meaning of "differential form", the only differential
forms that appear in an undergraduate DE textbook
are differentials, so there's no
good reason in a such a course, or in its textbook, to use the term
differential form for a differential.
There is also a pronunciation-difference in the
two usages of "differential form". The pronunciation of this
term in my notes is "differential form", with the
accent on the first word, providing a contrast with
"derivative form". In the other usage of "differential
form"—the one you're not equipped to understand, but
that is used in the book's definition-box on p. 59—the
pronunciation of "differential form"
never has the accent on the first word; we either say
"differential form", with the accent on the second
word, or we accent both words equally.
- The paragraph directly below the "Exact Differential
Form" box on p. 59 is not part of the current
assignment. However, for future reference, this paragraph is
potentially confusing or misleading, because while the first
sentence uses "form" in the way it's used on p.58 and in my
notes, the third sentence uses it with the other, more
advanced meaning. This
paragraph does not make sense unless the term "differential
form" has the meaning of a form of an equation (with the
standard-English meaning of "form") on line 2, but has the
meaning of
a differential on line 4.
Choosing to use the term "exact
differential form" in the first equation of this paragraph is,
itself, rather unusual.
When 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 a
type of differential, not a type of equation.
The standard terminology for
what the offending sentence calls "[differential equation]
in exact differential form" is exact equation
(or exact differential equation), just
as you see in the definition-box on p. 59. (The terminology
"exact equation" in the box has its own intrinsic problems,
but
is standard nonetheless.)
- In Example 1 on
p. 58, the sentence beginning "However" is not correct. In
this sentence, "the first form" refers to the
first equation written in the sentence beginning
"Some". An equation cannot be a total differential.
An equation makes an assertion; a total differential
(like any differential) is simply a
mathematical expression; it is no more an equation than
"\(x^3\) " is an equation. To correct this sentence, replace
the word "it" with "its left-hand side".
- The following is just FYI; it's not a problem with the
book: What the book calls the total differential of
a function F is what my notes call simply the
differential of F. Both are
correct. The word "total" in "total differential" is
superfluous, so I choose not to use it.
Comments, part 2. In my notes, you're going to
find section 3.3 more difficult to read than the
book's Section 2.4 (and probably more difficult than the earlier
sections of my notes). A
major reason for this is that a lot of
important issues are buried in a sentence on the book's p. 58 (the
sentence that begins with the words "After all" and contains equation
(3)). You may 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 main reason the
textbook is easier to read than my
notes is that these questions (whose answers are subtler and deeper
than you might think) aren't mentioned, which avoids the need to
answer them. The same is true of all the
DE textbooks I've seen; even with
the problems I've mentioned, our textbook is still better than any
other I've seen on the current market. But if you had a good
Calculus 1 class, you had it drilled into you that "\(\frac{dy}{dx}\)"
is not a real number \(dy\) divided by a real number \(dx\),
and you should be
confused to see a math textbook implying with words like
"After all" that's it's `obviously' okay to treat
"\(\frac{dy}{dx}\)" as if it were a fraction with real numbers
in the numerator and denominator. The Leibniz notation
"\(\frac{dy}{dx}\)" for derivatives has the miraculous feature
that the outcomes of certain symbol-manipulations
suggested by the notation can be justified (usually using
higher-level mathematics), even though the manipulations
themselves are not valid algebraic operations, and even though
it is not remotely obvious that the outcomes can be justified.
|
| F 2/9/24 |
Read Sections 3.3.2 and 3.3.3 of
my notes.
The rest of this assignment is being posted too
late for you to get it all done before class on Friday 2/9. Get
as much of it done by then as you can, and get the rest done
ASAP afterwards.
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, 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? That question is, essentially, what Sections
3.4–3.6 of my notes (which I'll have you read soon) 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, 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 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
|
| M 2/12/24 |
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.
Read the online handout
A terrible way to
solve exact equations. The example in this
version of the
handout is rather
complicated; feel free to read the simpler example in the
original version
instead. The only problem with the example in the original version is
that \(\int \sin x \cos x\, dx\) can be done three ways (yielding
three different antiderivatives, each differing from the others by a
constant), one of which happens to lead to the correct final
answer even with the "terrible method". Of course, if the terrible
method were valid, then it would work with any valid choice of
antiderivative. However, I've had a few students who were unconvinced
by this argument, and thought that because they saw a way to get the
terrible method to work in this example, they'd be able to do
it in any example. I constructed the more complicated
example to make the failure of the terrible method more obvious.
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.
With older editions of
the textbook, if I didn't get to this in class, I could tell students
to read the argument in the book, but that's no longer the case.
The argument in the
current edition glosses over some steps that need justification
(Why should the integal in equation (6)—which is what I'd write
as \(\int_{\rm spec} M(x,y)\ dx\)— even exist? And where is the
assumption that \(R\) is a rectangle being used? [Theorem 2 is false
if \(R\) is replaced by an
arbitrary open set.]), and all clues
to where
exactness is being used are buried in exercises. (For
the key step, the student is referred to exercises 35 and 36,
although exercise 31 handles this step much more simply. But either
way, that key step requires a particular theorem from Advanced
Calculus that you'd never have seen mentioned in Calculus 1-2-3.
And the book's suggested way of using this theorem
in exercise 35 involves an unnecessary step that would need to be justified by
otherwise unnecessary work that would at least triple the length of the
argument.)
.
In my notes, read
Section 3.3.4, the remainder of Section 3.3.5,
and Section 3.3.6.
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.
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.
- Read Sections 3.4, 3.5, and 3.6 . 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.
I am in the process of assembling most of the boldfaced conclusions
concerning differential-form DEs into
a summary for easier reference and comparison.
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.
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.
|
| W 2/14/24 |
2.4/ 10, 15, 23, 26 (these last two are
"schizophrenic IVPs")
Read The Math
Commandments.
|
| F 2/16/24 |
Do non-book problems 9 and
11. Update to problem 11: In part (d),
some important words ("whose differential is
\(M\,dx+N\,dy\) on this half-plane") that had been omitted
from each of the three bullet points have been inserted.
Similar
omissions have been corrected in parts (f) and (h).
|
| M 2/19/24 |
First midterm exam (assignment is to study for it).
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.
Violations of the third Math Commandment (or
any of the others) can be very costly on my exams, so I would
advise you to look over the review sheet. (However, you
can probably wait to review the items involving limits; these are
not as important for the first-midterm material as they can be
later in the course. Some of these limits are examples of the
"battles" referred to in the third commandment.)
|
| W 2/21/24 |
No new homework.
|
| F 2/23/24 |
No new homework.
|
| M 2/26/24 |
Read Section 4.1.
(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.
4.7 (yes, 4.7) / 1–8, 30 .
Problem #30 does not
require you to have read anything in Sections 4.1–4.7.
For problems 1–8, the only part of Section 4.7 that's needed
is the statement of Theorem 5
(p. 192), but Theorem 5 is
simply the 2nd-order case of the "Fundamental Theorem of Linear ODEs"
that I stated in class.
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.
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.
- More-general versions of Theorem 2 and Lemma 1
(pp. 160–161) are in Section 4.7. In the interests of
efficiency, I'll be covering those versions instead of
the ones in Section 4.2.
But to do the Section 4.2 exercises while
waiting until the more general versions have been
covered
(in which case you'd have a ton of exercises to do all
the once), just use
the versions in Section 4.2.
Unfortunately, hardly any of Section 4.2's exercises are
doable until the whole section has been covered, which takes more than
a single day (we have just started it in class). In order for you not
to have a single massive assignment when we're done covering Section
4.2, I recommend that, based on your reading, you try to start on the
exercises listed in the next assignment. Problems that you should be
able to do after doing the reading assigned above are
4.2/ 1, 3, 4, 7, 8, 10, 12, 13–16, 18, 27–32.
|
| W 2/28/24 |
4.2/ 1–20, 26, 27–32, 35, 46ab.
In #46, the instructions should say that the
hyperbolic cosine and hyperbolic sine functions can be
defined as the solutions of the indicated IVPs, not that
they are defined this way. The customary definitions are
more direct: \(\cosh t=(e^t+e^{-t})/2\) (this is what you're
expected to use in 35(d))
and \( \sinh t= (e^t-e^{-t})/2\). Part of what you're doing in
46(a) is showing that the definitions in problem 46 are equivalent
to the customary ones. One reason that these functions have
"cosine" and "sine" as part of their names is that the ordinary
cosine and sine functions are the solutions of the DE \(y''+y=0\)
(note the plus sign) with the same initial conditions at \(t=0\)
that are satisfied by \(\cosh\) and \(\sinh\) respectively. Note
what an enormous difference the sign-change makes for the
solutions of \(y''-y=0\) compared to the solutions of \(y''+y=0\).
For the latter, all the nontrivial solutions (i.e. those that are
not identically zero) are periodic and oscillatory; for the
former, none of them are periodic or oscillatory, and all of them
grow without bound either as \(t\to\infty\), as \(t\to -\infty\),
or in both directions.
Note: "\(\cosh\)" is
pronounced the way it's spelled; "\(\sinh\)" is pronounced "cinch".
|
| F 3/1/24 |
No new homework
|
| M 3/4/24 |
Read Section 4.3. Some comments and corrections:
- Anywhere in this
book that the term "complex roots" appears
(including in the title of Section 4.3 and the exercises),
this
term should be
replaced by "non-real roots", "non-real complex roots", or
"no real roots". As mentioned in class, 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.
- 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 anywhere in the
book.
The portion of this page from the sentence containing equation (4)
through the sentence containing equation (6), does not constitute
a derivation of equation (6). What this portion of the
page
provides is partial motivation for the (never stated)
definition of \(e^z\) for non-real complex numbers \(z\).
- 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 (and that \(e^z\)
has a Maclaurin seres).
A non-misleading way to introduce the
calculation of \(e^{i\theta}\) that's on p. 166 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
\(e^x\), and assume that it is legitimate to group the real and
imaginary terms into two separate series." Instead of the word
"identification" that's used in the line above the book's
equation (5) , we would then use the much clearer word
"definition".
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.
- In the paragraph "Complex Conjugate Roots" on p. 168, on the
first line ("If the auxiliary equation ..."), after
"\(\alpha \pm i\beta\)", the parenthetic phrase "(with
\(\beta\neq 0\))" should be inserted.
4.3/ 1–18.
Note: The book uses the complex
exponential function to derive the fact that in the
case of non-real characteristic roots \(\alpha\pm i\beta\), the
real-valued 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. It is, of course, possible to show that these two
functions are solutions by direct computation using only real-valued
functions.
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 does have some drawbacks, though:
- 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 the
relations between real and complex solutions of equation (2),
and the differentiation formula at the bottom of p. 166
(equation (7)).
(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
\(b\)). Doing this work is
very worthwhile, but
time-consuming.
In the paragraph after the box "Complex
Conjugate Roots) on p. 166, the authors allude to these issues,
but expect that most students will be perfectly happy letting the
wool be pulled over their eyes with a "proof by
notation" approach. Only if the student is "uneasy" about
conclusions based on notation, rather than on an honest
derivation, does the book encourage the student to take the trouble to check that these can
be justified. After all, why encourage students to try
to understand something when they're probably willing
to
accept it on blind faith?
|
| W 3/6/24 |
4.3/ 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.
Note: The DE in Example 4 should not
really be considered a "minor alteration" of the DE in Example 3. It
is true that the only difference 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. In this system, the amplitude of the
oscillations grow exponentially. This is displayed in Figure
4.7.
Example 3
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.
|
| F 3/8/24 |
4.7 (yes, 4.7)/ 25
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.
|
Second midterm: possible dates |
As of today (3/8/24), I'm hoping
(IN VAIN; SEE BELOW) to hold the second midterm either
on Friday Mar. 22 or Monday Mar. 25.
|
| M 3/18/24 |
Finish reading Sections 4.4 and 4.5.
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.
|
Second midterm actual date |
We'll have this exam on Wednesday, Mar. 27. This is much
later than I'd hoped. But even though I assigned all the relevant
MUC reading and some of the exercises for you to finish over spring
break, and even if I finish discussing the MUC in
class by Friday, Mar. 22, I think you need more time to get enough
practice with the method.
|
W 3/20/24
and
F 3/22/24
and
M 3/25/24
|
By the time this is posted, it will be too late for you to get much of
this done before the W 3/20 class, so I'm combining several
assignments. In principle, you should be able already to
do all the exercises in Sections 4.4 and 4.5 based on the assigned
reading, but you may have more confidence after you see me do examples
in class. However, it's hard for me to predict exactly which
types of functions \(g\) (as in "\(L[y]=g\)") I'll have time to cover
in examples by the end of Wednesday's class, so for now I don't want
to try say which exercises you should finish by the end of
Wednesday, which you should finish before Friday's class, and
which you should finish before Monday's class.
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.])
-
On the Miscellaneous Handouts page, I've added a section with
several MUC-related handouts. Shortly after Wednesday's class,
view the "granddaddy" file and read the accompanying "Read Me"
file, which is essentially a long caption for the diagram in
the "granddaddy file". (Feel free to view these sooner but I
think the diagram will make more sense after I've said certain
things in class.)
- 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 (in addition to the ones that were due Monday
3/18!):
Do 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? 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).
It's like building a motor skill. 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. That's just a rationalization for not doing
work you might find tedious. This strategy might help you
retain a skill for a week, but not for all the exams you'll need it
for,
let alone through the future courses (anywhere from zero to
several) in which you might 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?
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 3/27/24 |
Second midterm exam (assignment is to study for it).
|
| F 3/29/24 |
No new homework
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| M 4/1/24 |
No new homework
|
W 4/3/24 |
4.7/ 26, 29,
34a. (In #29, assume that the functions
\(p\) and \(q\) are linearly independent on the interval \(
(a,b)\) . In #34, assume that the interval of interest is the
whole real line.) Material covered that's been covered
in class recently is
sufficient to do problems 26 and 29 without reading any of
Section 4.7.
Read Section 4.7 up to, but not including, Theorem 7
(Variation
of Parameters). The only part of this that we have not already
covered in class is the part that starts after Definition 2 and
ends
with Example 3.
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.
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.)
|
F 4/5/24 |
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".
|
M 4/8/24 |
No new homework
|
W 4/10/24
and
F 4/12/24
|
In class I went over the following material very quickly. Go
over it again, more carefully:
- (a) Show that the Chain Rule is valid for
functions of the form \(t\mapsto f(h(t))\), where \(t\) is a real
variable, and \(h\) and \(f\) are, respectively, a real-valued and
a complex-valued differentiable function of a real
variable. (Recall
that the latter means that
particular, \(f\) can be written as \(u+iv\), where \(u\) and
\(v\)
are differentiable functions of a real variable, and we define
\(f'\) to be \(u'+iv'\).) In other words, show that for \(h\)
and \(f\) as above, $$\frac{d}{dt}(f(h(t))=h'(t)\, f'(h(t)).$$
(b) Let \(r=\alpha + i\beta\), where
\(\alpha\) and \(\beta\) are real. Recall that
for a real number \(t>0\)
the definition of \(t^r\) is
$$\begin{eqnarray*}
t^r&=&e^{r\ln t} \\
&=& e^{\alpha\ln t \ +\ i\beta\ln t} \\
&=& e^{\alpha \ln t}(\ \cos(\beta \ln t) + i\sin(\beta \ln t)\ )
\ \ \ \ \ \ \ (*)\\
&=& t^{\alpha}(\ \cos(\beta \ln t) + i\sin(\beta \ln t)\ ).
\end{eqnarray*}
$$
(Only the first equality above is the definition of \(t^r\). The
second just re-expresses \(r\ln t\) in terms of its real and
imaginary parts, and the third
then uses our definition of the
complex exponential function. The fourth equality, which is
not needed below, is just a reminder of the definition of
\(t^\alpha\) for an arbitrary real exponent \(\alpha\),
which we generalized to a complex exponent in the first equality.)
Using (*), show the following:
- For any complex numbers \(r\) and \(s\),
and any real \(t>0\),
$$t^{r+s} = t^r t^s\ \ \ \ \ \ (**).$$
Here and below, remember that there is no such thing as
"proof by notation". Even for arbitrary real
exponents
\(r\) and \(s\), without the definition of "\(t\) to an
arbitrary real exponent" in terms of the exponential
and natural log functions (the definition used in the
equation after (*) above), equation (**) is by no means
obvious when \(r\) and \(s\) are not integers.
Choosing the same notation
for "\(t\)
to a power" whether or not
exponent is an integer, cannot imply any
algebraic rules for non-integer exponents.
The fact that the integer-exponent rules extend to more general
exponents is beautiful and very convenient, but
it's something we have to derive; the choice of
notation can't make something true or false. You're
being shown power-notation that was chosen to
reflect and remind us
of various properties. The properties drive the
choice of
notation, not the other way around.
- For any complex number \(r\), and any real \(t>0\), the
complex number \(t^r\) is never the number 0.
- For any complex number \(r\), the function
\(t\mapsto t^r\) is differentiable
on the interval \( (0,\infty)\), and that
$$\frac{d}{dt}
t^r=rt^{r-1} \ \ \ \ \ \ (***).$$
(Part (a) above can be used to shorten your
work. At the last step of your derivation, you'll
need
to use (**) to simplify \(t^{-1}t^r\).)
(c) Using (***) and (**), check that for
the differential operator \(L\) defined by \( L[y]=at^2
\frac{d^y}{dt^2} +bt\frac{dy}{dt}+cy\) (where \(y\) is
any complex-valued function \(y\) defined on the interval
\((0,\infty)\), with independent variable \(t\)) satisfies $$
L[t^r] = [ar^2 +(b-a)r+c]\, t^r$$ for any complex number
\(r\).
(I just started going over the method for
the next group of exercises at the end of class on Monday. In
principle, you know all you need to know to do these exercises
based on what I presented, but [as of Tuesday 4/9] you have yet
to see me go all the way through a Variation of Parameters
problem. For this reason I'm combining the assignment due
Wednesday with the one due Friday. Based on the classroom
presentation
on Monday, and as much reading of Section 4.6 as you want, get as
many of the exercises done as you can before Wednesday's class,
and the rest before Friday's class. I may still add more
exercises for you to do by Friday.)
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.
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.)
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.
|
| M 4/15/24 |
4.7/ 24cd, 43, 44. For 24cd, instead of solving these DEs by the
method in problem 24's instructions, solve them using Variation of
Parameters.
Skim Section 6.1.
(We will not be covering Chapter 5.)
A lot of this is review of material we've covered already.
|
| W 4/17/24 |
Read Section 6.3.
Based on your reading of Sections 6.2 and 6.3, get a head
start on the exercises due Friday.
|
| F 4/19/24 |
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
comments about factoring
in the assignment that was due 3/25/24.
6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the
annihilator method for now;
just
use what we've done in class with MUC and superposition.
|
| M 4/22/24 |
No new homework. Review your notes and understand what we did last
week.
|
| W 4/24/24
(sort of) |
Here are some exercises that I could have assigned with a due-date
of 4/24, based on what we'd completed in the Monday 4/22 lecture:
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 3/25) 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 exercise in the 5–10
group is among the few such problems that are the reason I could
only say "usually 0 or 1," not "always 0 or 1."
The Section 6.3 exercises (including some of the ones on the "if
you want more practice" list further down on this page) contain
a few more of these exceptions.
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 rational root, but at least one
integer root of such
small absolute value that you'd guess it even without knowing the
Rational Root Theorem.
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In case you'd like some extra problems to practice with:
A the end of each chapter, there's a list of review problems for that
chapter. A
good feature of these lists is that, unlike the
exercises after each section, the location gives you no clue as to
what method(s) is/are likely to work, which is the same situation
you're in on an exam. However, since we didn't cover every section
of the chapters we covered material from, I need to tell you which
problems you should be able to do:
- Chapter 2:
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.
Even if you don't have time to work
through the problems on p. 79, they're good practice for figuring
out (or reminding yourself) of what 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. That's why I assigned several of my non-book problems during
the semester.
- Chapter 4:
You should be able to do exercises 1–36 on p. 231.
A negative feature of the book's exercises
on Cauchy-Euler DEs (in Chapters 4 and 6) is that none,
or almost none, of them
ask for solutions on \((-\infty,0)\).
That's why I assigned some non-book problems on that topic.
- Chapter 6:
You should be able to do exercises 1, 3, 4, 5, 7, 9, 10 on
pp. 343–344.
In addition, you should be able to do exercises
6.3/ 21–33. For 21–30, you may ignore the book's instructions
to use the annihilator method (as long as you remember the MUC).
Our use of the annihilator method was to show why the MUC
works. When actually solving a DE by the annihilator method, you still
end
up having to use the MUC, just packaged under a different name.
The only exercises in the book for which you have to use
annihilators are the ones like 6.3/ 11–20 and 6.Review/ 7,
exercises
in which you're asked to find an annihilator, not to solve a DE.
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Thursday 5/2/23 |
Final Exam
Location: Our usual classroom
Starting time: 3:00 p.m.
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