| F 1/13/23 |
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 the textbook, read Section 2.1.
In
my notes, read from where you left off in the last assignment
through Example 3.10 (p. 14). Also, I inserted a sentence on p. 4
(involving sine and arcsine) that you should read.
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.
In my notes, if you see "??" where
there ought to be a section number, the reason is that it's a section
that I haven't posted yet.
| W 1/18/23 |
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 Section 2.3, read the definition of "standard form"
(equation (4) on p. 49). Also, 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 is the same as the collection of
functions on the right).
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 (*).
- The "family of antiderivatives" interpretation of
"\(\int f(x)\,dx\)" was standard when I first learned
calculus, and I still regard it as the "correct"
interpretation. However, some present-day authors
(including, but not limited to, Nagle, Saff, and Snider) to
mean "any one member of this family of funtions,"
i.e. what I'm denoting "\(\int_{\rm spec} f(x)\,dx\)".
This other usage does have certain advantages;
e.g. it simplifies
some equations. However, it has other disadvantages. One of
these is that it leads many students to believe that, for
example, "\(\int x\, dx\)" represents exactly one function,
namely the function \(F\) given by \(F(x) =x^2/2\), and that
"\(\int x\, dx =x^2/2 +1\)" is wrong. Another
disadvantage is that the "family of antiderivatives" concept
(i) is important, certainly important enough to merit
a standard notation for it, (ii) is an object that
is uniquely attached to a given continuous function
(the function appearing under the integral sign) on an
interval, and (iii) is
omnipresent (although sometimes hidden in the
background), not just in the explicit context of
integration, but in the context of differential equations.
In
my notes, read
from Example 3.11 (p. 14) up through the current end of Section 3.3 (p. 21).
|
| F 1/20/23 |
In Section 2.3, read Example 2 to see one way of approaching
this IVP. I will add some comments and minor corrections on the
book's solution soon.
Comments and Corrections:
- 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 or 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
$$
\frac{dy}{dt}+ \frac{2}{\sec}y= 50\frac{{\rm kg}}{\sec} e^{-10t/\sec}.
\ \ \ \ \ \ \ (*)$$
This DE is in standard linear form, with coefficient
function
\(P(t)\) the constant function
\(2/\sec\), so one specific antiderivative is
\(\int_{\rm spec} P(t)\, dt =(2/\sec) t\) (a dimensionless
quantity, as is necessary for anything we're going
to exponentiate), 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
\(y(t)=
(\frac{185}{4}e^{-2 t/\sec} - \frac{25}{4}e^{-10 t/\sec}){\rm
kg},\)
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. But in students' work,
the relevant arithmetic still
needs to be done (including any conversions, if needed),
and the final answer should have the correct units
(e.g. "70 kg" rather than just "70", if the answer is a
mass).
In the assignment that was due F 1/20/23, see the comments
and corrections inserted after the first bullet point.
2.3/ 7–9,
12–15 (note which variable is which in #13!), 17–20,
22, 23, 28
Do non-book problem
2.
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 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 \(\frac{1}{x}\).
|
| M 1/23/23 |
2.3/ 25a, 27a, 31, 33, 35.
Corrections:
- #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 \(x=0\).
- #35: The term "a brine" in this problem is not proper English; it's
similar to saying "a sand". One should
either say "brine" (without the "a") or "a brine solution".
Another phrase that should not be used is the redundant "a
brine solution of salt" (literally "a concentrated
salt-water solution of salt"), which 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. As mentioned in class (in
somewhat different words), when solving differential equations, a tiny
algebra slip can make your answers utter garbage. For this
reason, there is usually no such thing as a "minor algebra error"
in solving differential equations. This is a fact of life you'll
have to get used to. The severity of a mistake is not determined by
the number of pencil-strokes it would take to correct it, or whether
your work was consistent after that mistake. If a mistake (even
something as simple as a sign-mistake) leads to an answer that's
garbage, or that in any other way is qualitatively very different from
the correct answer, it's a very bad mistake, for which you can
expect a significant penalty. A sign is the only difference between a
rocket going up and a rocket going down. In real life, details like
that matter!
I urge you to develop (if you haven't already) the mindset of
"I really, really want to know whether my final answer is
correct, without having to look in the back of the book, or ask my
professor." Of course, for many exercises, you can find answers in
the back of the book, and you are always welcome to ask me in office
hours whether an answer of yours is correct, but that fact won't
help you on an exam—or if you ever have to solve a
differential equation in real life, not just in a
class. Fortunately, DEs and IVPs have built-in checks that allow you
to figure out whether you've found solutions (though not
always whether you've found all solutions). If you make
doing this 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.
|
| W 1/25/23 |
In the 1/23/2023
version of my notes, read
Sections 5.1, 5.2 and 5.3. The most important of these is
Section 5.3; the two earlier sections are there in case
you need to review (or never learned) some terminology
that's needed in Section 5.3 and elsewhere. As discussed
in class, and again after Corollary 5.9, my notes' Theorem
5.6 ("FTODE") is what the textbook's Theorem 1 on
p. 11 should have said (modulo my having used
"open set"
in the FTODE instead of the book's "open rectangle").
1.2/ 18, 23–28, 31. Do not do these until
after you've read Section 5.3 in my notes.
Anywhere that the book asks you whether its Theorem 1 implies
something, replace Theorem 1 with the FTODE stated in my notes
(the same theorem I called the FTODE in class).
For 23–28, given the book's reference
to Theorem 1, 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
open 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 the 1/23/2023
version of my notes, read
Sections
3.2.4 and 3.2.5.
|
| F 1/27/23 |
This assignment is being posted later than I'd
planned. If you can't get it all done before the Friday 1/27 class,
I'll understand, but try to get as much of it done before that class as you
can.
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.
- 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, which I have not defined (and may
end up not using that specific term). The book has a "definition" of
implicit solution in Section 1.2, but the wording is ambiguous,
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.)
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 amount 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 Friday 1/27
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 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.
In the 1/26/2023
version of my notes, read the newly inserted
Section 3.2.3 ("Standard forms" and solutions in a region)
and Section 5.4 (The Implicit Function Theorem).
(This is easier to do now that
I've actually posted that version of the notes. Sorry!)
|
| M 1/30/23 |
Do non-book problems
3–6 .
For now (with the Monday 1/30
due date), the same instructions as for the previous assignment
apply.
Answers to these
non-book problems are posted on the
"Miscellaneous handouts" page. (Well,
they are now, at least, though they weren't when I first said
so ... .)
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).
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. (The expression \(\frac{1}{x}\) is the derivative
of \(\ln x\), not one of its antiderivatives; \(\ln |x|\) is
an antiderivative of \(\frac{1}{x}\), not its derivative.)
This does not mean you should study integration techniques to
the exclusion of material you otherwise would have studied to do
your homework or prepare for exams. You need to both review the old
(if it's not fresh in your mind) and learn the new.
In the 1/29/2023
version of my notes, read as much as you can (before class)
of Sections 3.2.7 (Implicitly defined functions), 3.2.8
(Implicit solutions, and implicitly defined solutions, of
derivative-form DEs), and 3.2.9 (General solutions of
separable DEs). This homework-update was posted very late, so I'll understand
if you can't get to this before class, but you'll need to read these
sections (and probably more) before Wednesday's class. The more you
can get to before Monday's class, the better.
|
| W 2/1/23 |
For exercises 2.2/ 7–14 and non-book problems 3–6,
complete the work you did in the last two assignments. (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.)
Do non-book problems
7 and 8.
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,
then 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.)
In my notes, dated
1/29/2023 or later, finish reading the sections "Implicitly
defined functions", "Implicit solutions, and
implicitly defined solutions, of derivative-form DEs"
and "General solutions of separable DEs".
(I'm not stating the section-numbers right now because my
current but not-ready-for-posting update has an additional
section. This will alter at least one section-numbers
in the 1/29/2023 notes. This update might
get posted on Tues. 1/31, in which case I'll be adding
to
the reading-portion of this assignment accordingly.)
|
| F 2/3/23 |
This assignment is being posted too late for you to get
much of it done before Friday's class. Do it on an "ASAP" basis.
In my notes, in addition to adding a lot of material since the last
version you saw, I've moved material around and have re-ordered
some sections. I anticipated that I might do this, which is why I
used section titles, not just section numbers, in the
last few assignments. If you're behind on the reading assigned in
the last week,
and are looking back at the assignments with due-dates of 1/27, 1/30,
or 2/1 to see what you were
supposed to read by those dates, use the section titles to see what
the corresponding sections are in the newest version of the notes. If
you're more behind than that, you'll need to figure out on our own
what the old assignments correspond to in the newest notes.
The update that I just posted (dated 2/2/2023) includes some
additions
to sections that I already asked you to read. I've put those additions
in blue
for now (along with the ones in blue from the last revision)
so that you can find them easily
if you read them soon. I'll be returning the blue items
to black
in a few days.
In the textbook, read Section 2.4 through the boxed
definition "Exact Differential Form" on p. 59. See Comments,
part 1, below.
In my notes, read the
following:
- Addition to Section 3.2.2 (Maximal and general solutions
of derivative-form DEs), currently on p. 19.
- Additions to Section 3.2.3 ("Standard Forms" and solutions in a
regions), currently on pp. 20–23).
- Additions to Section 3.2.6 (Implicit solutions, and
implicitly defined solutions, of derivative-form DEs),
if you didn't already read these in the
1/31/2023 revision posted Feb. 1. Currently these are on
pp. 35–43.
- Additions to Section 3.2.7 (General solutions in implicit
form [for a derivative-form DE], renamed from "General and implicit
solutions of a derivative-form DE in a region in \({\bf R}^2\)"),
currently on pp. 47–50.
- Additions to Section 3.2.10 (General solutions of separable DEs),
currently on pp. 63–68.
Theorems 3.42 and 3.43
in my notes are
closely related to the "Formal Justification of Method" on p. 45
of the textbook. You will find the book's presentation simpler
than mine, but this simplification comes at a high price:
-
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 it purports to establish neglects an important
issue. The question of whether the method gives all the
solutions, or even all the non-constant solutions, is
never even mentioned, let alone answered. An example in my
notes (currently numbered as Example 3.44, 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. (For the method to work reliably, we
actually need to assume even more, namely that \(p'\) is
continuous. But the discussion and example in my notes show
only that we need to assume that \(p'(r)\) exists at the
points \(r\) for which \(p(r)=0\).
- Section 3.3.1 (Differentials and differential-form DEs)
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.)
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
the combined graduate/undergraduate course Modern Analysis 2,
and occasional special-topics courses.) With the advanced
meaning of "differential form", the only differential
forms that appear in an undergraduate DE textbook
are differentials, so there's no
good reason in a such a course, or in its textbook, to use the term
differential form for a differential.
There is also a pronunciation-difference in the
two usages of "differential form". The pronunciation of this
term in my notes is "differential form", with the
accent on the first word, providing a contrast with
"derivative form". In the other usage of "differential
form"—the one you're not equipped to understand, but
that is used in the book's definition-box on p. 59—the
pronunciation of "differential form"
never has the accent on the first word; we either say
"differential form", with the accent on the second
word, or we accent both words equally.
- The paragraph directly below the "Exact Differential
Form" box on p. 59 is not part of the current
assignment. However, for future reference, this paragraph is
potentially confusing or misleading, because while the first
sentence uses "form" in the way it's used on p.58 and in my
notes, the third sentence uses it with the other, more
advanced meaning. This
paragraph does not make sense unless the term "differential
form" 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'll find the sentence plausible, but
you should be troubled by the fact that since \(\frac{dy}{dx}\) is
simply notation for an object that is not actually a real number
"\(dy\)" divided by a real number "\(dx\)", just how is it that an
equation of the form \(\frac{dy}{dx}=f(x,y)\) can be "rewritten" in
the form of equation (3)? Are the two equations equivalent? Just what
does an equation like (3) mean? In a derivative-form DE,
there's an independent variable and a dependent variable. Do you see
any such distinction between the variables in (3)? Just what
does solution of such an equation mean? Is such a solution the
same kind of animal as a solution of equation (1) or (2) on p. 6 of
the book, even though no derivatives appear in equation (3) on
p. 58? If so, why; if not, why not? Even if we knew what "solution
of an equation in differential form" ought to mean, and knew how to
find some solutions, would we have ways to tell whether we've
found all the solutions? Even for an exact equation, how do
know that all the solutions are given by an equation of the form
\(F(x,y)=C\), as asserted on p. 58?
The 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.
|
| M 2/6/23 |
In my notes
(with item-numbers and page-numbers taken from the version dated
2/4/2023,
look at the Table of Contents for Sections 3.3–3.6, to
get an idea off what you'll find where. (That's in case you want
to consult the notes when you're working on exercises from the
book. Later in this assignment, there's actual reading to do from
my notes.)
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. Sections 3.4–3.6 of my
notes are,
essentially, concerned with questions of when various mechanical
procedures will and won't give you a completely correct answer.
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")? For questions
answered in the back of the book: not all answers there are correct
(in general; 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).
2.4/ 1–8.
Note: (1) In class on Friday 2/3, we
did not get far enough to discuss how to tell whether a DE in
differential form is exact. For this, use the "Test for
Exactness" in the box on p. 60 of the book. We will discuss this
test in Monday's class. (2) For differential-form DEs, there is no
such thing as a linear equation. In these problems, you are meant
to classify an equation in differential form as linear if
at least one of the associated derivative-form equations (the ones
you get by formally dividing through by \(dx\) and \(dy\),
as if they were numbers) is linear. It is possible for one of
these derivative-form equations to be linear while the other is
nonlinear. This happens in several of these exercises.
For example, #5 is
linear as an equation for \(y(x)\), but not as an equation for
\(x(y)\).
In 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 Monday's class, I'll do some examples that
should help get you more confident in the method.
Don't invent a different method for solving
exact equations. 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
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 "\(y(x)= ...\)". But this exercise is an
example of what I call a "schizophrenic" IVP. In practice, if you are
interested in solutions with independent variable \(x\) and dependent
variable \(y\) (which is what an initial condition of the form
"\(y(x_0)=y_0\)'' indicates), then the differential equation you're
interested in at the start is one in derivative form
(which in exercise 22 would be \(x^2 +2y \frac{dy}{dx}=0\), or an
algebraically equivalent version), not one in differential
form. Putting the DE into differential form is often a useful
intermediate step for solving such a problem, but differential form is
not the natural starting point. On the other hand, if what you are
interested in from the start is a solution to a
differential-form DE, then it's illogical to express a preference for
one variable over the other by asking for a solution that satisfies a
condition of the form "\(y(x_0)=y_0\)'' or "\(x(y_0)=x_0\)''. What's
logical to ask for is a solution whose graph passes through the
point \((x_0,y_0)\), which in exercise 22 would be the point
(0,2).
2.4 (resumed)/ 21, 22
(note that #22 is the same DE as #16, so you don't have to solve a new
DE; you just have to incorporate the initial condition into your old
solution). Note that exercises 21–26 are what I termed
"schizophrenic" IVPs.
Your goal in these problems is to find an an
explicit formula for a solution, one expressing the dependent
variable explicitly as a function of the independent variable
—if algebraically possible—with the choice of
independent/dependent variables indicated by the initial condition.
However, if in the algebraic equation ``\(F({\rm variable}_1, {\rm
variable}_2)=0\)'' that you get via the exact-equation method (in
these schizophrenic IVPs), it is impossible to solve for the
dependent variable in terms of the independent variable, you have to
settle for an implicit solution.
In my notes
(with item-numbers and page-numbers taken from the version dated
2/4/2023):
-
Read Definition 3.56 (the last definition in Section
3.3.2) and the
paragraph that follows it. I covered the rest of Section 3.3.2 in the
Friday Feb. 3 class. (But if you missed that class, read all of
Section 3.3.2.)
- Read Section 3.3.3. It's OK to skip the second
page (unlees you missed Friday's class); I established the result
of that page in class, by a different method.
- Read Section
3.3.4 .
- Read Section 3.3.5
. In this section and the ones that follow, the portions
in magenta are optional reading. In a couple of
days, I'll be changing the currently-blue items in
Section 3.2's subsections to black, then changing all the
current magenta to blue.
- Read Section 3.3.6.
My notes don't
present the basic method for (trying) to solve exact
equations. I'll present that in class, but until I do, use what
you see in the book's Section 2.4.
- Skim
Section 3.3.7 up through statement 3.43;
the ideas are the same as for
derivative-form DEs.
Read statement 3.144 (p. 96) and Example 3.74
(pp. 101–102).
- Read Sections 3.4, 3.5, and 3.6 (Relation between
differential form and derivative form; Using differential-form
equations to help solve derivative-form equations; and
Using derivative-form equations to help solve differential-form
equations). 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.
|
| General info |
The date for your first midterm will be be Monday Feb. 13.
For any exam, I'll always give you essentially at least a week's notice
("essentially" meaning that, for example, I may let you know on a
Wednesday evening—via your homework page or an email— that
the exam will be the following Wednesday). Generally, two lectures
before the exam (this time, it might be
three lectures before), I will
give you a copy of
the corresponding exam from the last time I
taught this course.
(For these sample-old-exam purposes,
"this course" might either be honors or non-honors MAP2302; I'll
decide separately for each exam.)
I will give this out only
in class, or, for students with an
excused absence, in my in-person office
hours. I will not post or electronically distribute
any old exams or solutions.
|
| W 2/8/23 |
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.
[Note added later: I went through the argument in class on
Wed. Feb. 8.] 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 that's in the
current book 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.)
.
Do non-book problems 9 and
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. 77),
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.
Read Section 3.3.4 of my notes.
Catch up on any reading you haven't finished yet.
|
| F 2/10/23 |
Do non-book problem 11. This
example does not contradict anything we've learned, because the
region \(R\) has a hole (so, in particular, it's not a rectangle).
2.4/ 10, 15, 23, 26 (again, these last two are "schizophrenic IVPs")
Read The Math
Commandments.
|
| M 2/13/23 |
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.
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 2/15/23 |
No new homework.
But it wouldn't hurt for you to start reading 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.
|
| F 2/17/23 |
Read Section 4.1.
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 on Wednesday 2/15/23. (So, if you have
decent notes from that lecture, there's nothing in Section 4.7 you
need to read for this assignment.)
In problems
1–4, interpret the instructions as meaning: "State the largest
interval on which Theorem 5 guarantees existence and uniqueness of a
solution to the differential equation that satisfies [the given
initial conditions]."
|
| M 2/20/23 |
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 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 textbooks, you would rarely see this
writing mistake; in our edition of NSS, it's all over the
place.)
- On
p. 158, the authors mention that the "auxiliary equation" is also
known as the "characteristic equation". In class, I'll be using the
term "characteristic equation", which is more common.
- 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
waiting until I've covered 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 2/22/23 |
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 2/24/23 |
In Section 4.3, read 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.
Note: In the title of this section,
"complex roots" should be replaced by "non-real roots", "non-real
complex roots", or "no real roots". The same is true anywhere
you see the term "complex roots" in this book, including the
exercises assigned below. 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.
4.3/ 1–18.
In a few days, I'll have you read more of Section 4.3.
Note: The book uses the complex
exponential function (which we have not yet discussed in class; we
will discuss it soon if time permits) to derive the fact that in the
case of non-real characteristic roots \(\alpha\pm i\beta\), the
functions \( t\mapsto e^{\alpha t} \cos \beta t\) and \(t\mapsto
e^{\alpha t} \sin \beta t\) are solutions of the DE (2) on p. 166,
rather than showing this by direct computation using only
real-valued functions.
The complex-exponential
approach is very elegant and unifying. It is also useful for
studying higher-order constant-coefficient linear DEs, and for
showing the validity of a certain technique we haven't gotten to yet
(the Method of Undetermined Coefficients). It is definitely worth
at least reading about.
The drawbacks are:
- Several new objects (complex-valued functions in general,
and the derivative of a complex-valued function of a real
variable) must be defined.
- Quite a few facts must be established, among them 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
\(b\)). This is a very worthwhile exercise, but time-consuming.
- On exams in this class, all final answers must be expressed
entirely in terms of real numbers; complex numbers are
allowed to appear only in intermediate steps. (The instructions on all
your exams starting with the second midterm will say so.) Every
year, there are students who use the complex exponential function
without understanding it, leading them to express some final answers
in terms of complex exponentials. Such answers receive little if any
credit.
As the authors admit—sort of, in the sentence after equation
(9))—there are also some problems with the book's
presentation.
- Equation (4) on p. 168 is presented in a sentence that
starts with "If we assume that the law of exponents applies to complex
numbers ...". Unfortunately, the book is very fuzzy about the
distinction between definition and assumption, and never
makes clear that equations (4), (5), and (6) on p. 168 are not things
that need to be assumed. Rather, all these equations result
from defining \(e^{z}\), where \(z= a+ bi\), to be \(e^a(\cos b
+ i \sin b)\), a formula not written down explicitly in the book.
- A non-obvious fact, beyond the level of this course, is that
the above definition of \(e^z\) is equivalent to defining
\(e^z\) to be \(\sum_{n=0}^\infty \frac{z^n}{n!}\). This is a
series that—in a course on functions of a complex
variable—we might call the Maclaurin series for \(e^z\).
However, the only prior instance in which MAP 2302 students have
seen "Maclaurin series" (or, more generally, Taylor series)
defined is for functions of a real variable. To define
these series for functions of a complex variable requires a
definition of "derivative of a complex-valued function of a
complex variable". That's more subtle than you'd think. It's
something you'd see in in a course on functions of a complex
variable, but is beyond the
level of MAP 2302. So the sentence on p. 166 that's two lines
below equation (4) is misleading; it implies that we
already know what "Maclaurin series" means for
complex-valued functions of a complex variable (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 equation (5), we would then use the much clearer
word "definition".
|
| M 2/27/23 |
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.
|
| W 3/1/23 |
4.7 (yes, 4.7)/ 25
Read Section 4.4 up through Example 3.
Read Section 4.5 up through Example 2.
|
| F 3/3/23 |
Finish reading Sections 4.4 and 4.5.
As mentioned in class, 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.
Exercises due Friday are below (after my summary of various
facts about the method). Over the next few days, I'll be
assigning almost all the exercises in these two sections. The
ones due Friday are below, but feel free to get ahead by doing
more!
In class I used the term multiplicity of a root of the
characteristic polynomial. This is the integer \(s\) in the box on
p. 178. (The 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 in class about
multiplicity, it is imperative not to use the identical notation
\(r\) in \(t^me^{rt}\) as in in the characteristic polynomial
\(p_L(r)=ar^2+br+c\) and the characteristic equation
\(ar^2+br+c=0\). Replace the \(r\) in the box on p. 178 by the
letter \(\alpha\), so that the right-hand side of the first
equation in the box is written as \(Ct^m e^{\alpha t}\).
Recall
that \(ar^2+br+c\) can be factored as
\(a(r-r_1)(r-r_2)\), where \(r_1\) and \(r_2\) are
complex numbers, possibly real. The multiplicity \(s\) of
\(\alpha\) (as a root of \(p_L(r)\)) is the number of times
\(r-\alpha\) appears in this factorization. Thus:
- \(s=0\) if \(r-\alpha\) is not a factor of
\(p_L(r)\) (equivalently, if \(p_L(\alpha)\neq 0\));
- \(s=1\) if \(r-\alpha\) appears
exactly once in the factorization \(a(r-r_1)(r-r_2)\)
(equivalently, if the roots \(r_1,r_2\)
are distinct and \(\alpha\) is equal to one of them); and
- \(s=2\) if \(p_L(r)\) has a double root, and that
root is exactly \(\alpha\) (equivalently, if
\(p_L(r)=a(r-\alpha)^2\)).
Using \(\alpha\) instead of \(r\) in the first few lines of the
box on p. 178 also unifies the first half of the box (in which
cosine and sine don't appear) with the second half of the box:
\(t^m e^{\alpha t}\) is precisely the \(\beta=0\) case of
\(t^m e^{\alpha t}\cos\beta t\). Note also that \(t^m\) is the
\(\alpha=0\)
case of \(t^m e^{\alpha t}\).
For now, let's call any not-identically-zero function of the form
\(t^m e^{\alpha t}\cos\beta t\) or
\(t^m e^{\alpha t}\sin\beta t\)
(where \(C\) is a constant) an MUC basis function.
For MUC purposes, to each of the
these functions,
associate the complex number \(\alpha + i\beta\).
Here, \(\alpha\) and \(\beta\) can
be any real numbers, and \(m\) can be any non-negative integer. Any
of these three numbers could be zero (and so could any two, or all
three). Setting these numbers equal to zero, individually or in
combination, yields simpler functions to which associate a complex
number \(\alpha+i\beta\) the same way (but with the possiblity that
\(\alpha\) and/or \(\beta\) could be zero). Using the symbol
"\(\longleftrightarrow\)" to denote the association of an MUC basis
function with a complex number, we then have
\(t^m e^{\alpha t}\cos\beta t\longleftrightarrow \alpha +i\beta
\)
and
\(t^m e^{\alpha t}\sin\beta t\longleftrightarrow \alpha +i\beta\),
with the following special cases and subcases.
Note: In the right-hand side of a given non-homogeneous DE,
you'll usually see constants in front of the MUC basis
functions. The only MUC basis function that that might be confusing
for is the \(m=\alpha=\beta=0\) case below. A constant function
\(C\), not multiplied by anything (but
potentially added to something) is simply \(C\) times the
MUC basis function "\(1\)".
I've posted a more visual
presentation of this info (not using quite the same "MUC basis
functions", which I defined mostly to simplify writing this
in HTML)
in Canvas, under Files/MUC.
- Case \(\beta=0\)
\(t^m e^{\alpha t}\longleftrightarrow \alpha
\).
For each MUC basis function \(g\) (or each such function
multiplied by a constant), and each constant-coefficient linear
differential operator \(L\), there is a corresponding "MUC form of a
particular solution \(y_p\)," whose formula involves some number
of undetermined coefficients. The values of these undetermined
coefficients are then found by plugging the form of \(y_p\) into the
equation \(L[y]=g\). The number of undetermined
coefficients is determined completely by
\(g\); that number is not influenced by the order of \(L\). (All
that matters is that \(L\) be linear
and constant-coefficient.) Similarly the "naive guess" (see below)
is determined completely by \(g\).
If \(\alpha+i\beta\) is the complex number associated with the MUC
basis function \(y_p\), then the MUC form of \(y_p\) is
\(t^s
\tilde{y}_p\), where \(s\) is the
multiplicity of \(\alpha+i\beta\) as a characteristic root,
and \(\tilde{y}_p\) is the "naive guess": a function-form that depends
only on \(g\), not on \(L\).
In the box on p. 178 of the book, the "naive guess" is the function
you'd get by omitting the factor \(t^s\),
The "MUC eligible functions"—the functions \(g\) for which
the Method of Undetermined Coefficients gives a way of finding
a particular solution of \(L[y]=g\)— are the linear
combinations of MUC basis functions (i.e. functions \(g\) of the
form \(C_1g_1 +C_2 g_2 +\dots + C_n g_n\), where each of the function
\(g_j\) is an MUC basis function, and each \(C_j\) is a constant).
Exercises:
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.
Use the "\(y=y_p+y_h\)" approach
discussed in class , plus superposition (problem 4.7/ 30, previously
assigned) where necessary, plus your knowledge (from Sections 4.2 and 4.3)
of how to solve the associated homogeneous equations for all the DEs
in these problems.
Note that the MUC is not
needed to do exercises 1–8, since (modulo having to use
superposition in some cases) the \(y_p\)'s are handed to you on a
silver platter.
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 3/6/23 |
In class, for the sake of simplicity and
time-savings, 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. (I'm pleased that on the first exam, anyone who did
such name-change in an intermediate step, did remember in
their final answers to switch the variables back to what they
originally were. Some past students have simply written a note
telling the instructor how to interpret their new
variable-names. No. [Not if you want 100% credit
for an otherwise correct answet to
that problem. Writing
your answer in terms of the given variables accounts for
part of the point-value and time I've budgeted
for.])
In Canvas, under Files/MUC, view
the "granddaddy" file and read the "Read Me" file.
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.
| W 3/8/23 |
4.4/ 23 (posted now under the assignment due 3/3/23, but
was accidentally omitted from that assignment originally).
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).
As mentioned in class,
in a constant-coefficient differential equation \(L[y]=g\),
the
functions \(g\) to which the MUC applies are the same regardless of
the order of the DE, and, for a given \(g\), the MUC form of a
particular solution is also the same regardless of this order. The
degree of the characteristic polynomial is the same as the order of
the DE (to get the characteristic polynomial, just replace each
derivative appearing in \(L[y]\) by the corresponding power of
\(r\), remembering that the "zeroeth" derivative—\(y\)
itself—corresponds to \(r^0\), i.e. to 1, not to \(r\).)
However, a polynomial of degree greater than 2 can
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 homogeneous, constant-coefficient,
linear DE of order greater than two, the
corresponding characteristic polynomial has at
least one root that is an integer of small absolute value
(usually 0 or 1). For any cubic polynomial
\(p(r)\), if you are able to guess even one root, you can factor
the whole polynomial. (If the root you know is \(r_1\), divide
\(p(r)\) by \(r-r_1\), yielding a quadratic polynomial
\(q(r)\). Then \(p(r)=(r-r_1)q(r)\), so to complete the
factorization of \(p(r)\) you just need to factor \(q(r)\). You
already know how to factor any quadratic polynomial, whether or
not it has easy-to-guess roots.)
- For problem
38, note that if all terms in a polynomial \(p(r)\)
have even degree, then effectively \(p(r)\) can be treated as a
polynomial in the quantity \(r^2\). Hence, a polynomial of the form
\(r^4+cr^2+d\) can be factored into the form \((r^2-a)(r^2-b)\),
where \(a\) and \(b\) either are both real or are complex-conjugates
of each other. You can then factor \(r^2-a\) and \(r^2-b\) to get a
complete factorization of \(p(r)\). (If \(a\) and \(b\) are not real,
you may not have learned yet how to compute their square roots, but
in problem 38 you'll find that \(a\) and \(b\) are real.)
You can also do problem 38 by extending the
method mentioned above for cubic polynomials. Start by guessing one
root \(r_1\) of the fourth-degree characteristic polynomial \(p(r)\).
(Again, the authors apparently want you to think that the way to find
roots of higher-degree polynomials is to plug in integers, starting
with those of smallest absolute value, until you find one that works.
In real life, this rarely works—but it does work in all the
higher-degree polynomials that you need to factor in this book.) Then
\(p(r)=(r-r_1)q_3(r)\), where \(q_3(r)\) is a cubic polynomial that you
can compute by dividing \(p(r)\) by \(r-r_1\). Because of the
authors' choices, this \(q_3(r)\) has a root \(r_2\) that you should be
able to guess easily. Then divide \(q_3(r)\) by \(r-r_2\) to get a
quadratic polynomial \(q_2(r)\)—and, as mentioned above, you
already know how to factor any quadratic polynomial.
- For
problem 40, you should be able to recognize that \(p_L(r)\) is \(r\)
times a cubic polynomial, and then factor the cubic polynomial by
the guess-method mentioned above (or, better still, recognize that
this cubic polyomial is actually a perfect cube).
- 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.) In the notation used in
the last couple of lectures, using \(s\) for the multiplicity of a
certain number as a root of the characteristic
polynomial, \(V\neq 1\) puts you in the
\(s= 0\) case, while \(V = 1\) puts you in the
\(s= 1\) case.
|
| F 3/10/23 |
Second midterm exam (assignment is to study for it).
|
| M 3/20/23 |
In Section 4.2, re-read Theorem 2, Lemma 1, and their proofs
(pp. 160– 161).
|
| W 3/22/23 |
Do non-book problem 12.
4.2/ 35, 36
4.4/ 33–36. (See comments for 4.5/ 37–40.
For three out of four of these exercises, you should find the
small-font, green, parenthetic comment
in the assignment due 3/8/23
very useful. For remaining exercise, the sentence beginning
"For any cubic polynomial ..." [a few lines later in
the same assignment] should be helpful.)
4.7/ 26, 29, 31, 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.) The material covered in class on Monday 3/20 is
sufficient to problems 26, 29, and 31 without reading any of
Section 4.7. Material covered in class several weeks ago
(relating the solutions of a non-homogeneous linear equation to
solutions of the associated homogeneous equation) is all that's
needed for 34a.
|
| F 3/24/23 |
No new homework.
|
| M 3/27/23 |
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.
4.7/ 9–14, 19, 20
Do non-book problem
13. (You'll need this before trying exercises below.)
4.7/ 15–18, 23ab.
Reminder about some terminology. As I've
said in class, "characteristic equation" and "characteristic
polynomial" are things that exist only for constant-coefficient
DEs. This terminology should be avoided in the setting of
Cauchy-Euler DEs (and
was avoided for these DEs in early editions of our
textbook). The term I used in class for equation (7) on p. 194,
"indicial equation", is what's used in most textbooks I've seen,
and really is better terminology—you invite confusion when
you choose to give two different meanings to the same
terminology. Part of what
problem 23 shows is that the indicial
equation for the Cauchy-Euler DE is the same as
the characteristic equation for the associated
constant-coefficient DE obtained by the Cauchy-Euler
substitution \(t=e^x\). (That's if \(t\) is the independent
variable in the given Cauchy-Euler equation; the substitution
leads to a
constant-coefficient equation with independent variable \(x\).)
In my experience
it's unusual to hybridize the terminology and call the book's
Equation (7) the characteristic equation for the Cauchy-Euler
DE, but you'll need to be aware that that's what the book
does. I won't consider it a mistake for you to use the
book's terminology for that equation, but you do need to
know how to use that equation correctly (whatever you call
it), and need to
understand me when I say "indicial equation".
In our textbook, p. 194's equation (7) is actually introduced
twice for
Cauchy-Euler DEs, the second time as Equation (4) in Section
8.5. For some reason—perhaps an oversight—the authors
give the terminology "indicial equation" only in Section 8.5,
rather than when this equation first appears in the book's first
treatment of Cauchy-Euler DEs, i.e. in Section 4.7.
It's also rather unusual and ahistorical
to use the letter \(t\) as the independent variable in a Cauchy-Euler
DE, even though we're certainly allowed to use any letter we
want (that's not already being used for something else). The reason
we use `\(t\)' for constant-coefficient linear DEs (as well as some
others, especially certain first-order DEs), is that when these DEs
arise in physics, the independent variable represents time.
When a Cauchy-Euler DE arises in physics, almost always the
independent variable is a spatial variable, for which a
typical a letter is \(x\), representing the location of
something. In this case, the common substitution that reduces a
Cauchy-Euler DE to a constant-coefficient DE (for a different
function of a different variable) is the substitution
\(x=e^{<\mbox{new variable}>}\) rather than \(t=e^x\). Earlier
editions of our textbook used \(x\) as the independent variable in
Cauchy-Euler DEs, and made the substitution \(x=e^t\), exactly the
opposite of what is done in the current edition. (Again,
we're allowed to use whatever variable-names we want; the
letters we use don't change the mathematics. It's just that in
practical applications it's usually helpful mentally to use
variable-names that remind us of what the variables represent.)
|
| W 3/29/23 |
(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 \(r\), the function
\(t\mapsto t^r\) 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 \( L[y]=at^2
\frac{d^y}{dt^2} +bt\frac{dy}{dt}+cy\) for any complex-valued
function \(y\) (with independent variable \(t\) in the
interval \((0,\infty)\), satisfies $$ L[t^r] = [ar^2
+(b-a)r+c]\, t^r$$ for any complex number \(r\). (I sketched
this argument in class, but want you to go through it more
carefully on your own.)
|
| F 3/31/23 |
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 exercise:
- 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.
Redo 4.7/40 by starting with the substitution
\(y(t)=t^{1/2}u(t)\)
and seeing where
that takes you.
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| M 4/3/23 |
4.7/ 43, 44
Read Section 6.1.
(We will not be covering Chapter 5.)
A lot of this is review of material we've covered already.
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.
Do non-book problem
14.
Read Section 6.2.
|
| W 4/5/23 |
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 4/7/23 |
6.2/ 1, 9, 11, 13, 15–18. The characteristic polynomial for #9
is a perfect cube (i.e. \( (r-r_1)^3\) for some \(r_1\)); for #11 it's
a perfect fourth power.
For some of these problems and the ones
below from Section 6.3, it may help you to first review my
comments about factoring
in the assignment that was due 3/8/23.
6.3/ 1–4, 29, 32. In #29, ignore the instruction to use the
annihilator method (which we are skipping for reasons of time); just
use what we've done in class with MUC and superposition.
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| M 4/10/23 |
Read Sections 7.1 and 7.2.
|
| W 4/12/23 |
Third midterm exam (assignment is to study for it).
|
| F 4/14/23 |
No new homework.
|
| M 4/17/23 |
7.2/ 4, 6–8, 10, 12,
13–20,
21–23, 26–28, 29a–d,f,g,j.
In the instructions for
1–12, "Use
Definition 1" means "Use Definition 1", NOT Table
7.1 or any other table of Laplace Transforms.
For 13–20,
do use Table 7.1 on p. 356 (as the instructions say to do),
even though we haven't derived
all of the formulas there yet in class, or discussed
linearity of the Laplace Transform (Theorem 1 on p. 355) yet in
class.
On your
final exam, you'll be given
this Laplace
Transform table. Familiarize yourself with where the entries
of Table 7.1 (p. 356) are located in this longer table. The
longer table comes from an older edition of your textbook, but is
very similar to one you can still find on the inside front cover
or inside back cover of hard-copies of the current edition, and
somewhere in the e-book (search there on "A Table of Laplace
Transforms").
Warning: On line 8 of this table, "\( (f*g)(t)\)"
is not \(f(t)g(t)\); the symbol "\(*\)" in this line denotes an
operation called convolution
(defined in Section 7.8 of the
book, which I doubt we'll get to), not simple multiplication.
For the ordinary product \(fg\) of functions \(f\)
and \(g\), there is no simple formula that expresses
\({\mathcal L}\{fg\}\) in terms of \({\mathcal L}\{f\}\) and
\({\mathcal L}\{g\}\).
In Section 7.3, read from the beginning up through Example 1
(p. 362).
In Section 7.4, read the review of the "method of partial
fractions" (more accurate name: the
[real] partial fractions decomposition ["PFD"] of a [real]
rational function) on pp. 370–374. For now,
in the examples, ignore the parts of these examples that mention
"\({\mathcal L}^{-1}\{F\}\)". That's the notation for
the inverse Laplace Transform, an operation that takes us
from functions in the "\(s\)-world" to functions in the
"\(t\)-world". The inverse transform will be of critical
importance to us soon, but we're not there yet. I'm
putting the partial-fractions review (alone) into the current
assignment because we won't have time for that review in class,
and I'll need you ready to rock-'n'-roll as soon we get to Section
7.4.
7.4/ 11, 13, 14, 16, 20
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| W 4/19/23 |
Read the rest of
Section 7.3.
7.3/ 1–10, 12–14, 20, 31
|
| F 4/21/23 |
Read Section 7.4. This time, when you get to the review of
partial fractions, don't ignore the computations of \({\mathcal
L}^{-1}(F).\)
7.4/ 1–10, 21–24, 26, 27,
31
7.5/ 15, 17, 18, 21, 22. Note that in these problems, you're being
asked only to find \(Y(s)\), not \(y(t)\).
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| M 4/24/23 |
Read Section 7.5 up through Example 2.
Read Section 7.6,
7.5/1–8, 10, 29.
To learn some shortcuts for the partial-fractions work that's
typically needed to invert the Laplace Transform, you may want
first to read the web handout
"Partial fractions and
Laplace Transform problems".
7.6/ 1–10, 11–18
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| W 4/26/23 |
7.6/ 19–32, 36ac. In 21–24, you
may skip the "Sketch the graph" part of the exercises.
Repeating what I said at
the end of Monday's class: For all of the above problems (or
those of a similar type) in which you solve an IVP, write your
final answer in "tabular form", by which I mean an expression
like the one given for \(f(t)\) in Example 1, equation (4),
p. 385. Do not leave your final answer in the form of
equation (5) in that example. On an exam, I would treat the
book's answer to exercises 19–33 as incomplete, and would
deduct several points. The unit step-functions and "window
functions" (or "gate functions", as I call them) should be
viewed as convenient gadgets to use in intermediate
steps, or in writing down certain differential equations (the
DEs themselves, not their solutions). The purpose of these
special functions is to help us solve certain IVPs
efficiently; they do not promote understanding of solutions.
In fact, when writing a formula for a solution of a DE, the use
of unit step-functions and window-functions
often obscures understanding of how the solution behaves
(e.g. what its graph looks like).
For example, with the least
amount of simplification I would consider acceptable, the
answer to problem 23 can be written as
$$ y(t)=\left\{\begin{array}{ll} t, & 0\leq t\leq 2, \\
4+ \sin(t-2)-2\cos(t-2), & t\geq 2.\end{array}\right.
\hspace{1in} (*)$$
The book's way of writing the answer obscures the fact that the
"\(t\)" on the first line disappears on the second
line—i.e. that for \(t\geq 2\), the solution is purely
oscillatory (oscillating around the value 4); its magnitude does
not grow forever.
Note. In equation (*), observe that
I overdefined \(y(2),\) giving it a value on the first
line and then again on the second. The only reason this is
okay is that both lines give the same value for \(y(2)\), a
reflection of the fact that \(y(t)\) is continuous.
Since solutions \(y(t)\) of differential equations are always
continuous, we are guaranteed that if our tabular form for
a piecewise-expressed solution \(y(t)\) of a DE (or IVP) is
correct, then at any "break-point" \(t_1\) we will have
\(\lim_{t\to t_1-} y(t) = y(t_1) = \lim_{t\to t_1+} y(t),\) so
we can "overdefine" \(y(t_1)\) as in equation (*) without fear
of contradicting ourselves. This provides a useful
consistency-check on our tabular-form answer: At a "break
point" \(t_1\), if overdefining \(y(t_1)\) leads to two
different values of \(y(t_1)\) on the two lines on which
\(y(t_1)\) is defined, then our answer cannot be
correct (and we should go back and find our
mistake(s)). This consistency-check is very easy to do,
so we should always do it.
In exercise 23, using trig identities the
formula for \(t\geq 2\) can be further simplified to several
different expressions, one of which is \(4+
\sqrt{5}\sin(t-2-t_0)\), where \(t_0=\cos^{-1}(\frac{1}{\sqrt{5}}) =
\sin^{-1}(\frac{2}{\sqrt{5}})\). (Thus, for \(t\geq 2\), \(y(t)\)
oscillates between a minimum value of \(4-\sqrt{5}\) and a maximum
value of \(4+\sqrt{5}\).) This latter type of simplification is important
in physics and electrical engineering (especially for electrical
circuits). However, I would not expect you to do this further
simplification on an exam in MAP 2302.
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