| Date due |
Section # / problem #'s |
| F 1/8/10 |
Read the syllabus and the
web handouts
"Taking notes in a college
math class" and "What is a solution?".
Read Section 1.1 and do problems 1.1/ 1-16.
Do non-book problem #1.
|
| M 1/11/10 |
1.2/ 1, 3-5, 14, 15, 19-22. "Explicit solution" is synonymous
with "solution". I will say more about the terminology "explicit
solution" and "implicit solution" (which we have not used yet in
class) at a later time.
Note: Many of your homework assignments will be a lot longer
than the ones I've given so far. I don't want anyone to feel after
Drop/Add that he/she wasn't warned. Often, most of the book problems
in a section aren't doable until we've finished covering practically
the entire section, at which time I may give you a large batch to do
all at once. Heed the suggestion above the assignment-chart: "If one
day's assignment seems lighter than average, it's a good idea to read
ahead and start doing the next assignment, which may be longer than
average."
|
| W 1/13/10 |
Read the material on "implicit solutions" in Section 1.2,
pp. 8-9.
1.2/ 2, 10, 11, 30
Do non-book problem #2.
Read sections 2.1 and 2.2. (we are skipping sections 1.3 and 1.4).
In your reading, make sure you understand the discussion following
Example 3 (bottom of p. 44 to top of p. 45).
|
| F 1/15/10 |
1.2/ 18,23-29
2.2/ 1-5,8,9,11,17-19. "Solve the
equation (or IVP)" means "Find the general solution of the equation
(or IVP)", i.e. the collection of all solutions to the DE (or
IVP). For IVP's, the "general solution" is usually a single
function (this the case whenever the hypotheses of the Fundamental
Existence/Uniqueness Theorem—Theorem 1 on p. 12—are
met); when no initial conditions are imposed, the general solution is
usually an infinite collection of functions.
|
| W 1/20/10 |
2.2/13,23, 27abc, 28-31, 33, 34
Exercises 3-6 on the updated non-book problems
page. (These have changed since the last time you looked at this
page.)
In doing the exercises from Section 2.2 or the non-book problems
above, you may find that the hardest part is doing the integrals. I'm
intentionally assigning problems that require you to refresh your
basic integration techniques. Remember my warning from the first day
of class (which is also on the class home
page):
You will need a good working knowledge of Calculus 1 and 2. In
particular, you will be expected to know integration techniques ...
.If you are weak in any of these areas, or it's been a while since
you took calculus, you will need to spend extra time reviewing or
relearning that material. Mistakes in prerequisite material will be
graded harshly on exams.
Your integration skills need to be
sufficient for you to get the right answers to problems such as
the ones in this homework assignment, not merely to go through the
motions. 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
``∫ lnx dx=1/x'', or ``(d/dx)(1/x)=
lnx'', even if the rest of your work is correct. (1/x
is the derivative of lnx, not an antiderivative;
lnx is an antiderivative of 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 both to review the old
(if it's not fresh in your mind) and learn the new, even if
this takes a lot of time.
|
| F 1/22/10 |
Read Section 2.3. When you apply the method that's in the box on
p. 51, don't forget the first step (writing the equation in "standard
form"). Be especially careful to identify the function P
correctly; its sign is very important. For example, in 2.3/17,
P(x)=-1/x, not just 1/x.
2.3/ 1-6, 7-9, 13-15, 17-20, 33. In #33, note 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 (that's why I warned you above to be careful
when identifying the function P. 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.
|
| M 1/25/10 |
On this day, remind me to discuss whether to keep the first midterm on
its originally scheduled day (Mon. Feb. 1), or move it to a later day.
2.3/ 22,23 (re-read Example 2 on p. 52 first), 24, 27a, 28,
30-32, 35.
Do non-book problem
#7.
As I mentioned in class several lectures ago, I am not satisfied
with the book's treatment of "implicit solutions" and equations in
differential form. The good news is that I'm writing my own notes on
these topics for you. The bad news is that the notes are already longer
than I expected, and I haven't even gotten to the part where I will
define "implicit solution". However, I'm assigning you now to
read the part that I've written so far, so that when the rest is
written you won't have to read everything all at once. I will want you
to have read the completed set of notes before your first exam (which,
of course, means that I will have to finish writing the notes pretty
soon). The notes are here.
(Continuation of 2.3/ 33b). Consider the differential equation
(*) xy'-2y=3x
(the same DE as in 2.3/ 33b).
- Without finding any formulas for solutions of (*),
show that every solution of (*) defined on a domain that includes 0
satisfies y(0)=0.
-
Let C1 and
C2
be constants, and define a function φ on the whole real line
by
φ(x) = C1x2 -3x
for x≥0,
φ(x) = C2x2 -3x
for x<0.
- Show that for every choice of the constants
C1 and
C2, the function φ above is a solution of (*) on
the whole real line.
- Show that the general solution of (*)
on the interval (0,∞) is given by the formula
y=Cx2-3x, where C is an arbitrary
constant.
- Show that the general solution of (*)
on the interval (-∞,0) is given by the same formula,
y=Cx2 -3x. Why are these solutions not the
same functions as in the preceding part of this problem, even though
the formulas are the same?
- Show that the collection of functions φ defined earlier
in this problem, treating C1 and
C2 as arbitrary constants (i.e. allowing all real
numbers for their values), is the general solution of (*)
on the whole real line.
Note: in this problem, the domain-interval ``the whole real line'' may
be replaced by any open interval containing 0; the facts in parts 1
and 4 are true for any such interval.
|
| W 1/27/10 |
Read Section 2.4 through the "Test for Exactness" on p. 61.
2.4/ 1-8
[This part is not due till Friday since I am not posting it until
late Tuesday night, but if you see it early enough you can get a
headstart.]
Read the updates to the notes posted in the previous
assignment. The updates are as follows:
- A footnote has been added to the last part of Definition 2.10
(p. 10).
- Definition 2.11 (p.11) has been expanded. This, and the remaining
changes, affect the page-breaks of the last few pages of the first
version of these notes. (So the corresponding pages may look
different, but the only new material is what I'm telling you here.)
- Definition 2.12 (p.11) has been altered (the first part now
defines solution curve, not solution)
and expanded.
- On p. 12, the paragraphs immediately before and after Remark 2.13
are new, and a typo in Remark 2.13 has been corrected.
- Everything from ``Now we return to Definition 2.12 ...'' on
p. 14 to
the end of the notes is new.
|
| F 1/29/10 |
Read the updates to the notes. For updates through
Tues. (1/26), see the list given with the homework due 1/27. If I make
any other updates before Friday, I will similarly list them here, so
that you don't have to re-read the whole set of notes.
Here is information I should have given regarding 2.4/ 1-8
in the previous assignment:
Classify an equation in differential form as linear if at
least one of the related derivative-form equations (the ones you get
by formally dividing through by dx and dy) are
linear. It is possible for one of these derivative-form equations to
be linear while the other is nonlinear. This happens in exercises
1,2,5,6 and 7. For
example, #5 is
linear as an equation for x(y), but not as an equation for
y(x).
2.2/6, 16. An equation in differential form is called separable
if, by the operations of addition/subtraction of differentials, and
multiplication by functions (other than the constant function 0), you
can arrive at an equation of the form g(x)dx=h(y)dy (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.
2.2/ 22. Note that although the differential equation
doesn't specify independent and dependent variables, the
initial condition does. Thus your goal in #22 is to produce an
explicit solution "y(x)= ...". But this exercise is an example
of what I call a "schizophrenic" IVP. In practice, if you are
interested in solutions with independent variable x and
dependent variable
y (which is what an initial condition of the form
``y(x0)=y0'' indicates),
then the differential equation you're interested in at the
start is one in derivative form (which in exercise 22 would
be x2 +2y dy/dx=0, or an
algebraically equivalent version), not one in differential
form. Putting the DE into differential form may be 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 you should
not express a preference for one variable over the other by asking for
a solution that satisfies a
condition of the form
``y(x0)=y0''
or ``x(y0)=x0''. Instead,
you should ask for a solution whose graph passes through the point
(x0, y0), which in exercise 22
would be the point (0,2).
Read the portion of Section 2.4 from ``Method for Solving Exact
Equations'' (p. 63) through the end of the section.
2.4/11, 12, 14, 16, 17, 19, 20-22, 27a, 28a, 32 (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 "schizophrenic IVPs". In all
of these, the goal would be to find an explicit formula for a
solution—if algebraically possible—with the choice
of independent/dependent variables indicated by the initial condition.
However, if the equation ``F(variable1,
variable2)=0'' that you get via the exact-equation
method (in these exercises) is impossible to solve for the dependent
variable in terms of the independent variable, you have to settle for
the implicit solution.
|
| M 2/1/10 |
Read everything in Section 2.4 that you have not yet read.
Read the online handout "A terrible method for solving
exact equations". The parenthetical "we proved it!" on the
handout does not yet apply, since I haven't yet done the proof in
class. (In your reading of Section 2.4, you've seen at least part of
proof, but the book leaves the key part of the argument as an exercise
that relies on a theorem that you are unlikely ever to have seen.)
Do non-book problems
8 and 9. These two problems have been added to the
non-book-problems page since the last time you looked at it. Note:
problem 9 has parts (a) through (j).
Read the updates to my notes. Here are the changes
since the last update:
- The definition of the term ``algebraically equivalent''
(which is in the same location on p. 8 as before) has been put inside
a boldfaced, numbered definition so that it could be more easily found
and referred to later. This affects the numbering of later items
(old Definition 2.8 = new Definition 2.9) and the page-breaks beyond
this point.
- The paragraph after Remark 2.14 (old Remark 2.13) has been
lengthened.
- A footnote has been added on (new) p. 15. This affects the
numbering of later footnotes.
- On p. 16, an existence/uniqueness theorem for DEs in differential
form has been added, along with a paragraph before and after it.
- The title of Subsection 2.2.2 has had words added for
clarification (p. 17).
- Everything from the bottom of p. 21 (starting with the
paragraph above Definition 2.21) up to the beginning of Section 2.3 on
p. 23 is new.
- The last paragraph of the notes is new.
|
| W 2/3/10 |
So that you may celebrate Groundhog's Day with friends and family,
there are no new exercises. But check back later on Tuesday to see if
there are updates to my notes for you to read.
For students who want a supply of exercises to practice with:
If you
have done
all your homework (and I don't mean "almost all"), you should
be able to do all the review problems on pp. 81-82 except #s 9,
11, 12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35, 37, and the last part
of 41.
Read The Math
Commandments.
|
|
General info |
One student asked how students in my MAP 2302 classes have performed
on the first exam in the past. I don't remember most statistics of
that sort of the top of my head, but for anyone who's interested,
here's how to find it:
Go to my home page, http://www.math.ufl.edu/~groisser
Under ``Course materials'', click on the ``Past classes'' link.
On the Past Classes page, click on the link to any of my
previous MAP 2302 classes. This will take you to the home page for
that class, which has links to grade-scale pages. Each grade-scale
page has some statistics on each exam, and a link to the list of
scores.
|
| F 2/5/10 |
First midterm exam (assignment is to study for it)
|
| M 2/8/10 |
Read section 4.1. (We're skipping Sections 2.5 and 2.6, and all of
Chapter 3.)
|
| W 2/10/10 |
4.1/ 1-10. Typo correction: In #10a, 2nd line, and in #10d, 2nd line,
the function
written after "B" should be "sin", not "cos".
|
| General info |
The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list
of scores so that you may see the grade distribution. Exams will be
returned Friday 2/12/10.
Here are some comments on the
exam. |
| F 2/12/10 |
Read Section 4.2 through Theorem 2 on p. 172.
4.2/ 1,3,4,6-9,11,12,13-17,21-25.
Add as part(c) to #21: Now solve the same DE by the
"integrating factor" method we learned in Chapter 2. Do you
expect to get the same general solution that you got using the
method in parts (a) (b)? Do you get the same general solution
both ways?
|
| M 2/15/10 |
Read the remainder of Section 4.2.
4.2/ 2,5,10,18-20, 27-32. Note: The answer to #27 in the
back of the book is wrong.
Read my comments on the
exam (this is the same as the link in the "General info" entry
above the assignment due 2/12/10).
Re-do all the exam problems on which you did not get a perfect
score. For all but #7, you should be able to check by yourself
whether your new answers are correct (some of the answers that would
be harder for you to check by yourself are in my posted comments on
the exam), but you are welcome to check your answers against my
solutions during office hours. For #7, the final answer is 32/9
gm/L. As a matter of policy, I do not post solutions on the internet.
Solve the initial-value problem for the DE in exam problem #4,
but with the following initial conditions: (c) y(0)=3π/4,
and (d) y(0)=3π/2.
|
| W 2/17/10 |
4.2/ 26,33,35,36,37,43,46ab. Note: The answer to #35a in the
back of the book is wrong.
Do non-book problems
10 and 11. The operator L1+L2
is defined by (L1+L2)[f]
=L1[f]+L2[f].
|
| F 2/19/10 |
4.3/ 1,2,4,6,7,9-12,17,18,21-23,28
|
| M 2/22/10 |
4.3/ 3,5,8,13-16,24-26,32,33 (students in electrical engineering may
do #34 instead of #33),36
|
| W 2/24/10 |
Read Section 4.7 through Theorem 6.
4.7/ 1-4,5-8,10-13,15,17,19,20,23,27-29.
In 1-8, in addition to doing what the book
says, use Theorem 5 to write down the largest interval on which you
are guaranteed, without having to solve the IVP, that a solution
exists. An important feature of linear equations is that you can
predict the domain of the solution (more precisely, predict that the
domain of the solution is
at least a certain interval) from the coefficient functions
and the initial value of the independent variable alone, without
having to solve the equation or use the initial values of the
dependent variable and its derivative(s). For nonlinear equations,
often the domain depends on the initial value of the depenedent
variable and its derivatives, and the domain is usually hard to
determine without solving the IVP.
Sometimes a linear equation, especially one
that is not in standard form (the form in Theorem 5, in which the
top-derivative term has coefficient 1) has one or more solutions with
a domain that is larger than what Theorem 5 guarantees. The solution
for Exercise 19 is an example.
|
| F 2/26/10 |
4.4/ 9 (note that -9 = -9e0 t),
10,11,14,17
Add parts (b) and (c) to 4.4/ 9-11,14,17 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 17: y(0)=1, y'(0)=2.
4.5/ 3-8,25,26
4.7/ 24ab
|
| M 3/1/10 |
4.4/ 13,18
4.5/ 1,2,20,27,28,29,41
non-book problem 14
|
| W 3/3/10 |
4.5/ 42. Part (b) of this problem, 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.
4.5/ 45. This is a nice problem that requires you to
combine several things you've learned.
The strategy is similar to the approach in
Exercise 41. Because of the "piecewise-defined" nature of the
right-hand side of the DE, there is a sub-problem on each of three
intervals:
Ileft = (-∞ -L /(2V)],
Imid = [-L /(2V), L /(2V)], and
Iright = [-L /(2V), ∞). The solution
y(t) defined on the whole real line restricts to solutions
yleft, ymid, and
yright on these intervals.
You are given that
yleft(t) is identically zero. Use the
terminal values yleft(-L /(2V)),
y'left(-L /(2V)), as the initial values
ymid(-L /(2V)),
y'mid(-L /(2V)). You then have an IVP to solve on
Imid. 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 c1,
c2. Using the IC's at t=-L /(2V), you obtain
specific values for c1 and
c2, and plugging these back into the general
solution gives you the solution ymid of the relevant IVP on
Imid.
Now compute the terminal values
ymid(L /(2V)),
y'mid(L /(2V)), and use them as the initial
values yright(L /(2V)),
y'right(L /(2V)). You then have a new IVP to
solve on Iright. The function
yright is what you're looking for.
If you do everything correctly (which may
involve some trig identities, depending on how you do certain steps),
under the simplifying assumptions m = k = 1 and L =
π, you will end up with just what the book says:
yright(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 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 Imid, you have to handle the
cases V≠ 1 and V = 1 separately. (If we were not
making the simplifying assumptions m = k = 1 and L =
π, these two cases would be πV/L ≠ (k/m)1/2
and πV/L = (k/m)1/2.) In the notation introduced
last week, using s for the multiplicity of a certain number as
a root of the characteristic polynomial, V≠ 1 puts you in the
s = 0 case, while V = 1 puts you in the
s = 1 case.
The book's answer to part (a) is correct only for
V ≠ 1. The value of A for
V=1 is π/2. (I'm just telling you this so you can check your
answer; you're still supposed to figure out this answer yourself.)
|
| F 3/5/10 |
Second midterm exam (assignment is to study for it)
|
| M 3/15/10 |
Enjoy your spring break!
|
| General info |
The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list
of scores so that you may see the grade distribution. |
| W 3/17/10 |
4.4/ 1-8, 12,15,16, 19-26
4.5/ 9-16
|
| F 3/19/10 |
4.4/ 27-32
4.5/ 17-19, 21,22,23,24,30, 31-36
|
| M 3/22/10 |
4.6/1, 3-18, 19 (first sentence only)
4.7/24cd, 37-44. (In all of these, assume the domain interval
is {t > 0}.) Note: to apply Variation of Parameters as
presented in class, you must first put the DE in "standard 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).
Note that it is possible to solve all the DEs
in 24cd and 37-43 either by the Cauchy-Euler substitution applied to
the inhomogeneous DE, or by using Cauchy-Euler just to find a FSS for
the associated homogeneous equation, and then using Variation of
Parameters for the inhomogeneous DE. Both methods work. I've
deliberately assigned exercises that have you solving some of these
equations by one method and some by the other, so that you get used to
both approaches.
Redo 4.7/44 by starting with the substitution
y(t)=t–1/2u(t) and seeing where
that takes you.
|
| W 3/24/10 |
Read Sections 6.1 and 6.2.
|
| F 3/26/10 |
6.2/ 1,11,13,14,15-18
Read Section 6.3
|
| M 3/29/10 |
Read sections 7.1 and 7.2.
7.2/1-8, 10, 12
(note: "Use Definition 1" means "Use Definition 1", NOT the
box on p. 358 or any other
table of Laplace Transforms)
|
| W 3/31/10 |
7.2/ 13-20 (use table on p. 384; we'll derive these in class Wed.),
21-28, 29a-d,f,g,j, 30
Read Sections 7.3 and 7.4. When we get to 7.4 in class, I will
not have time to review the method of partial fractions; I will assume
you know the method when I do examples.
|
| F 4/2/10 |
7.3/ 1-10,12,14,19,25,31. In class, we have derived only the first three
lines of the Table 7.2 so far, but I want you to start getting
practice using all the properties listed in the table.
|
| M 4/5/10 |
7.4/ 1-10,11,13,15,16,20,21-24,26,27,31
7.5/ 15,21,22
|
CORRECTION! |
The last fraction I wrote on the board Monday (after class) as part of
the answer to problem 6 had the numerator "s". The numerator
should have been "s-3".
|
| W 4/7/10 |
Third midterm exam (assignment is to study for it).
You will
be given a Laplace Transform table to use on the exam; you do not have
to memorize any transforms or inverse transforms. (But you will be
expected to know how to make use of the information in the table.)
|
| General info |
The grade scale for the third midterm is now posted on your grade-scale page, with a link to the list
of scores so that you may see the grade distribution. |
| F 4/9/10 |
No new homework.
|
| M 4/12/10 |
7.5/1-8,10,29. To learn some shortcuts, you may want
first to read the web handout "Partial fractions and Laplace
Transform problems"
(pdf file).
Read Section 7.6 through Example 5.
|
| W 4/14/10 |
7.6/ 1,2. If you feel that you are ready to tackle harder problems
based on the reading that was due Monday but that I have not yet
covered in class (I will on Wednesday), start on the homework due
Friday, since that will be a long assignment.
|
| F 4/16/10 |
7.6/3-10,11-18,29-32,33-40. For all the problems in
which you solve an IVP, write the final answer in "tabular form". (For
those of you who missed class, by "tabular form" I mean an expression
like the one given in Example 1, equation (3), for the function
f. I.e. do not leave your final answer in the form
that's at the end of Example 1, after you've inverse-transformed
Y(s). The unit step-functions should be viewed as convenient
gadgets to use in intermediate steps to work through a problem
efficiently.)
Read Section 8.1.
|
| Note about next week
|
Because of the amount of material left to cover (through section 8.4,
with some omissions), I estimate that I'm going to need to cover new
material on Wed. Apr. 21, the last day of class. I'll then hold an
attendance-optional review session (Q&A format, as always), on Friday
Apr. 23 at our usual class time, 11:45-12:35. This does have the
side-benefit of putting the review two days closer to the final exam
than if we had the review on the last day of class.
|
| M 4/19/10 |
Read sections 8.2, 8.3, and 8.4.
8.2/1-6,7,8,9,10,11-14,17-20,23,24,27,28,37.
Note: "convergence set" in the book is what I
initially called "domain of convergence" and, later, "interval of
convergence". Any time the book's problems tell you to find the
convergence set, find only the open interval of convergence; I
don't want you to spend time trying to decide whether the series
converges at the endpoints. For this class, 100% of the way we'll
apply power-series ideas to solving DEs involves only the open
interval of convergence.
|
| W 4/21/10 |
Based on your reading and what we've done in class so far, do as many
of the problems below as possible by Wednesday. I will do several
examples in class on Wednesday, but if you wait to get started on
these, you will have too much to do before Friday's review.
8.3/1,3,5-10,11-14,18, 20-22,24,25.
8.4/15, 20 ("Equation (16)" is the first un-numbered equation
after Equation (15)),21,23,25
|