May 5 2008 

April 25 2008 

April 23 2008 
 Reminder: The final exam is this Tuesday April 29
6pm8pm (SPECIAL TIME) in LCB 222.
 Here is the practice final solution: fexam_practice_sol.pdf. Some of the
problems that I gave were particularly long, the final will not be harder
than this. Errata: (corrected in online version) In question 5d I
accidentally flipped the sign of the n>1 Fourier coefficients of
F(t). This means the a_{n} should have a minus sign
and x_{p}(t) should be corrected accordingly.

April 21 2008 

Here is the solutions to Exam2: exam2_sol.pdf. Although I did
distribute photocopies of this during class, I just
realized I did not post them on the website. Sorry for the
delay!

April 20 2008 
 Here is the practice final exam: fexam_practice.pdf. The real final will be
no longer/harder than this one, but other topics may be included. The
review sheet for the final is here: revsheetf.pdf. Solutions to the practice final
will be given in class on Tuesday.

April 18 2008 
 For practice final exams, please visit the Spring 2006
Math 2280 website, which includes corrections.
I should be
able to post this weekend the practice final that we will do together
on Tuesday.
 Here is an animation in Maple of the onedimensional wave equation
(vibrating strings) that I showed in class: weq.mw
(PDF: weq.pdf).
 If you want to have endless fun with strings, the vibrating string
Java applet that we saw in class is available from the website: http://www.falstad.com/loadedstring/.
To visit the other physics applets (including one illustrating Fourier series) just click
on the "More acoustics applets" link.
 HW9 solutions are here: hw9sol.pdf.

April 15 2008 
 Homework 10 due Tue April 22 but accepted until Fri April 25.
It is a relatively short HW and the solutions will be released
on April 25 evening.
9.1: 13, 19
9.2: 12, 17
9.3: 1, 2, 12
9.4: 1
9.5: 3
 Final exam Tue April 29 6pm8pm in LCB 222
We will be doing a review for the final on our last day of
class (Tue April 22). This will consist on going over a
practice final that I will post this Friday. The final is all
inclusive and the material covered is up to section 9.5.
 Maple files from today's class

April 9 2008 
 Project #2 is optional and will count for extra credit. You
have until Friday April 25 to turn it in. It consists of the following:
 Read Application 9.3 (pp608609), it explains how to find the
Fourier series of rather complicated functions using Maple. The file app_9_3.mw (PDF: app_9_3.pdf) shows how to do this for the given
example. Please find the Fourier series for the functions specified in
13 (p9.4) using the above Maple code as a model. Try to figure out the
pattern in the Fourier series by looking the first few terms in the
series.
 Read Application 9.5 pp628630, it explains how to use the results
from Section 9.5 (that we will see shortly) to figure out the
temperature at any point x of a heated rod for some time t. Although the
textbook uses Matlab, there is a Maple implementation here: heatedrod.mw (PDF: heatedrod.pdf) that you can use as a model.
Please complete 12 at the end of this Application section (p630).
 Here is the Maple code we saw in class, where we approximated several
periodic functions with Fourier series. It may also help for the project:
gibbs.mw (PDF: gibbs.pdf).
 Here is also some Maple code to compute the Laplace transform and to
transform a rational expression into partial fractions: ex_7_3_6.mw (PDF: ex_7_3_6.pdf).

April 4 2008 
 Homework assignment #9 covering chapter 7 (Laplace
transform) is due Thursday April 10. Exercises in parenthesis are
optional and do not count for extra credit. Total 18 problems (short
ones).
7.1: (15), (16), 18, (26), 28, (31)
7.2: 2, 4, 6, 13, 17, (20)
7.3: 7, (8), (13), 14, 27
7.4: 3, 7, 10, 15, 21
7.5: 2, 13, 33

April 2 2008 
 One of you found a sign mistake in problem 3 of the pratice exam 2
solutions. The updated solutions are here (the only change is in page 6).
exam2_practice_sol.pdf

April 1st 2008 

March 28 2008 
 Solutions to HW8 are now posted: hw8sol.pdf
 The practice midterm #2 and review sheet are here: exam2_practice.pdf and revsheet2.pdf.
If you time yourself doing this exam, keep in mind that it is longer than
the actual one.

March 25 2008 
 Midterm #2 Timeline
 Thu March 27:
 HW8 due. In order for all to benefit from the
solutions to HW8 as early as possible, they will be
posted after class on March 27. Thus there will be
no extensions for HW8.
 A practice midterm will be sent out together with a
review sheet.
 Tue April 1: HW8 returned graded. Correction to
midterm #2 in class (about 1h).
 Thu April 3: Midterm #2 (about 1h). Material
covered: Chapters 46, EXCEPT §4.2 and §6.5 (which
we skipped)
 HW7 Solutions are posted here: hw7sol.pdf
 Problem 4.3.10 is solved here: ex_4_3_10.mw (PDF: ex_4_3_10.pdf)
 PPLANE: If you would like to use pplane in
Matlab as I did in class today, just download the
appropriate file for your particular Matlab
version.
Printing in matlab can be achieved through the FILE menu OR
to print to a eps file named "filename.eps"
print('deps','filename.eps');
% help print for more options
To view the file, go to a terminal and
gv filename.eps
To print to the default printer, go to a terminal
and
lpr filename.eps
If you have problems printing, you can always print the
figures to a file (as described above) and email them to me
(with filenames containing the problem number, please).
 Office Hours changes:
 Fri March 28: cancelled
 Extra office hours 2pm4pm on April 1 and 3.

March 14 2008 
 Homework 8 is due Thursday 03/27 and consists of
6.1: 6, 7, 11, 14, 20, 24
6.2: (5), 6, (7), 8, 10, (12), (16), 25, 28, 31
6.3: 8, (9), 10, 14, (15), 16, 17
6.4: (12), 13, (14), 15, 16
note: problems in parenthesis are optional and do not count for
extra credit.
 If you need to see me during the spring break, please send me an
email to set up an appointment.
The regular office hours during the
spring break are cancelled.
 Midterm exam #2 will be on Thursday April 3rd and will
cover Chapters 46.
 HW5 solutions hw5sol.pdf and HW6
solutions hw6sol.pdf are now
available.

March 11 2008 
 The office hour for Wed March 12 CANCELLED. If you have questions
about the homework, please send me an email and I will respond as fast as
possible.
 Here are some Maple files that we saw in class today:
 ex_5_6_2.mw, shows how to solve the three
tank cascade problem with salty water coming in.
(PDF: ex_5_6_2.pdf)
 ex_5_6_4.mw is an implementation in Maple
of the variation of parameters method for systems. note: could be
useful for the homework!
(PDF: ex_5_6_4.pdf)
 ex_6_1_1.mw, shows the phase portrait of a
competing species population (squirrel/rabbits) and the phase portrait of
its linearization around one of the equillibria.
(PDF: ex_6_1_1.pdf)

March 7 2008 
 HW7 (due Thursday March 13 2008) is relatively short. For the
problems 5.5.36 (and 5.4: 10,12) you are strongly encouraged to use Maple (see
below for useful worksheets). For the other systems, please do them
by hand (verifying using Maple if possible!). Of course, if you
prefer you can use some other software to do this.
5.4: 3, (10), (12)
5.5: 2, 6, 21, 27, 31, 36
 Here are some Maple worksheets that will be useful for the homework.
 ex_5_4_6.mw, an example from the book
§5.4, which shows how to deal with multiple eigenvalue solutions when
the matrix x' = Ax is defective. (PDF: ex_5_4_6.pdf)
 prob_5_4_31.mw, a problem from the
book which shows a rather more complicated case. (PDF: prob_5_4_31.pdf).
 matexp.mw, shows how to compute matrix exponentials by using fundamental matrix solutions and using the builtin commmand MatrixExponential. (PDF: matexp.pdf)

March 1st 2008 
Here are some clarifications on the programming assignment 4.3.10
 I mistyped the true solution (a typo over a typo, sorry). This
is what the true x(t) is:
x(t) = (1/9)*(2*exp(2*t)  2*exp(t)  6*t*exp(t))
Just to double check, numerical values (x(1),y(1)) are:
[1.31500851888888870,1.02538370055555550]
 I've seen several people doing the following to define the
slope function
q:=Vector([0.,exp(t)]);
f:= (t,x) > A.x + q;
The problem here is that the value of t in q is taken
to be the last assigned value for t (think of t as a global
variable) and NOT as the value you give when you write f(t,x).
So when you run RungeKutta it needs evaluations of f at
t, t+h/2 and t+h, and with the above syntax
your f will be evaluated only at t (which breaks down
the numerical method)
The alternative I suggest is to write:
q:= t > Vector([0.,exp(t)]);
f:= (t,x) > A.x + q(t);
In these two expressions t is an anonymous variable that
will not be affected when you assign t.

February 28 2008 
 This is something you need for exercise 4.1.13: fieldplot.mw. This maple worksheet shows how to
plot the direction field for a system of DEs. I thought I had posted this
before. Since I didn't, you can turn in this exercise next Thursday March
6.
 The office hours for this Friday February 29 are cancelled
 HW6 is due on Thursday March 6
5.2: 2, 8, 12, 27
5.3: 2, 8, 15
 For HW5, problem 4.3.10 there is a typo in the book in the true
solution. The y(t) is right as it is. The right x(t)
should be,
x(t) = (1/9)*(2*exp(2*t) + 2*exp(t) 
6*t*exp(t)). This problem is now due on March 6 as well.

February 21 2008 
 Your midterm exam #1 is graded now. The solution is here: exam1_sol.pdf
 The Homework #5 is due Thursday February 28, except as marked.
3.5: 49 (variation of parameter)
3.6: 2, 11, 16
4.1: 2, 3, 13, 16, 26
4.3: 10 (see note below)
5.1: 13, 18, 22, 27, 31, 36
Note on problem 4.3.10:
 This problem involves Maple programming. If you feel you will do this
better in Matlab or Mathematica or Octave or Sage, feel free to use these
other languages. The reference code I give is in Maple though.
 For this problem only, you may work in groups of up three people, in
which case only one writeup is needed
 The due date for this problem only can be extended in case you are
having problems.
 The code we saw in class shows different numerical methods for solving
first order systems of DEs, is is here: sdesol.mw
(PDF: sdesol.pdf).

February 15 2008 
 The practice exam 1 correction is available here:
exam1_practice_sol.pdf.
In the printed version of these solutions that I distributed in class there
were sign mistakes on the real parts of roots of characteristic
polynomials. This is now corrected.
 The HW4 solutions are here: hw4sol.pdf.
Your homeworks are graded.
You can come by on Friday or Monday to pick them up.
 Office hours: There will be office hours on Mon Feb 18, but not
on Fri Feb 22 or Mon Feb 25.

February 12 2008 
 The review sheet for exam 1 is here: revsheet1.pdf. A practice midterm exam #1
is now available: exam1_practice.pdf. I
strongly encourage you to work through it before next class. We will
solve it together next Thursday and solutions will be posted here. This
practice exam is somewhat longer than the real exam, so if you can do it in
an hour, you are in good shape.
 Solutions to project 1 are here:
proj1p1.mw,
proj1p2.mw,
proj1p3.mw,
proj1p4.mw,
proj1p5.mw
proj1p6.mw
(PDF: all included proj1.pdf)
 The maple file demonstrating the book's examples on resonances is here:
resonances.mw.
 Solutions to HW4 will be posted on Thursday after class so that you
can review this material before the exam. If you did not turn in your HW4
today in class: I will not accept any homeworks after Thursday evening and
even if you do turn it in before, I cannot guarantee that you will get it
back graded before the exam.

February 7 2008 
 The notes from last class (Feb 5) are posted here notes080205.pdf. Sorry for my writing but I
wasn't expecting to put them online.
 HW3 solution is here: hw3sol.pdf

February 5 2008 
 HW4 due Tue Feb 12 2008 consists of the
following exercises. As always the exercises in parenthesis
are optional (i.e. they will not be graded but solutions
will be posted).
3.1: 4, (5), 19, 29, (31), 38, 43
3.2: 8, 13, 21
3.3: 9, 10, 33, 39
3.4: 5, 6, (13), 14, 16
3.5: (3), 5, 7, 32, 35
Grading: 5 points per exercise (total 100pts)
Note: HW4 should be available graded no later than Mon Feb
18, the solutions will be posted earlier than that, so that
you can study for the exam.
 First midterm: MOVED to Tue Feb 19 (instead of
Feb 14) in class. The material to cover is: 1.11.5, 2.12.6,
3.13.5. There will be a review session on Feb 14 where we
will go over a mock exam.
 There is a Maple worksheet on the website that gives
examples of overdamped, underdamped and critically damped
free motions: damp.mw (PDF:
damp.pdf)
 Solutions for HW2 are posted here: hw2sol.pdf

January 24 2008 
 The first computing project can be downloaded here project.pdf. It is due on Tuesday
02/05.
 If you have questions about the project please do not hesitate
to write.
Do not wait to the last minute to start working on the
project!
 The version of Maple I recommended earlier was not the most recent one
available in the computer lab.
It is better to use this one:
xmapleV11.02
(this is a command to be written in a terminal. To open a terminal simply
select the corresponding entry in the popup menu that appears when you
right click on your desktop)
 The Maple worksheets that are referenced in the project are: uspop.mws, denumsol.mw,
denumsol2.mw and unstable.mw. Each problem can be done by modifying
one of these worksheets. To avoid unecessary complications, my advice is to
have one worksheet per problem.
 Problems 1, 2 and 4 are straightforward adaptations of the Maple
worksheets uspop.mws, denumsol.mw, and denumsol2.mw.
I am saying "straightforward"
because I heard some of you had already done some of these problems during
the lab.
 Problems 3, 5 and 6 require minor modifications of denumsol.mw.
 For those programming inclined, the worksheet unstable.mw shows how to use procedures in
Maple.
You are not required to use procedures.
 The office hour of Monday 01/28 is cancelled. Please feel free to come
to my office on Tuesday 01/29 afternoon.

January 22 2008 
 Here is Homework #3, due on Thursday 01/31. The problems in
parenthesis are optional.
2.3: 2,(3),4,11,13,(14),15,(16),17,(18)
This is a smaller homework assignment because the first computer
project will be assigned next Thursday 01/24 and due Tuesday 02/05.
 The solutions for HW1 are here: hw1sol.pdf. Please let me know if you have any
concerns about the grading.
 Section 2.3: The crossbow bolt Maple worksheet we saw in class
is here: crossbow.mw(PDF:
crossbow.pdf)
 Sections 2.42.6: The numerical methods handout is here: nummethods.pdf and the accompanying Maple
worksheet here: denumsol.mw (PDF:
denumsol.pdf).
The discussion on numerical methods will continue in class on Thursday
01/25. During the second hour we will be in the computer lab to experiment
with these methods. This will help you with your computer project. Please
come to class as usual, we will all migrate to the lab during the midclass
break (or earlier).
 All the Maple worksheets in this site have PDF versions for easy
viewing even if you do not have Maple around.

January 18 2008 
 The doomsday/extinction example for which I had a sign error in class is avaiable here: doomsday.pdf.

January 17 2008 
 Maple version 11 is available in the Sun environment (the Math department
computer lab).
To use this particular version, simply open a terminal and type
the command:
xmapleV11
In the Union Computer Lab, it is available on both
Macs (Applications > Science & Engineering) and
PCs (Start > Programs > Class Software).
The particular way to access Maple v11 may have changed since the instructions I found were written, so you may have to do some searching in the applications menu.
Files posted here will be compatible with this version of Maple.
 Maple tip: Try the following command to plot several solutions to a DE on its slope field.
Here the DE is y'=x+y1 and we plot solutions for which
y(0) = 1, y(0) = 0 or y(0) = 1.
with(DEtools);
DEplot(diff(y(x), x) = x+y(x)1, y(x), x = 3 .. 3,
{y(0) = 1, y(0) = 0, y(0) = 1}, y = 3 .. 3,
linecolor = black);
 The broken airconditioner example (from §1.5, pp5860) is
available here: ac.mws (PDF:
ac.pdf)
 The US population example (from §2.1, pp9092) is
available here: uspop.mws (PDF:
uspop.pdf)
 Homework #2 is due on Thursday January 24 in class.
Optional exercises are in parenthesis.
1.5: 5, 13, 33, 38, 41
2.1: (3), 4, 13, 15, 22
2.2: 6, 10, 13, 16
Grading: all exercises are worth 5 points,
except: 1.5.38 (15pts), 1.5.41 (10pts), 2.1.13 (10pts) and 2.1.22 (10pts).
Total 90 points.

January 10 2008 
 Here is HW assignment #1, due on Thursday January 17 in class.
Exercises in parenthesis are optional (solutions will be provided as
well)
1.1: 6, 15, (16), 19, 27, (32), 35
1.2: 6, (8), 14, 21, 32
1.3: 12, (13), 14, 23, 26, (27)
1.4: 12, 22, 38 (see page 37 for the DE satisfied by continuously
compounded interest), 44, 65 (CSI problem)
Note: for problems 1.3.23 and 1.3.26 you may use dfield as we did in
class. Unfortunately printing from a Java applet can be challenging. So I suggest:
 dfield has a Matlab version with a GUI that is identical to the Java
version and where printing should be straightforward. It is available for download.
 With Maple slope fields can be plotted as in this example sfield.mws (PDF:
sfield.pdf). Update: Now compatible with
Maple versions 8 and up.
 With Mathematica, it is explained in p3031 of your book.
Grading: Each exercise is worth 5 points (total 90 points).
Computer accounts: if you are registered in a math class, an account has
been generated for you. Contact me if you do not know how to access it.
 An email has been sent to the class mailing list. Please contact me if
you did not receive it.
 Homework assignments will be given on this webpage and by email
sometime between the end of the class and the next day. Other
announcements will go here also.

January 8 2008 
Covered 1.1 and 1.2. To complement this material, please read the examples:
1.1.7, 1.1.9, 1.1.10, 1.2.2 and 1.2.3.
