Math 2280-2 Introduction to Differential Equations
Spring 2008

Last modified: $Date: 2008-05-05 00:43:06 -0600 (Mon, 05 May 2008) $

Class information

Time: 4:35pm-6:30pm
Days: Tuesdays and Thursdays
Where: LCB 222
Textbook: Differential Equations and Boundary Value Problems, Computing and Modelling, Edwards and Penney (fourth edition, Prentice Hall)
Prerequisites: Math 2270 (linear algebra) and either of Math 1260 or Math 2210 (multivariate calculus).

Contact information

Instructor:Fernando Guevara Vasquez
Office: LCB 212
Office hours:MWF 10am-11am or by appointment
(please see announcements below for any changes)
Phone: 801-581-7467
Email: fguevara(AT)math(DOT)utah(DOT)edu


The syllabus was distributed in class on 01/08 and is available for download here syllabus.pdf.



The material covered by an exam and the conditions of the exam will be announced no later than a week before the exam. I will only accept well-documented reasons (such as medical) for missing an exam, and you must contact me within 24 hours of the exam to make arrangements for a retake.

Office hours changes

Normal office hours are MWF 10am-11am.
The following office hours are cancelled

Homework assignments and other announcements

May 5 2008
April 25 2008
April 23 2008
  • Reminder: The final exam is this Tuesday April 29 6pm-8pm (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 an should have a minus sign and xp(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 one-dimensional wave equation (vibrating strings) that I showed in class: (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: 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 6pm-8pm 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 (pp608-609), it explains how to find the Fourier series of rather complicated functions using Maple. The file (PDF: app_9_3.pdf) shows how to do this for the given example. Please find the Fourier series for the functions specified in 1-3 (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 pp628-630, 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: (PDF: heatedrod.pdf) that you can use as a model. Please complete 1-2 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: (PDF: gibbs.pdf).
  • Here is also some Maple code to compute the Laplace transform and to transform a rational expression into partial fractions: (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 4-6, EXCEPT §4.2 and §6.5 (which we skipped)
  • HW7 Solutions are posted here: hw7sol.pdf
  • Problem 4.3.10 is solved here: (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"
    % 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 2pm-4pm 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 4-6.
  • 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:
    •, shows how to solve the three tank cascade problem with salty water coming in.
      (PDF: ex_5_6_2.pdf)
    • 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)
    •, 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.
    •, 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)
    •, a problem from the book which shows a rather more complicated case. (PDF: prob_5_4_31.pdf).
    •, 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:


  • 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 Runge-Kutta 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: 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: (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:,,,, (PDF: all included proj1.pdf)
  • The maple file demonstrating the book's examples on resonances is here:
  • 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: 1.1-1.5, 2.1-2.6, 3.1-3.5. 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: (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,, and 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,, and
    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
  • For those programming inclined, the worksheet 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.pdf)
  • Sections 2.4-2.6: The numerical methods handout is here: nummethods.pdf and the accompanying Maple worksheet here: (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 mid-class 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:


    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+y-1 and we plot solutions for which y(0) = -1, y(0) = 0 or y(0) = 1.

           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 air-conditioner example (from §1.5, pp58-60) is available here: ac.mws (PDF: ac.pdf)
  • The US population example (from §2.1, pp90-92) 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 p30-31 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.