Pendulum without damping: vector field and some trajectories in the phase plane. Note the three critical points at (0,0), (,0), (2,0).

MATH 246, Fall 2006: Differential Equations for Scientists and Engineers

News

• The Final Exam is on Thu, Dec. 14, 1:30-3:30pm in the following rooms:

  section	TA	room
  02*1	Miguel Pauletti	ARM 0131
  02*2	Prashant Athavale	CSI 1115

• Assignment 4, due Thu, Dec. 9
  Hints:
  1(a): Pick sufficiently many initial points in the rectangle [0,3]×[0,2] (modify the statements for a1=... , for a2=... in the code appropriately). For each initial point, plot the trajectories both for increasing time (e.g, [0,20]) and decreasing time (e.g., [0,-20]).

  Ignore the warning messages from ode45. You can use the command

    warning('off','MATLAB:ode45:IntegrationTolNotMet')
  

  at the beginning of your m-file to suppress these warnings. Then your code should run without any warnings or errors.

  1(b): You should first do (c) to find all critical points and their types. There are two stable critical points which are possible limits for t going to infinity. The rectangle [0,3]×[0,2] is divided into two regions, consisting of initial points which converge to the first critical point, and initial points which converge to the second critical point. Sketch these two regions by hand. The boundary between these two regions is a trajectory which goes into a saddle point.

• Using Matlab for Autonomous Systems
• Table: Critical points of autonomous systems
• What I covered in class so far (12/06): 1.1-1.3, 2.1, 2.2, 2.4, 2.6, M5.3, M6.3, 2.5, 2.3, 2.7, 8.1-8.3, 8.6, 3.1, 3.2, 3.5, 3.4, 3.6, 3.3, 3.7, 3.8, 3.9, 6.1, 6.2, 6.3, 6.4, 6.5, 7.1, 7.5, 7.6, 7.8, 9.1, 9.2, 9.3, 9.4, 9.5 (numbers are sections in Boyce & DiPrima, numbers with M are sections in "Differential Equations with Matlab")
• Coming up next: Review
• What you should do now:
  • Do the recommended problems for sections 1.1, 2.1-2.6, 2.7, 3.1, 3.5, 3.4, 3.6, 3.7, 3.8, 3.9, 6.1, 6.2, 6.3, 6.4, 6.5, 7.5, 7.6, 7.8, 9.1, 9.3
  • Read the following instructions and try them out on a computer:
    Using Matlab for Autonomous Systems
  • Using Matlab for Higher Order ODEs and Systems of ODEs: Solving ODEs numerically and symbolically, plotting the solutions, phase plane plots, vector fields.
• Math 246 Tutoring is available Mon 9-10, Tue 10-12, Tue 1-2 in MTH 0301, see the Tutoring Schedule.
• Math 246 Matlab Tutoring is available Mon 6-8, Tue 6-8, Thu 6-8 in the PG2 WAM Lab, see the Matlab Tutoring Schedule.

Sections and TAs

Syllabus

Information about time & place, instructor, textbooks, syllabus, grading policy, Matlab assignments, dates of exams, final exam. This was handed out in the first class.

• Detailed List of Topics

Recommended Problems from the Textbook

How to Hand in Matlab Homework

Read the section 4.4 ``Presenting Your Results'' in "Differential Equations with Matlab". You will only receive credit for your homework if you follow the following rules:

• Printouts of interactive input at the Matlab prompt will not be accepted! You must prepare an m-file problemX.m containing the commands for each problem.
• All printed numerical values must be labeled, all graphics must be labeled
• Print out only those values which were asked for in the problem!
• Describe your observations and answer all questions asked in the problem.

The easiest way is to use the publish command (available in Matlab 7 ). Your m-file contains the main title (starting with %%), then a "cell title" (starting with %%) followed by comment lines (starting with %), followed by Matlab commands, followed by additional "cells" in the same format. The publish command then generates an html file (which you can print from a web browser) or a Microsoft Word file containing input, output, and graphics. (see also p. 37, 41 in "Differential Equations with Matlab").
Example: From the m-file publish_example.m we generate the web page publish_example.html with the command publish publish_example

If you don't use publish: For each problem hand in

1. a printout of the file problemX.m containing the Matlab commands, and comments answering the questions asked in the problem. Also include comments for each plot command indicating which of the attached graphics it produced.
2. a printout of the file problemX.txt containing Matlab output (using the Matlab diary command). Use the command echo on to print out all Matlab commands together with the output.
3. a printout of all graphics. Clearly label all graphs and put a comment in the file problemX.m so that it is clear which Matlab command produced which graphic.

   (On Glue, WAM typing print figXY.ps in Matlab produces a postscript file figXY.ps of the current figure which you can then send to the printer.)

4. additional (possibly handwritten) pages containing answers to questions asked in the problem (unless they are contained in the comments of the file problemX.m).

Using Matlab for Differential Equations

You are expected to read these documents carefully and try out the explained commands and examples on the computer (you can copy and paste the commands from the browser into the Matlab window).

• Using Matlab for First Order ODEs: How to plot direction fields, solve ODEs numerically and symbolically, plot solutions
• Using Matlab for Higher Order ODEs and Systems of ODEs: Solving ODEs numerically and symbolically, plotting the solutions, phase plane plots, vector fields.
• Using the Euler Method in Matlab
• Solving ODEs using the Laplace Transform in Matlab
• Using Matlab for Autonomous Systems

Matlab Information

• Matlab is available in many computer labs on campus:
  • On Windows machines select Matlab from the Start menu to start the integrated enviroment (with editor and debugger).
  • On Macintosh OS X computers : Click on the face icon on the bottom left to open the Finder. Select "Applications" on the left in the Finder, then double-click on "Matlab" on the right in the Finder.
  • On WAM or Glue Unix machines type tap matlab and then matlab & . This starts the integrated enviroment (with editor and debugger).
    In order to start Matlab in a terminal window type matlab -nodesktop . In this way you can also use Matlab remotely (but you won't see any graphs unless you have an X-server running on your local machine). You can use your favorite editor (nedit, pico, vi, emacs) for writing m-files.
• Learning Matlab:
  • Read chapters 2, 3, 4, 8 in "Differential Equations with Matlab" and try the commands on the computer.
  • For an introduction to standard Matlab commands read the Matlab Primer (PDF file) and try the commands on the computer.
    
  • Introduction to numerical and symbolic Matlab commands. Read this and try the commands on the computer.
  • For more information about symbolic Matlab commands read in the Symbolic Math Toolbox Documentation the chapters ``Getting Started'', ``Calculus'', ``Simplifications and Substitutions'' (also available as PDF file which is better for printing).
• Common Problems with Matlab
• Matlab documentation, Symbolic Math Toolbox Documentation
  Matlab help:  

  Note that some commands such as diff have different meanings for numerical and symbolic objects.

Using Computers on Campus

• Computer labs where Matlab is available
• Currently available WAM/Glue Unix machines
• How to get a WAM/Glue account, Printing account
• Text editors available on WAM, Glue (for writing m-files): nedit, pico, vi, emacs
• How to use a PC, Macintosh computer, Unix workstation in the campus labs