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 2005: Differential Equations for Scientists and Engineers

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Table for type and stability of critical points
Matlab Assignment 3, due Dec. 9 .
- 1(a): Use ode45 to solve the ODE for t in [0,50]. For a=8 you should find that y(t) converges to k·2 as t goes to infinity with some integer k. This means that the pendulum makes k full revolutions before settling down at the bottom. 1(b): Plot the five solutions which give 0,1,2,3,4 full revolutions (not the one with a=8). See this Matlab Example. 1(c): Find values a₀, a₁ with a₁-a₀<0.05 so that the pendulum makes zero revolutions for a=a₀, and one revolution for a=a₁. Make a phase plane plot which shows these two solutions together. 1(d): You can find the derivatives by hand, and also the entries of A=J(0,0) and A=J(,0). Use eig to find the eigenvectors and eigenvalues.

What I covered in class so far (12/08): 1.1, 1.2, 1.3, 2.1, 2.2, 2.4, M5.2, 2.5, M6.3, 2.3, M5.3, 2.6, 2.7, 8.1, 8.2, 8.3, 8.6, M7.1, M7.2, M7.3, M7.4, M10.1, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 6.1, 6.2, 6.3, 6.4, 6.5, 7.5, 7.6, 9.1, 7.8, 9.3 (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 7.8, 9.1, 9.3 Read the following instructions and try this on a computer: Using Matlab for Higher Order ODEs and Systems of ODEs: Solving ODEs numerically and symbolically, plotting the solutions, phase plane plots, vector fields. Using Matlab for Autonomous Systems Read chapters 2, 3, 4, 7 in "Differential Equations with Matlab". Tutoring is available Tue 10-12, Wed 11-12, Fr 9-10 in MTH 0301, see Tutoring Schedule.

Sections and TAs section discussion session TA 0211 Fri 8:00 MTH 0305 Zhiwei Chen zchen@math 0231 Fri 10:00 MTH 0305 0241 Fri 11:00 MTH 0305 0221 Fri 9:00 MTH 0305 Benjamin Trahan btrahan@math 0232 Fri 10:00 MTH 0307 0242 Fri 11:00 MTH 0307 Syllabus Information about time & place, instructor, textbooks, syllabus, grading policy, Matlab assignments, final exam. This was handed out in the first class. Detailed Syllabus Recommended Problems from the Textbook How to Hand in Assignments 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 willnot be accepted! You must prepare an m-file problemX.m containing the commands for each problem. All output and graphics must be clearly labeled (see below for instructions) Print out only those values which were asked for in the problem! For each problem hand in 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. 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. 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.) additional (possibly handwritten) pages containing answers to questions asked in the problem (unless they are contained in the comments of the file problemX.m). If you use Matlab 7 you can also use the publish command (see p. 37, 41 in "Differential Equations with Matlab") to generate an html file and print this from your web browser. This printout replaces 1.-3. from above. 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

section	discussion session	TA
0211	Fri 8:00 MTH 0305	Zhiwei Chen zchen@math
0231	Fri 10:00 MTH 0305
0241	Fri 11:00 MTH 0305
0221	Fri 9:00 MTH 0305	Benjamin Trahan btrahan@math
0232	Fri 10:00 MTH 0307
0242	Fri 11:00 MTH 0307