Problem Sets

Fall 2005

• M427K-C
  
• Text: Polking, Boggess and Arnold - Differential Equations with Boundary Value Problems [Prentice-Hall, ISBN 0-13-186236-7, 2nd edition] (used copies of 1st edition are ok)
  
• Computer accounts can be obtained at the computer lab [RLM 7.122] with ut EID.
  
• Problem Set #1: Read Sections 2.1 (first-order equations), 2.2 (separable equations) and 2.4 (linear equations); page 35, #13-18 (no interval of existence), #23-24 [note there should be no prime on N in the second line of problem 23]; page 55, #14-15. [due 9/8/05]
  
• Note that problem sets are usually due on Thursdays. Questions regarding graphs require some type of graphical hardcopy, unless there is explicit discussion of sketches done by hand.
  
• Problem Set #2: page 55, #3-8, 16-21 (no interval of existence), #28; page 75, #35-40 [note that classroom technique is more straightforward than method outlined on bottom half of page 74]. [due 9/15/05]
  
• Work on items [1],[2],[3] from the lab notebook set below.
• Problem Set #3: Problems on exact equations: page 75, #9, 10, 12, 15, 16, 19; on page 76, look for a trick to do easily #44. Begin reading chapter 4 on 2nd-order equations, sections 1,2 and 3; page 156, #1-8, 11, 12, 16, 25, 26, 27, 32. [due 9/22/05]
  
• First exam: Friday, 10am - September 23, 2005
  
• Problem Set #4: Read sections 4.3 and 4.5; page 156, #28-31, 33-36; page 172, #18-23 [due 9/29/05].
  
• Problem Set #5: Page 177, #8,9,10,13,14; page 214, #11,12,14,15,17 Note that the answer given for problem #9 on page 177 is wrong! [extension: due 10/11/05].
  
• Problem Set #6: Pages 214-215, #32, 34-36 [due 10/13/05]
  
• Problem Set #7: Read chapter 9 [pages 372-381]; page 390, #7-12 [due 10/20/05]
  
• Work on items [4], [5] from lab notebook section below.
  
• Prepare for second exam on Friday October 21st.
  
• Problem Set #8: Page 390, #22-27.
  
• Problem Set #11: Page 644, #13-18 including questions asked to make and explain plots * But note that the boundary data is changed to be u(0,t) = u(L,t) = 0, and it is not required that the initial data u(x,0) = f(x) should satisfy the boundary data [due 12/1/05].
  
• Lab Notebook [6] which is due 12/6/05.
  
• Problem Set #12: Turn in Lab Notebook [6] as this problem set [due 12/6/05].

Diff Eq Lab Notebook

• Lab Notebook -- It should include all the graphs that you have done in the lab or using a computer, together with analytical discussion.

• Keeping a lab notebook is part of good science. Computer lab assignments/projects should be written up, and include "pasted in" hard output, in the lab notebook with analytical discussion.
• [1] Obtain computer account for math computer lab or access to a computer network running matlab. Run matlab and determine if the system has the Polking program dfield; if not, download this program or get the system administrator to do that; see how to obtain a graph like Figure 7 on page 24 or top of page 16. Make "hard copy" and paste in your diffeq notebook. Discuss any interesting aspects of this graph.
• [2] Make dfield plots for the following four equations: (i) dy/dx = x/sqrt(x^2+y^2); (ii) dy/dx = y/sqrt(x^2+y^2); (iii) dy/dx = -x/y; (iv) dy/dx = -y/x. Discuss the behavior of solutions for these dfield plots, especially for the first cases the long-term behavior for large x (both positive and negative).
• [3] Use dfield to attempt to analyze Example 6.25 from the text (page 68). Explain the symmetry of the solutions, and why dfield does not produce a graph exactly like Figure 3 from the text.
• [4] Convert problems 3, 9, 15 and 17 on page 156 to systems of two first-order differential equations, and then make phase plane plots using the Polking program pplane. Discuss varying aspects of your graphs.
• [5] Use pplane to make phase plane plots for the following problems on page 390: #7-10, #16-19, #31-32.
• [6] Make phase plane plots for problems #1-8 on page 468. Find and classify all equilibrium points.

• Information concerning VNC [Virtual Computing Network]
  
• ODE Software for MATLAB from Rice [created by John Polking]
  
• This software consists of two programs: dfield to study direction fields for first-order equations; and pplane to study phase plane plots for first-order systems in two variables. These programs run in matlab, but they must be downloaded for your system unless you are working in the mathematics computer lab. Polking has developed a third program odesolve which allows you to compute phase plane plots for non-autonomous systems.
  
• On the Rice web page there is a java applet for dfield and pplane that should run if your browser is java-enabled. But there may be problems with printing.
  
• An interesting math web page math world sponsored by Wolfram.
  
• Some interesting books: Martin Rees, Our Cosmic Habitat; Steve Lohr, Go To; James Gleick, Isaac Newton.
  
• An interesting article appears in Scientific American (Aug 03): J. Bekenstein, "Information in the holographic universe"
  
• What I am currently reading: Orhan Panuk, Istanbul; Proof, a play by David Auburn; Mehrling, Fischer Black and the revolutionary idea of finance
  

• First exam: Friday, September 23rd
  
• Second Exam: Friday, October 21st
  
• Third Exam: Friday, November 18th
  
• Final Exam: Monday, December 19th [9-12n]