Homework is due at the beginning of class on Thursdays.

Due October 4: Chapter 1 review problems: 5, 6, 9, 11-22, 42, 43a
Chapter 2 review problems: 2, 3, 4, 9, 12-24

Due October 11: Section 1.5) 1, 2, 5-9, 10d, 18. Section 3.1) 5, 7, 10, 14-17, 20-23. Additional exercise (equivalence of integral and differential expressions).

Due October 18:

• Section 3.2) 1, 2, 14. Section 3.4) 3, 6, 7. Section 3.5) 5, 7, 17, 19, 20. Section 3.8) 1, 5, 6, 8, 11, 14, 15.
• In addition, in all of the above except 3.5) 20 and 3.8) 1 and 8, compute exp(t A) for the given matrices; for the two dimensional problems, sketch the image of the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1) under exp(t A) for some reasonable positive value of t.
• Show that:
  • If A is an n x n matrix and U is an invertible n x n matrix, then exp(t A) = U exp(t inv(U) A U) inv(U), where inv(U) denotes the inverse of U.
    Hint: Use the series expansion of the matrix exponential.
  • If B and C commute, i.e. BC = CB, then exp(t (B + C)) = exp(t B) exp(t C).
    Hint: Show that d/dt exp(t (B + C)) = (B + C)exp(t (B + C)) and d/dt (exp(t B) exp(t C)) = (B + C) exp(t B) exp(t C), and that both exp(t (B + C)) and exp(t B) exp(t C) equal the identity matrix when t = 0; invoke uniqueness of solutions.
  • If B and C don't commute, then exp(t (B + C)) need not equal exp(t B) exp(t C).
    Hint: If you're not sure how to construct an example, try B and C with only one nonzero entry.

Due October 25: Homework 4 handout

Due November 1: Section 4.1) 2, 9, 15, 31, 33. Section 4.2) 5, 11, 16

Due November 8: Section 4.3) 1, 9, 15, 18-22. Section 4.4) 3, 4a, 6, 7b

Due November 15: Section 5.1) 1-5, 14, 15, 17, 30. Section 5.2) 1, 2, 4, 12, 13, 15, 21-23

Due Wednesday, November 21, by 11:00 AM (drop off at my office, 359B, JBE): Section 5.3) 1-3, 9, 11, 12, 14.
In addition: Compute the linearization of the vector field for Euler's equations for a free rigid body at the equilibria, and determine the eigenvalues of the corresponding matrices.

Due November 29: Section 5.4) 1, 3, 11, 13-16, 20, 21

Due December 6: Section 5.6) 1-8. Chapter 5 review problems) 6, 7, 25, 27, 28

Specifications and recommendations

Check your work! Whenever possible, verify that your analytically derived solution really is a solution. Sketches of phase portraits don't need to be exact or artistic, but should capture key qualitative features. If you can't draw well, supplement your sketches wirh descriptive text. Late homework will be discounted and, at the discretion of the grader and/or the instructor, may not be accepted. This policy can be toughened up at any time at the instructor's discretion.

Your homework should be neatly written and well-organized, with the pages securely fastened together and your name on every page. Many of the exercises involve several nontrivial steps; make it clear to your readers (and yourself!) what it is you're doing at each step.

Clearly number the exercises and try to submit them in numerical order; if any problems are out of sequence, indicate that at the beginning of the assignment. (You don't need to solve them in order, just submit them in order.) The grader should not have to hunt through several pages to find a particular problem.

Computer use

A CD with a very useful software package is included with the text. Some exercises will make explicit use of this software. You are strongly encouraged to use this software in developing your intuition regarding ODEs; many advanced techniques in the analysis of differential equations involve the interplay of numerical and analytic (paper and pen) methods.

If a problem asks you to plot something and describe the qualitative features of the plot, you must describe them -- the plot alone won't count for much partial credit.

You may use symbolic computation packages (e.g. Maple or Mathematica to carry out the routine calculations in the homework; please check with the instructor or TA if you have any doubt as to what would be considered routine in a given context. If you use such software, please write up the key steps of your solutions by hand, indicating which parts were carried out by computer and attaching a hardcopy of your computer work.

Computer difficulties do not justify late or incomplete assignments.
 

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