Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the Hirsch, Smale and Devaney text. Unless otherwise noted, each part (a), (b), (c), etc. of a problem should be answered. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.
Important: Please list the names of any students or faculty who you worked with on the assignment. Also be sure to cite any texts, websites, manuals, etc. you may have used.

Chapter 7 Exercises (pp. 156 - 157)
Problems:   1a, 1b, 1c, 3, 6

Hints and Notes: For problem #3, convert the second-order equation into a first-order system and apply the Picard iteration method to the system (an example is given on pp. 145 - 146 of the text.) For problem #6, for which values of a does the Existence and Uniqueness Theorem apply? If it does apply, give the unique solution. If it doesn't apply, how do you construct more than one solution through x(0) = 0?

Chapter 8 Exercises (pp. 184 - 187)
Problems:   1 (ii), 1 (iii), 5

Hints and Notes: For problem #1 (ii), use polar coordinates to study the nonlinear system. For problem #5, the equilibrium points will depend on the parameter a. Use the trace-determinant plane to analyze the behavior of the associated linear system at the equilibrium points. Be sure to describe any bifurcations you find.

Additional Problem:
Given the "nice" differential equation X' = F(X), where F(X) is a C1 function, prove that the flow phi_t(x) is continuous. (See class notes from 11/8 for setup and hints.)