Ordinary Differential Equations, MATH 304

Homework Assignment #3

Due Friday, September 26, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to Differential Equations, 4th ed., by Blanchard, Devaney, and Hall. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You should do all of the problems; however, you only need to write up solutions for the starred (*) problems. You should also write up the solution to the additional problem based on the supplementary reading. Be sure to write up solutions neatly, making sure to show your work.

You are encouraged to work on these problems with other classmates, and it is ok to use internet sources for help if it's absolutely necessary; however, the solutions you turn in should be YOUR OWN WORK and written in YOUR OWN WORDS. Complete solutions to all problems will be posted the same day the homework assignment is due.

Important: Please list the names of any students you worked with on the assignment.

Section 1.6
Problems:   31, 32*, 34, 35, 37, 38, 40*, 43*

Note: For problem #43, be sure to explain your answers carefully.

Section 1.7
Problems:   2*, 3, 4*, 11, 13*, 18, 19

Note: For problems #2, 3 and 4, sketch the entire bifurcation diagram, as done in class.

Section 1.8
Problems:   4, 5, 8, 10*, 20*, 21, 30*

Additional Problem: Consider the periodic differential equation dy/dt = f(t,y) with period T, where f and f_y are continuous functions on the entire ty-plane. Show that the Poincare Map P is a strictly increasing function, that is, if a < b, then P(a) < P(b). Hint: Try proof by contradiction. (See class notes from 9/22.)