Assignment 1: Due Jan 24 (Thursday!)
page 15, #1,5,6
page 24, #1-5
A. Use dfield (or any program which draws direction fields) to draw
graph a direction field for
y' = (10 - y)(y)(2-y)
B. Identify the equilibrium (fixed points) for the equation and decide whether nearly solutions are approaching or leaving the equilibrium points. Graph the three solutions which passe through the point (0,1), (0,4) and (0,11).
C. Solve the initial value problem y'+y = exp(-rt) with y(0) = 0 for r a constant greater than zero. Pay special attention to r = 1. Show that the solution is a continuous function of r as well as t.
Assignment 2: Due Jan 31
page 38, #15, 38, 42
page 47, #8, 18, 30
page 60 #4, 8
Consider carefully the real-world implications of that last problem!
There is no homework due the week of February 7. Study for the midterm!
The "Miscellaneous problems" on page 131 (ignore pages 132-3) are good
practice, although a bit repetitive. In addition, here are some problems
from the sections you haven't worked on yet. Some of these may show up
again on next week's homework.
page 88, #16, 18, 20, 21
page 99, 1-5, 19, 21
page 118, problems 15-18 (this gives a slightly different proof of the
existence and uniqueness theorem)
page 129, problems 1-13 all cover linear difference equations. This is
overkill, but you should do some of these.
Assignment 3: Due February 14
page 88, #16, 20,
page 99, 1,4, 21
page 129, #15
Assignment 4: Due February 21
A. Find a solution on [0, infinity) to y' + y = g(t); y(0)=0, where g(t) = 1 if 1 < t < 2 and g(t)=0 otherwise. Make a sketch of both the input function g(t) and your solution y(t).
B. Find the linearization of the equation y' = y(-1+4y-3y^2) about each of the fixed points
C. Find the general solution of the 2nd order linear equation y'' + 6y' + 10 y=0.
D. Find the solution of the 2nd order linear equation y'' - 2y' + 5y =0 with y(0)=0 and y'(0)=1.
E. Solve the inhomogeneous equation y'' + 4y = 4 with y(0)=0 and y'(0)=0.
F. Describe the solutions to the equation y'' + by' + y=0 as a function of the parameter b. Find the value of b at which a bifurcation occurs, and describe in words what happens.
G. Given the situation of problem F, with initial conditions y(0)=1 and
y'(0)=0. Plot y(10) as a function of b. Plot y(20) as a function of b.
Approximately what value of b minimizes y(20)^2 + y(20.1)^2.
Assignment 5: Due February 28
Page 222, # 17, 19Assignment 6: Due March 20
Page 383 #2, 5, 8, 10, 16, 18Assignment 7: Due Tuesday, April 1
Page 410 #2, 10, 12, 14, 28Assignment 8: Due Tuesday, April 8
Page 439 #1, 2Assignment 9: Due Tuesday, April 15
Page 528, #17a,b,c,d (skip e)Assignment 10: Due THURSDAY, April 24
Page 272, #19, 20Assignment 11: Due THURSDAY, May 1 Use these to study for the midterm, and then turn them in the day after.
Page 586, #28, 29