M427K Homework Assignments, Spring 2008

Updated March 26!

Unless noted otherwise, all computer-based problems are meant to be done using MATLAB. If you do not already have easy access to a computer with MATLAB, you can have a math department account. See me to arrange this.

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, 19
Page 230, # 1, 2, 8, 12, 22, 37, 40
Page 235, # 2, 4

Because of the midterm on March 7, there is no homework to be turned in on Thursday, March 6. However, I recommend that you work problems 2, 5, 8, and 10 on page 383.
New suggestions on March 4: page 383, problems 15, 19, 21, 23 (all odd so you can look up the answers),
Page 399 #11, 15, 17. Feel free to use MATLAB to diagonalize the matrices.

Assignment 6: Due March 20

Page 383 #2, 5, 8, 10, 16, 18
Page 399 #12, 16, 18. Feel free to use MATLAB to diagonalize the matrices.

Assignment 7: Due Tuesday, April 1

Page 410 #2, 10, 12, 14, 28
Page 421, #17,
Page 429, #17, 19, 20

Assignment 8: Due Tuesday, April 8

Page 439 #1, 2
Page 492, #2, 17,
Page 501, #4, 7
Page 511, #13, 19
Page 534, #4, 13

Assignment 9: Due Tuesday, April 15

Page 528, #17a,b,c,d (skip e)
Page 260, #21
Page 266, #19, 23, 26. Because of problem 26, Legendre polynomials appear frequently in quantum mechanics, describing the dependence on angle of wavefunctions.

Assignment 10: Due THURSDAY, April 24

Page 272, #19, 20
Page 278, #10, 21
Page 284, #11, 12, 13. All of these equations occur in quantum mechanics.
Page 292, #16
Page 585, #14.

Assignment 11: Due THURSDAY, May 1 Use these to study for the midterm, and then turn them in the day after.

Page 586, #28, 29
Consider the following functions on [0,1], extended to be periodic with period 1. How do their Fourier coefficients scale with n?
(a) f(x) = x(x - 1/2)(x-1)
(b) g(x) = sin(pi x)
(c) h(x) = 2/[2 + exp(2 pi i x)]
For parts (a) and (b), you do NOT have to compute the Fourier coefficients themselves! Just figure out how smooth the function is at x=0. For part (c), it's possible to find all of the Fourier coefficients without ever doing an integral. You may find the identity 1/(1+z)=1 -z + z^2 - z^3 + ... to be useful.
(d-f) View f(x), g(x), and h(x)-h(0) as functions on [0,1] with Dirichlet boundary conditions. Indicate how fast the Fourier sine series for each function decays. Which kind of Fourier series is better?