Week 1
Monday
During Class, load the Maple
file
Reading Assignment for Tuesday: Read chapter 7.
Tuesday
We shall work on WorkSheet
1 during the class period.
For Thursday, please make sure that you understand the definition,
the properties, and how to use the Laplace transform in solving DEs.
So please go back over the worksheet and start doing some of the exercises
in chapter 7.
Thursday
We will work on using the properties of the Laplace transform to solve
a differential equation. Note that this will involve finding the
inverse Laplace transform using the partial fraction decompostion.
Friday
More problems, perhaps involving the Heaviside function and the Dirac
function. Also will look at the Laplace transform of the convolution.
Week 2
Monday
Quiz time.
Discussion of convolution and transfer function.
Tuesday
More on the Laplace transform
Thursday
Use salt tank problems to motivate systems of DEs
Friday
Go through example 8.6. Use a matrix factorization to show how to uncouple
a system of DEs
Introduce eigenvalues and eigenvectors
Read 8.1 for Monday
Week 3
Monday
More eigenvalues and eigenvectors.
Assignment for Tuesday: Please look at the Maple
file for the Maple syntax which produces eigenvalues and eigenvectors.
Be sure to read 8.1 and 8.2.
Tuesday
Take Home Quiz 2 due.
During class - More on eigensystems and decoupling
Thursday
In class Quiz 3 - over everything.
Friday
Last day before Holiday break.
During class - 8.3 (Please note that matrices in this section
are symmetric and that symmetric matrices have nice eigensystem properties.
Namely, if A is symmetric then its eigenvalues are always real and the
eigenvectors for different eigenvalues are always orthogonal (i.e. perpendicular).
Furthermore, symmetric matrices are never deficient in eigenvectors.
These properties will also imply that it will be possible to choose the
matrix S in the decomposition so that the inverse of S is its own
transpose.
I created a Maple worksheet which illustrated the properties of the eigensystem of a symmetric matrix.
Tuesday. Begin Chapter 9. Will go over exercise 174 in class.
Thursday. More on Chapter 9.
Friday: More on Chapter 9. Quiz 4 due.
Week 5
Monday: How to convert complex solutions to real solutions.
Tuesday: Review and questions.
Thursday: Exam 1 This exam will cover the material in chapters 7, 8, and 9. From chapter 7, you should know the definition of the Laplace transform, its main properties, and how to use it to solve differential equations. From chapter 8, you need to know how to find (by hand) eigenvalues and eigenvectors of 2 x 2 matrices. You need to know how to diagonalize a matrix and you need to know the special properties of symmetric matrices. Note that I will not ask a rotation of axes problem. From chapter 9, you need to know how to write a linear system of differential equations in matrix form and how to use the diagonalization of the matrix to convert a coupled system into an uncoupled system. Given the eigensystem of a matrix, you need to know how to write down the general solution and how to use the initial conditions to find the constants in the general solution. On the exam you will not be asked to do any conversion from complex to real, but you do need to know the structure of the solution when there are complex eigenvalues. You should also be able to discuss the stability of critical points of homogeneous and non-homogeneous linear systems.
Friday: Converting second and higher order ordinary DE's to systems - see 9.9 on page 160.
Note: Midterm grades were based on 30% quizzes and 70% exam 1
Week 6
Monday: Questions over exercises 180 - 184
During class: Work on WorkSheet 6 and begin Chapter 10
Assignment for Tuesday: Read pages 183-184.
Take Home Quiz 5 due Friday: I want 4 pages (you can use 2 sided if you wish) handed in - provide Phase Planes (along with a few trajectories so you may need to use DEplot rather than dfieldplot) for the non-linear systems in example 10.4,exercise 200, exercise 201, and exercise 202 (assume gamma =3). In the first three, you may want to restrict your attention to the first quadrant. Can you determine the equilibrium points (approximately) and their stability from the plane phase analysis?
Tuesday: More on non-linear systems. WorkSheet 7
Thursday: WorkSheet 8 on linearizing non-linear systems and determining stability.
Friday: Take Home Quiz 5 due AND In-class Quiz 6 will be given.
Questions will consist of (1) converting between ordinary DE's and systems
of DE's (2) Structure of solution to non-homogeneous systems of linear
differential equations (3) phases plane analysis of systems (linear and
non-linear), (4) linearizing non-linear systems (ie finding
tangent planes) and (5) determining the stability of equilibrium solutions.
.
Week 7
Monday: Hand back quizzes 5 and 6.
Hand out take home quiz 7 (due Friday) (You will need the Maple
document on solving systems of DE's numerically)
During class work on WorkSheet 9.
Tuesday: Work on finding the phase plane for the pendulum.
Thursday: Show some transparencies of predator - prey trajectories (
compare non-linear and linear solutions).
Begin work on WorkSheet 10 - more on the pendulum.
Friday: I will be out of town on Friday so we will not have class. Thus you can have a slight extension on Quiz 7. Please put it in the middle bin outside my office by 2 pm on Sunday. Although WorkSheet 10 is not to be turned in, I expect you to know this material. If you have any questions, please ask them on Monday. The plan is to begin the approximation section of the course on Monday.
Week 8
Monday: Begin Chapter 12.
Tuesday: More on infinite series
Thursday: Begin Taylor series
Friday: Quiz in preparation for next week's exam.
Week 9
Monday: Here is a Maple
document which gives some commands for generating
Taylor series. Today a few more words on Taylor series and then begin
Fourier series.
Tuesday:
Thursday: EXAM 2 This
exam will cover
-- Converting ordinary DEs to systems
-- Stability of critical points
-- Finding and determining critical points
-- Solution structure of x'=Ax+b (will ask you to find
the eigenvalues of A by hand)
-- Series (convergence/divergence).
-- Taylor series (finding and knowing radius of convergence)
-- You should know the Taylor series for common functions (geometric,
sin(x),cos(x),exp(x))
Friday: More on Fourier series
Week 10
Monday: Still more on Fourier series. Work on Worksheet 13.
You are to work through worksheet 13 for Tuesday and produce some graphs.
Tuesday: Review of worksheet 13 and handout worksheet 14. You might find the following Maple files useful - the first produces coefficients for the Fourier series of a nicely defined function; the second one is for piecewise defined functions.
Thursday:
Friday: