(Summer Session II, 2005)
This is an introductory course on (Ordinary) Differential Equations - frequently called as ODEs. Topics will include homogeneous and nonhomogeneous linear equations, systems of equations, the Laplace transform, and if time permits, series solutions to equations and nonlinear systems. The prerequisite is completion of one of the standard calculus sequences (either MAT 125-127 or MAT 131-132). The 200-level courses MAT 203/205 (Calculus III - Multivariable Calculus) and MAT 211 (Linear Algebra) are not required, but it is strongly recommended that you have had some exposure to basic ideas in multivariable calculus (esp. multivariable chain rule) and linear algebra (esp. eigenvalues and eigenfunctions). The course will be augmented by the use of math software such as Mathematica, Maple or Mupad. A few sessions will be held in the SINC site.
There will be an early exam in the first week
of class (Thursday, July 14) on multivariable chain rule and diagonalizing 2 by
2 matrices (yes, only 2 problems). Check back for more informationon the early
exam. 4 computer projects will be assigned only from 1.1-3.6 which is about
first half of the course.
Computer Project
Project 1
You will be given a problem number through email(The
2 people who never showed up in class, should contact me and let me know their
email address). Turn to page 76 of your text book(Chapter 1 review problems) and
solve the equation by hand or computer. Now, using any of Maple,Mathematica,Matlab
or Mupad, plot the slope field and at least 2 solution curves on the same picture.
The choice of the range of x and y is up to you. Print it out, or email me the
file (in the original format).
Project 2
Again, you will be given a problem
number via email. Turn to page 41, and first solve the differential equation
by hand (you do not need to submit this). Then numerically solve the corresponding
D.E. using Euler, Improved Euler and Runge-Kutta Method using step size h=0.05
in a interval of length 2. You are free to choose your initial condition and
the interval. Then make a table that compares all three methods and the actual
solution in the format of Figure 2.6.2 in page 135.
Project 3.
You
will be given two problem numbers via email. Turn to page 375-376 and plot the
directional fields and the solution curves of the corresponding problems. You
should find all the critical points points by hand and the critical points should
be included in the picture. Then for each critical point, state what type of
critical point it is ( eg. stable proper node which is a sink... read
through 6.1 to understand the concept.). For maple there is a useful command
called DEplot. Here is an example-(although I would prefer
to have more solution curves...) of how it is used (written by Jonathan Inbal).
Homework
|
Due Date |
Exercises |
|
July 20 |
1.1: 9,11,15,21 |
|
July 27 |
1.3: 9 |
|
Aug 8 |
3.2: 9, 21 |
|
Aug 18 |
5.5: 3, 5, 15, 17 |
| Homework | 10% |
| Computing Project | 10% |
| Early Exam (July 14 in class) | 5% |
| First Midterm (July 27 in class) | 20% |
| Second Midterm (August 8 in class) | 20% |
| Final (Tuesday, May 14 8:00-10:30am) | 35% (cumulative) |
No make-up exams will be given. If a midterm exam is missed because of a serious (documented) illness or emergency, your semester grade will be determined on the basis of other work done in the course. Exams missed for other reasons will be counted as failures.
Resources: If you have questions regarding the course material at any time during the semester, you are encouraged to send email, either to myself or to the grader. It is not always easy to explain mathematical concepts via email, however, so if your question is a mathematical one, you should probably ask me in person during my office hours. Another excellent source of help is the Mathematics Learning Center (A-125 in the Physics Building), which is staffed by advanced math majors and graduate students daily. For a schedule of their hours, check their website.
Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748/TDD. DSS will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.
| Day | Topics |
|---|---|
| July 11 | Materials for the Early Exam -
Multivariable Chain Rule, Diagonalizing 2 by 2 Matrices. click 1.1: Mathematical Models |
| 13 |
1.4: Separable Equations 1.5: Linear First-Order Equations |
| 14 |
1.6: Substitution Methods and Exact Equations Early Exam |
| 18 |
Word Problems from 1.4-1.6 1.3: Slope Fields and Solution Curves 2.1: Population Models |
| 20 | Computer Session 2.2: Equilibrium Solutions |
| 21 | 3.1: Second-Order Linear Equations 3.3: Homogeneous Equations with Constant Coefficients |
| 25 |
Computer Session on 2.4-2.6 (Not included
in Exam) 2.4: Euler's Method 2.5: More on Euler's Method 2.6: The Runge-Kutta Method |
| 27 - Midterm I (6:00-7:40) | |
| 28 | 3.2: General Solutions of Linear Equations 3.5: Nonhomogeneous Equations |
| August 1 |
4.1: First-Order Systems of Equations
4.2: The Method of Elimination |
| 3 |
3.7: Electrical Circuits 5.1: Matrices and Linear Systems 5.2: The Eigenvalue Method |
| 4 |
Computer Session 6.1: Stability and the Phase Plane (Not included in Exam) |
| 8 -2nd Midterm | |
| 10 | 5.4: Multiple Eigenvalue Solutions (Not
included in Exam) 5.5: Matrix Exponentials |
| 11 | 7.1: Introduction to Laplace Transforms 7.2: Transformation of Initial value Problems |
| 15 | 7.3: Translation and Partial Fractions 7.4: Derivatives, Integrals, and Products of Transforms |
| 17 | Review - You DO NOT want to miss this! |