Introduction to Ordinary Differential Equations Fall 07, Tue, Fri 12:30 – 2:00  HNS 106

Instructor: Leon Kaganovskiy
Office: HNS 110 or HNS 204 (Physics Computer Lab, 2nd floor)
Office Hours:

Mon: 11-12:30 and 4-6,   Tue :  4-6 (office/computer lab HNS 204),  Wed: 11-12:30 and 4-6,  Thu :  research day,  Fri   :  4:30-6:30 (office/computer lab HNS 204)

I am available at other times by appointment.  

email: lkaganovskiy@ncf.edu - best way to ask a question.    Also if you need me ASAP:  (941) 366-6134;  (941) 961-3896

 

Course Goals and Objectives: 

This course focuses on Differential Equations and Computational Methods using Matlab/Maple. It is intended for Mathematics and Science students who are going to apply these techniques in their coursework.   Reflecting the shift in emphasis from traditional methods to new computer-based methods, we focused on the mathematical modeling of real-world phenomena as the goal and constant motivation for the study of differential equations.  Topics covered include standard methods for 1st order equations (separation of variables, integrating factor for linear equation etc…), population models, equilibrium and stability, acceleration-velocity models, numerical methods – Euler, Improved Euler, Runge-Kutta, linear equations of higher order, resonance in mechanical and electrical systems, systems of ODEs – eigenvalues and numerical methods, nonlinear systems with applications to population modeling and nonlinear mechanics, chaos, and Laplace transform techniques.   I will use Edwards and Penney, Differential Equations and Boundary Value Problems, as well as their Application Manual which will let  us employ Maple and Matlab to investigate computer approaches to solving ODEs analytically and numerically in almost all sections of the textbook.

 

Minimal Prerequisites:

Prerequisites: Calculus I and II.

Grading Policy: The final grade will be based on tests and problems, as follows:

  1. There will be two exams which will count 35% and 45 % of the final evaluation.

1st exam – 1 week before Spring break, Final – exam week.     Exact dates and times will be announced in class.

 

  1. Homework assignments and computer projects will count 20% of the course grade. Late homework will be accepted only by special permission of the instructor.

Attendance Policy:

There are no specific attendance credit points, but you are responsible for attending all the classes and keeping abreast of all the material presented in class.

Special Need Students:

Students with the need for special accommodations must work with the Counseling and Wellness Center, which will establish the specific accommodations and communicate them to me.

Academic Dishonesty Policy:

Any suspected instance of plagiarism will be reported to the office of the Provost and handled in accordance with the College’s policy.

 

Books:    (http://wps.prenhall.com/esm_edwards_bvp_4/0,13817,4668829-,00.html)

Differential Equations and Boundary Value Problems  4th Edition (newest)

 

EDWARDS & PENNEY

ISBN-10:      0132364425
ISBN-13: 9780132364423
Publisher: Prentice Hall
Copyright: 2008                          
Format: KIT/PACKAGE

 


Maple Tutorial

Matlab code for dfield6 and pplane6.   Both programs should be saved as dfield6.m  and  pplane6.m ,  respectively  into Matlab program work directory ( or any other place as long as Matlab path is set to find it).

To use them just type dfield6 or pplane6 in the Matlab prompt.

 

Topics to be covered and Homework Assigned  (exact due dates will be announced in class).

This course plan may be modified during the semester. Such modifications will be announced in advance during class periods, and the students are responsible for keeping abreast of such changes. The WWW page for the course will also be used to list assignments and other notes, and students are responsible for checking this web page regularly.

1.1 Differential Equations and Mathematical Models

            HW: 9, 16, 20, 22, 26, 28, 33-36 (just write the equation), 45

1.2 Integrals as General and Particular Solutions

            HW: 3, 4, 9, 16, 17, 26, 30, 33, 38

1.3 Slope Fields and Solution Curves

            HW: 4, 6, 11, 13, 14, 23, 24, 27

            Application A, p 30 in the textbook

1.4 Separable Equations and Applications

            HW: 5  9  17  20  30  41  42  47  52  54  55  56  64  65

            Application - population modeling with logistic equation

1.5 Linear First-Order Equations

            HW: 7  12  16  25  34   37   41   42  

            Application 1.5    Understand and confirm in Maple their formulas and use your choice of min and max temperature for your own investigation.



2.1 Population Models

            HW:   11  12  13  15  18  21  22  23  26  34

            Application is not very interesting => no application assignment for this section 

2.2 Equilibrium Solutions and Stability

            HW: 13  14  16  18  20  21  22  23  29

2.3 Acceleration-Velocity Models

            HW:  2   3   4   5   10  11  21  25  29 

2.4 Numerical Approximation: Euler’s Method

Application:   Use Matlab code to do famous numbers problem.   Plot the results and create table like Fig 2.5.1 on page 126.

HW:  5  26  30  ( for 26 and 30 do the same as in application)

2.5 A Closer Look at the Euler Method

Application:   Use Matlab code to do famous numbers problem.   Plot the results and create table like Fig 2.5.1 on page 126.

HW:  5  25  ( for 25 do the same as in application)

2.6 The Runge-Kutta Method

Application:   Use Matlab code to do famous numbers problem.   Plot the results and create table like Fig 2.5.1 on page 126.

Do application B with skydiver, plot resulting velocities and positions.

 

3.1 Introduction: Second-Order Linear Equations

            HW:  10  12   14   18   20-25  33  35  37  39  41  52 

            Application 3.1

3.2 General Solutions of Linear Equations

            HW:  1  5  7  9  16  19  24  27  28  36  

            Application 3.2

3.3 Homogeneous Equations with Constant Coefficients

            HW:  4  5  9  12  14  24  30 ( may find roots in Maple)   41  42   52   55

            Application 3.3

3.4 Mechanical Vibrations

            HW:  3  5  6  7  12  15  18  22   31

3.5 Non-homogeneous Equations and Underdetermined Coefficients

            HW:  10  12  14  22  24  26  28  36  45 

3.6 Forced Oscillations and Resonance

            HW:  4  7  10  11  14   21  22   24   26

            Application 3.6

 

4.1 First-Order Systems and Applications

HW:  5  10  18  21  22  26  30           

4.3 Numerical Methods for Systems

            Application:  p 107, investigation A and C, repeat investigation on p 120 with several different IC, try to discover new behavior.

 

5.1 Matrices and Linear Systems

            HW:  17  18  20  22  26  31  35  37

            Application.

5.2 The Eigenvalue Method for Homogeneous Systems

            HW:  3  6  10   12  14  20   24   28  31

            Application.  Brine Tank investigation.   Do both open and closed.   I advise to use Matlab approach; it is more clear.     

5.3 Second-Order Systems and Mechanical Applications - extra material

5.4 Multiple Eigenvalue Solutions      -  extra material

5.5 Matrix Exponentials and Linear Systems   - extra material

5.6 Non-homogenous Linear Systems   -   extra material

 

6.1 Stability and the Phase Plane

            HW:  6  8  9-12  19  20   23-26

            Application.    

6.2 Linear and Almost Linear Systems

            HW:  20  24   26  28   30   32  34   36 (do not integrate, use qualitative approach)

Application:  MapleApproach       Do NOT do the problems in the Application, apply Maple Approach to 6.2. 20, 22, 24, 26, 28

Make sure to select appropriate range of x and y in the phaseportrait and select interesting initial conditions.

6.3 Ecological Models: Predators and Competitors

            4-25 – Just use Maple worksheet!

            Application.    Employ MapleApproach worksheet above and pplane7 – I want to see both or will take points off !!!!

6.4 Nonlinear Mechanical Systems

            HW:  2  5  7  14  15  18

            Application

6.5 Chaos in Dynamical Systems

 

7.1 Laplace Transforms and Inverse Transforms

            HW:  8  10  14  18  20  21  24  28  36  39  41

            Application

7.2 Transformation of Initial Value Problems

            HW:  5  9  12  15  20  22  24  28  31  34  36

            Application  -  Here is extended Maple worksheet which has solution of the system of ODE.

7.3 Translation and Partial Fractions

HW:  8  10  30  36  39

            Application

7.4 Derivatives, Integrals, and Products of Transforms

HW:  2   10  11  16  18   22  26  28  30  32

7.5 Periodic and Piecewise Continuous Input Functions

HW:  4  7  8   15  17   21  25  26  27  33  40 (can employ Maple)

            Application

7.6 Impulses and Delta Functions

 HW:  3  7  10  11  14  16  18  19