Introduction to Differential Equations and Applications

Syllabus

(1) First Order Differential Equations: Solution techniques for linear, separable equations and exact equations. Modeling of problems in mechanics. Remarks on existence and uniqueness of solutions.
BD sections: 1.3, 2.1,2.2,2.4,2.6
EXAM #1 on material from (1)-30-50 minutes

(2) Second Order Differential Equations: Analytic techniques for homogeneous equations with constant coefficients. Linear independence and characteristic equation. Nonhomogeneous equations and variation of parameters. Problems in mechanical vibrations and electric circuits.
BD all of chapter 3.
EXAM#2 on material from (2) -30- 50 minutes

(3)Series Solutions of Second Order Linear Equations : Power series expansion near regular and singular points. Bessel, Legendre and Hermite equations and their occurrence in mathematical physics.

Special homework on material from (3)

(4) Nonlinear 2nd order equations, Euler equations Examples from electrostatics and other branches of physics.

(5)The Laplace Transform Definition of Laplace transform and application to initial value problem. Step functions, discontinuities, impulse functions, and the convolution integral

(6)Systems of First Order Linear Equations Brief review of matrices and system formulation. Eigenvalues and linear dependence. Interpretation of eigenvalues in physical systems.

(7) Nonlinear Differential Equations and Stability Introduction to phase-plane analysis techniques and critical points. Applications to nonlinear systems such as the predator-prey model. Periodic solutions, limit cycles. Solution of 2x2 systems by the exponential matrix; Lyapunov Stability.

Final exam (110 minute written examination on material from (4)-(7))

Learning Objectives and Instructor Expectations

The contents of the course are themselves demanding, this means you will have to invest a significant amount of time in this course. Thus, class participation, independent reading from the textbook and working out homework problems is essential.

Schedule and Homework

Follow links in the table below to obtain a copy of the homework in PostScript (.ps) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. click here.

Grading, Exams and Office Hours

Exams#1,2:
30-50-minute written examination each that will cover material as described in (1) and (2) of the Syllabus respectively

Special Homework
It will cover material from (3) of the Syllabus.
Practice exams for each exam will be posted on this site in due time. Before every exam there will be a review session. During the exams, you are allowed the use of a crib sheet (8.5 by 11 - one-sided). No calculators for the exam.

Final Exam:
The final exam will last 110-minutes, testing your understanding of the material we covered in sections (4), (5), (6) and (7) of the syllabus and emphasize basic techniques as applied to simple, fundamental problems. There will be no deliberately obscure questions in exams to test your mental dexterity. Bring with you: (1) a double-sided sheet of notes (2) the table with the method of undetermined coefficients (3) the table of Laplace Transforms.

Important Note on Office hours
Office hours are hours during which I am guaranteed to be in my office, answering your questions and dealing with problems you may have in this course. Office hours are not time during which you do your homework in my office. Rather, you should use this time to ask questions about problems which you have tried to work out, but got stuck at some point. In other words, you should come to office hours prepared, just like you should come to class prepared.

Extra credit
(1) I might assign some homework problems of extra difficulty for those wishing to explore the techniques further and test their understanding.
(2) Since most of you come from diverse areas of science and engineering, economics etc. where ODEs are frequently used, in every homework, you are strongly encouraged to explore your area and find one (1) problem that uses the techniques of this week's homework (which I will be describing each week). You will get extra credit for a sound explanation of the origins of the problem and a full solution This part of the coursework, prepares you to explore literature independently of what I say in class, and to learn how to think on your own. Extra credit will be generous here.

Tutorials