Syllabus for AS.110.302.01
Differential Equations and Applications
Summer 2010
Course Information:
Lecture: M,T,W,Th 9:00-11:30am
Classroom: To be determined
Lecturer: Peng Shao
Office:
Email:
Web Page:
Office Hours:
211 Krieger Hall
pshao@math.jhu.edu
http://math.jhu.edu/~pshao/ta/302
MW 1:00--1:50pm, and by appointment
Textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Problems, 9th edition. The price of the textbook is not trivial and the differences between editions sometimes are not huge, so if you have an older edition of this book it is acceptable
to use it as an alternative textbook. But keep in mind that the exercises between different
edition might be different so make sure you are doing the correct exercises if you use older
edition. No execuse like "I use an old edition and I didn't notice the new edition's problem
3.2 has part (d)" will be accepted.
Course Introduction: Even though the official name of this class is called Differential
Equations, a special kind of equations called ordinary differential equations will be the
main topic of this class. An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function
of a single independent variable. In the simplest form, the unknown function is a real or
complex valued function, but more generally, it may be vector-valued or matrix-valued:
this corresponds to considering a system of ordinary differential equations for a single
function. Ordinary differential equations are further classified according to the order of
the highest derivative with respect to the dependent variable appearing in the equation.
Ordinary differential equations are of prominent importance for pure mathematics, applied mathematics, physics, economics and engineering. The research towards ordianry
differential equations can be generally classified into two classes. The first class is the
quantitative analysis which involves techniques and knowledges required to solve an ODE
explicitily in analytical form, or at least numerically accurate. The second class is the qualitative analysis which is, when solving an ODE explicitly is theoretically/practically inaccessible, to investigate the exsitence, uniqueness, pattern of the singularities and dynamic
properties of the solution(s).
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During this summer course it is of course impossible to cover all of these branches of ODE
research. Also considering many of the students who will enroll in this class possibly major in physics or engineering, the lecturer believes it is better to focus on the quantitative
analysis of three most common classes of ODE: first-order linear ODE, second-order linear ODE and systems of first-order ODE. Also we will study two important classical tools:
the Laplace transform and series method. In the last few lectures we will discuss one important aspect of qualitative analysis: the stability of non-linear ODE. If time allowed we
will try to introduce some basic results about the existence and uniqueness of the solution
of some typical ODEs. Basically speaking we may cover Chapter 1, 2, 3, 6,7,8,10 from the
textbook.
Part of the lectures are left for self-reading and the students may be tested the understanding to these knowledge by homework assignments.
Prerequisites: Knowledge from Calculus I,II,III and Linear Algebra will be highly recommended even crucial. We do not need too much results from Complex Analysis except
some basic understanding to Euler formula. No computational knowledge such as the
ability to use mathematical software is required.
Tips: ODE is an advanced mathematical class for undergraduate level, which means the
students are supposed to be quite familar to and skilled in the basic mathematics knowledge they have learned. Also this course involes significant amount of computation and
according to the lecturer's personal experience nothing but this significant amount of computation can help you to master the knowledge. Therefore the students are suggested to
keep away from the calculators and mathematical software as much as they could and
complete the homework on their own---don't forget you will find no calculators during
quizzes/tests.
Since summer course is short and intensive, the students are suggested to read the contents from textbook before the lecture, or else they may very likely find themselves completely lost in a jungle of math, ten minutes after the beginning of the lecture. A math
course should have been a lot of fun.
Supplementary Materials There are tons of excellent textbooks about ODE you can
find from the library, Amazon, or even ebay. Personally I would recommend the interested
students to check out the following items as supplementary materials
1. An Introduction to Ordinary Differential Equations (Dover Books on Advanced Mathematics), by Earl A. Coddington.
2. Ordinary Differential Equations by H. Pollard and M.Tenenbaum
Acutally I didn't read through these books, but they are both highly rated on Amazon.com.
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And look at their price! They are just slightly more expensive than a pile of blank printer
papers. These two books are old but not out of date because the contents we will cover in
this summer course have not changed greatly for more than 100 years.
3. Ordinary Differential Equations by V. Arnold
Even though Arnold claims this book assuming "irreducible mininum of absolutely essential material" I still think this book may be too challenging for undergraduates. But if you
want to learn ODE, especially the phase flow analysis from a "coordiante-free" viewpoint,
then this book is the bible. This is my personal favorite book on ODE.
Don't be freaked out, none of these supplmentary materials are needed for this course at
all , especially the book 3. But I would suggest you buying book 1 because I may choose
some contents from it for the lecture, or you can use it for self-reading. Sometimes I feel
it helpful to have at least two different books on the same topic and of similar level, by
comparing the difference between them one can learn much.
Homework and Quizzes: Homework will be assigned as the following:
Monday:
required to turn in on Thursday, usually consists 3-4 questions
Thursday: required to turn in on the next Monday, usually consists 6-7 questions
Each question is worth 10 points, therefore in one week you can earn as many as 100 points
from the homework.
Due to the shortness for this summer course we can hardly provide any homework review
sections, but the students are welcome to meet the instructor or TA(if there is any) during
office hours.
Late homework will not be accepted (unless it is the result of an officially excused absence).
And also you may notice that homework will have a great weight in your final grades. This
is because this course is so intensive that I have to make sure everyone in the class keeping
the pace. Collaboration on homework are highly encouraged. However, when it comes
time for the students to write up the solutions, the instructor expect them to do this on
their own. To make sure of this, there will also be pop-quizzes in which I will pick up some
questions(usualy 1-2) from old homework you turned in, to make sure you have a clear
idea how to complete the homework problems on your own. Missed quizzes may not be
made up (unless this is the result of an officially excused absence). Each question in the
quizzes is also worth 10 points.
Exams: There will be one (mid-term) exam and one final. The final exam will be cumulative, but with an emphasis on the material covered since the mid-term. Each exam/final
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will be worth 150 points.
Grading:
Homework, Quizzes
The mid-term Exam
Final Exam
10 points/question
150 points
150 points
The final grade will based on the accumulation of all points the students earned in the
entire course.
Calculator: No calculator is allowed in the quizzes/tests.
Schedule: This is only a tentative schedule and is subject to further modification as the
class progresses.
Day
Jun/1
Book Section
1.1-1.3
S.R.1
2.1
Jun/2
2.2-2.4
Jun/3
2.5-2.6
Jun/7
2.8
1
Topic
We will learn some basic notations and terms about ODE,
and several common simple ODE raised from natural sciences. Then the classification of ODE is introduced.
The textbook explains the way to solve a variabl-cofficient
first-order ODE in a very clear and self-contained way, so
the lecturer feels proper to leave it as a self-reading topic.
We will extend the techniques the students learned in the
May/26 course and the self-reading, to a general method
called seperation. Then we will use this method to solve
some ODE raised from natural sciences, but more complicated. In the end we will cover the differences between linear and non-linear ODEs. Also an important existence/uniqueness theorem is introduced. The students are
encouraged to read the proof in seciton 2.8 by themselves in
advance.
We will learn how to solve two kinds of important ODEs:
autonomous equations and exact equations, and their application to population dynamics.
Today's course is devoted to the long proof of existence/uniqueness theorem we learned on May/27 course.
Self Reading.
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S.R.
3.1
Jun/8
3.2-3.3
Jun/9
3.4-3.6
S.R.
4.1
Jun/10
4.2-4.4
Jun/14
6.1-6.2
Jun/15
6.3-6.4
Jun/16
6.5-6.6
Jun/17
Ch 5
Jun/18
Midterm
S.R.
7.1-7.2
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The students are encouraged to read this section by themselves as the solution to a simple case of homogeneous
function(constant-coefficient) is rather elementary, but
contains most of the ideas we will us on Jun/2 course. Also
self-reading on 3.2 is highly encouraged.
Today's course is devoted to the fundamental solution to homogeneous second-order ODE and the linear independence
of functions.
Today we will learn some complicated examples about how
to solve homogeneous ODE, when the characteristic function has complex/repeated solutions. This lecture demands
some understanding to Euler formula. If time allowed we
will learn some basic knowledge of inhomogeneous ODE
The students are suggested to read the topic about general
theory of solving higher-order ODE by themselves as this is a
tiny topic. Additionally interested students are encouraged
to read the last few sections in Chapter 3, which contain several important and interesting problems about vibrations.
We will discuss methods we will need to solve some higherorder ODEs.
We will learn the definition, especially when it is welldefined, of Laplace transformation and its application to initial value problem.
We will use the methods we learned yesterday to focus on
an interesting example, ODE whose forcing function is discontinuous, after we introduce the step function.
We will introduce a very insteresting and important function
called δ-function. Then we will introduce the convolutional
integrals.
We will pick up some useful techniques about the series
method.
The test will cover all materals we learned from Chapter
1,2,3,4,62 . Note that today is Friday.
The students are suggested to make a self-reading on section
7.1 which includes several concrete examples about systems
of ODEs. Also they are required to go though every detail on
section 7.2 to review some basic properties and facts about
matrices.
The contents of Chapter 5, will not be covered in midterm/final
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Jun/21
7.3-7.5
Jun/22
7.6-7.8
Jun/23
9.1-9.2
Jun/24
9.3-9.4
S.R.
9.5
Jun/28
Jun/29
9.6
9.7-9.8
Jun/30 Ch 10
Jul/1
Review
Jul/2
Final
After we learn some basic facts about systems of linear algebraic functions we will go on to introduce the basic ideas to
solve a system of first-order ODE. Then we will try to learn
how to solve a homogeneous first-order ODE system, given
the fact that all of eigenvalues of the constant coefficient matrix are distinct and real.
We will continue our study to discuss the solutions of some
more complicated system of homogeneous first-order ODE,
when the eigenvalues of the matrix behave worse.
We will introduce some basic ideas about how to investigate
the stability of an ODE and second- order linear homogeneous systems with constant coefficients
We will study almost linear systems and then apply the
knowledge about stability to a concrete problem: Competing Species.
The students are encouraged to read the Predator-Prey
Equations during this weekend.
We will learn Lyapunov theory.
In this section we discuss further the possible existence of
periodic solutions of some second order autonomous systems, then we discuss the interesting phenomenon of chaos.
The lecturer tentatively uses this day to introduce some
results from chapter 10 about the partial differential
equations, especially those equations such as heat
equations/wave equations which are closely related to
physics/engineering and have many applications.
Of
course due to the intensiveness of this course the lecturer
may also use today as a making-up of the old materials which we didn't cover in full details. Again, today's
materials are not covered in the final, of course.
Today we will go over all of the knowledge we have learned
during this summer course and prepare for the final exam.
All materials we learned in this course, except Ch 5,10, are
covered. Good Luck!
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