MA2AA1 (ODE’s): Introduction
Sebastian van Strien (Imperial College)
email address: <s.van-strien@imperial.ac.uk>
web address: http://www2.imperial.ac.uk/~svanstri/
January 7, 2013
Organisation of this Module
Relevant material
Recommended Books
Exam Revision
The what and why of ODE’s
Differential Equations MA2AA1
Sebastian van Strien (Imperial College)
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Organisation of this Module
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The lectures for this module will take place Monday 2-3,
Tuesday 11-12 and Friday 2-3 in Clore.
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Each week I will hand out a sheet with problems. It is very
important you go through these thoroughly, as these will
give the required training for the exam and class tests.
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Support classes: Monday 3-4, from January 14.
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There will be two class tests. These will take place in
week 4 and 8. Each of these count for 5% .
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Questions are most welcome, after lectures or during office
hour: 12-1 Tuesday office 6M36 Huxley Building.
Differential Equations MA2AA1
Organisation of this Module
Sebastian van Strien (Imperial College)
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Relevant material
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There are many books which can be used in conjunction to
the module, but none are required.
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The lecture notes displayed during the lectures will be
posted on my webpage:
http://www2.imperial.ac.uk/~svanstri/ Click
on Teaching in the left column. The notes will be updated
during the term.
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The lectures will also be recorded. See my webpage.
This module will discuss:
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techniques for solving differential equations explicitly (E);
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more purely mathematical issues (M);
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a few applications (A).
Differential Equations MA2AA1
Relevant material
Sebastian van Strien (Imperial College)
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Recommended Books
Listed below are some often-recommended books which you
might want to consult.
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Simmons + Krantz, Differential Equations: Theory,
Technique, and Practice, about 40 pounds. This book
covers a significant amount of the material we cover. Some
students will love this text, others will find it a bit
longwinded. (A,E)
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Hirsch + Smale (or in more recent editions): Hirsch +
Smale + Devaney, Differential equations, dynamical
systems, and an introduction to chaos.
Quite a few exercises and lecture notes can be freely
downloaded from the internet.
Differential Equations MA2AA1
Recommended Books
Sebastian van Strien (Imperial College)
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Other recommended books
1. Tenenbaum, Pollard, Ordinary differential equations. Dates back
to the 60’s. This 700 page book is published by Dover and costs
about 15 pounds. It is well extremely well written and contains
solutions to the exercises. It does not cover all the material of
the lectures though. (A,E)
2. Braun, Differential Equations and Their Applications: An
Introduction to Applied Mathematics. A book with many
applications. (A,E,M)
3. Teschl, Ordinary Differential Equations and Dynamical Systems.
We will be following these notes quite closely. These can be
downloaded for free from the authors webpage. (M)
4. Arnold, Ordinary differential equations. This book is an absolute
jewel and written by one of the masters of the subject. It is a bit
more advanced than this course, but if you consider doing a
PhD, then get this one. You will enjoy it. (M)
Differential Equations MA2AA1
Recommended Books
Sebastian van Strien (Imperial College)
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Exam Revision
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The main way to revise for the tests and the exam is by
doing the exercises.
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Over the last 5 years this module was taught in four
different ways. As a result previous exams are NOT
indicative for the examination this year.
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However, I will hand out a practise examination later this
term.
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Moreover, the exercises ARE indicative for what the exam
this year will be like.
Differential Equations MA2AA1
Exam Revision
Sebastian van Strien (Imperial College)
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What are ODE’s (Ordinary Differential Equations)
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ODE’s are problems where one tries to solve a function
x : R → Rn which satisfies a functional relation of the type
F (t, x, x 0 , x (2) , . . . , x (k ) ) ≡ 0 and where t ∈ R.
Here x 0 is short for x 0 (t) (the first derivative w.r.t. to t) and
x (j) is short for the j-th derivative of x w.r.t. to t. Usually t is
called the independent variable and x the dependent
variable (or the state variable).
dx
d 2x
Sometimes we write ẋ =
and ẍ =
.
dt
dT 2
An example is the law of Newton: mẍ(t) = F (x(t)) ∀t.
Here F is the gravitational force. Using the gravitational
force in the vicinity of the earth, we approximate this by
mẍ1 = 0, mẍ2 = 0, mẍ3 = −g.
This has solution
 
0
g  2
0 t .
x(t) = x(0) + v (0)t −
2
1
Differential Equations MA2AA1
The what and why of ODE’s
Sebastian van Strien (Imperial College)
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According to Newton’s law, the gravitational pull between
two particles of mass m and M is F (x) = γmMx/|x|3 . This
gives
mẍi = −
γmMxi
for i = 1, 2, 3
(x12 + x22 + x32 )3/2
Now it is no longer clear how to solve this equation or even
whether there are solutions to mẍ = F (x).
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Sometimes we have additional parameters in the problem:
(2)
(k )
F (t, xλ , xλ0 , xλ , . . . , xλ , λ) ≡ 0.
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In ODE’s the independent variable is one-dimensional.In a
Partial Differential Equation (PDE) such as
∂u ∂u
+
=0
∂t
∂x
the unknown function u is differentiated w.r.t. several
variables:
Differential Equations MA2AA1
The what and why of ODE’s
Sebastian van Strien (Imperial College)
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Why ODE’s (Ordinary Differential Equations)
Why ODE’s?
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In many problems in physics, biology, economics and so
on, time evolution is studied and some state variable
x ∈ Rn depends on one variable, usually time t, e.g.:
dx
d 2x
= f (x, t) or 2 + V (x) = E.
dt
dt
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2
d x
For brevity one often writes ẋ = dx
dt and ẍ = dt 2 . One also
(n) for the n-th derivative of y .
uses y 0 = dy
dx and y
We will find that many ODE’s cannot be solved explicitly, and
therefore we need to address issues such as
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existence and uniqueness of solutions
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methods for determining the qualitative behaviour of
solutions.
Differential Equations MA2AA1
The what and why of ODE’s
Sebastian van Strien (Imperial College)
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So let’s begin with the fun!
Differential Equations MA2AA1
The what and why of ODE’s
Sebastian van Strien (Imperial College)
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