0Introduction
IA Differential Equations
0 Introduction
In this course, it is assumed that students already know how to do calculus.
While we will define all of calculus from scratch, it is there mostly to introduce
the big and small
o
notation which will be used extensively in this and future
courses (as well as for the sake of completeness). It is impossible for a person
who hasn’t seen calculus before to learn calculus from those few pages.
Calculus is often used to model physical systems. For example, if we know
that the force
F
=
m¨x
on a particle at any time
t
is given by
t
2
−
1, then we
can write this as
m¨x = t
2
− 1.
We can easily integrate this twice with respect to
t
, and find the position
x
as a
function of time.
However, often the rules governing a physical system are not like this. Instead,
the force on the particle is more likely to depend on the position of the particle,
instead of what time it is. Hence the actual equation of motion might be
m¨x = x
2
− 1.
This is an example of a differential equation. We are given an equation that
a function
x
obeys, often involving derivatives of
x
, and we have to find all
functions that satisfy this equation (of course, the first equation is also a
differential equation, but a rather boring one).
A closely related notion is difference equations. These are discrete analogues
of differential equations. A famous example is the Fibonacci sequence, which
states that
F
n+2
− F
n+1
− F
n
= 0.
This specifies a relationship between terms in a sequence (
F
n
), and we want to
find an explicit solution to this equation.
In this course, we will develop numerous techniques to solve different differ-
ential equations and difference equations. Often, this involves guessing of some
sort.