18.03 Lecture #11 Oct. 2, 2009: notes
Topic for today is an introduction to higher order ordinary diFerential equations. The big topic
covered in the lecture but not in the reading is Euler’s method for ±nding approximate numerical
solutions.
Today we put away the childish things with which we have so far been concerned, and begin
to speak and reason like adults: about higher derivatives.
Recall the notation
y
m
for the
m
th derivative of a function
y
(of a single independent variable
x
). An
n
th order ordinary diFerential equation means an equation involving the ±rst
n
derivatives
of
y
, and also the independent variable
x
. Our experience with ±rstorder diFerential equations
suggests that we can avoid some nasty behavior if we assume that our equation is solved for the
highest derivative: that is, if it is in the standard form
y
(
n
)
(
x
) =
F
(
x,y,y
(1)
,y
(2)
,... ,y
(
n
−
1)
)
,
(Standard
n
th order ODE)
Here
F
is some function of
n
+ 1 variables. If the derivatives are written in Leibnitz notation, the
same equation looks like
d
n
y
dx
n
=
F
(
x,y,
dy
dx
,
d
2
y
dx
2
,... ,
d
n
−
1
y
dx
n
−
1
)
,
(Leibnitz
n
th order ODE)
Solving
such an equation means ±nding a function
y
that makes the equation true.
Example.
If
n
= 1, the equation above becomes
y
′
=
F
(
x,y
), the familiar standard form for a
±rstorder ODE. In this case, in order to solve the equation, we need some additional information:
an
initial condition
y
(
x
0
) =
y
0
.
This example suggests that solving (Standard
n
th order ODE) will also require some additional
information. What additional information? The punch line is that we need to know some “starting
point”
x
0
, and
n
numbers: the value of
y
at
x
0
, and also the values of the ±rst
n

1 derivatives at
x
0
. That is, we need to be given
y
(
x
0
) =
y
0
, y
(1)
(
x
0
) =
y
(1)
0
,... ,y
(
n
−
1)
(
x
0
) =
y
(
n
−
1)
0
.
(Initial conditions)
I’ll give two reasons that this is reasonable extra information to ask for. The ±rst reason is
computational: using (Initial conditions), it’s possible to compute approximate numerical solutions
to (Standard
n
th order ODE). The second reason is physical: in some large class of examples of
physical systems described by a diFerential equation, (Initial conditions) is exactly what you should
expect to know at the beginning in order to predict the future behavior of the system.
Interlude from mathematics: in the theoretical study of diFerential equations, one seeks to prove mathematical
theorems shaped like this: “there is a unique function
y
de±ned near
x
0
and satisfying both (Standard
n
th order
ODE) and (Initial conditions).” If you can prove such a theorem, then the conclusion is that you were asking a
reasonable question. Such theorems can be motivated by examples from physics, but examples from physics can