1.1: Differential Equations and Mathematical Models
±
Basic Definitions
An ordinary
differential
equation
is an equation that relates an unknown function, one independent variable, and one or more of the
functions derivatives.
A partial
differential
equation
is a differential euqation which has two or more independent variables.
Notation
:
F
H
x
,
y
H
x
L
,
y
'
H
x
L
,
y
''
H
x
L
, ...,
y
^
H
n
LL
= 0
This represents an
n
th
order differential equation with independent variable
x
.
y
H
n
L
H
x
L
=
G
I
x
,
y
H
x
L
,
y
'
H
x
L
, ...,
y
I
n

1
M
H
x
LM
This is called the normal
form
for a
n
th
order differential equation with independent variable
x
.
Initial
Value
Problems
are differential equations with extra information or initial conditions:
y
H
a
L
=
b
0
,
y
'
H
a
L
=
b
1
,
y
''
H
a
L
=
b
2
, ...,
y
I
n

1
M
H
a
L
=
b
n

1
.
A s
o
l
u
t
i
o
n
to
F
H
x
,
y
H
x
L
,
y
'
H
x
L
,
y
''
H
x
L
, ...,
y
^
H
n
LL
= 0 is a function
u
H
x
L
that is continuous on an interval
I
such that
u
',
u
'', .
..,
u