SECTION 4.1 Nth ORDER LINEAR STUFF
THE OPERATOR.
L
[
y
] =
y
(
n
)
+
p
1
(
t
)
y
(
n

1)
+
p
2
(
t
)
y
(
n

2)
+
· · ·
+
p
n
(
t
)
y
, or
L
[
y
] = (
D
n
+
p
1
(
t
)
D
n

1
+
p
2
(
t
)
D
n

2
+
· · ·
+
p
n

1
(
t
)
D
+
p
n
(
t
))[
y
]
.
PROPERTIES OF THE OPERATOR.
First, as for order two, the operator
L
is linear,
that is,
L
[
c
1
y
1
+
c
2
y
2
+
· · ·
c
n
y
n
] =
c
1
L
[
y
1
] +
c
2
L
[
y
2
] =
· · ·
c
n
L
[
y
n
]
.
Second, operators with constant coefficients behave just like polynomials. To see what
this means, let’s compute (3
D
+ 2)[(2
D

1)[
y
]].
INITIAL CONDITIONS.
y
(
t
0
) =
y
0
,
y
0
(
t
0
) =
y
1
, . . . ,
y
(
n

1)
(
t
0
) =
y
n

1
EXISTENCE AND UNIQUENESS.
When
p
1
(
t
)
, p
2
(
t
)
, . . . p
n
(
t
) and
g
(
t
) are all con
tinuous on an interval I and
t
0
is any point in I, then there is a unique function
y
(
t
) that
satisfies the initial conditions above and the differential equation
L
[
y
] =
g
(
t
) on I, and this
function
y
(
t
) is defined throughout the interval I.
GENERAL SOLUTIONS, HOMOGENEOUS CASE.
Solutions
y
1
(
t
)
, y
2
(
t
)
, . . . y
n
(
t
)
of
L
[
y
] = 0 form a
fundamental set
of solutions on an interval
I
if every solution on the
interval
I
can be written as
c
1
y
1
+
c
2
y
2
+
. . .
+
c
n
y
n
for some choice of constants
c
1
, c
2
, . . . , c
n
.
Note that a fundamental set of solutions has as many functions as the order of the differential
equation. If the Wronskian of
n
solutions is not zero somewhere in the interval I, then the
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 Spring '08
 Fonken
 Differential Equations, Equations, Derivative, fundamental set

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