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CHAPTER 6
Linear Diﬀerential Systems
6.1 HigherOrder Linear Diﬀerential Equations
This section is a continuation of Chapter 3. As you will see, all of the “theory” that we
developed for secondorder linear diﬀerential equations carries over, essentially verbatim, to
linear diﬀerential equations of order greater than two.
Recall that a Frst order, linear diﬀerential equation is an equation which can be written
in the form
y
±
+
p
(
x
)
y
=
q
(
x
)
where
p
and
q
are continuous functions on some interval
I
. A second order, linear
diﬀerential equation has an analogous form.
y
±±
+
p
(
x
)
y
±
+
q
(
x
)
y
=
f
(
x
)
where
p, q
, and
f
are continuous functions on some interval
I
.
In general, an
n
th
order linear diﬀerential equation
is an equation that can be written
in the form
y
(
n
)
+
p
n

1
(
x
)
y
(
n

1)
+
p
n

2
(
x
)
y
(
n

2)
+
···
+
p
1
(
x
)
y
±
+
p
0
(
x
)
y
=
f
(
x
)
(L)
where
p
0
,p
1
,.
.
.
n

1
, and
f
are continuous functions on some interval
I
. As before,
the functions
p
0
1
.
.
n

1
are called the
coeﬃcients
, and
f
is called the
forcing
function
or the
nonhomogeneous term
.
Equation (L) is
homogeneous
if the function
f
on the right side is 0 for all
x
∈
I
.In
this case, equation (L) becomes
y
(
n
)
+
p
n

1
(
x
)
y
(
n

1)
+
p
n

2
(
x
)
y
(
n

2)
+
+
p
1
(
x
)
y
±
+
p
0
(
x
)
y
= 0
(H)
Equation (L) is
nonhomogeneous
if
f
is not the zero function on
I
, i.e., (L) is nonhomoge
neous if
f
(
x
)
±
= 0 for some
x
∈
I
. As in the case of second order linear equations, almost
all of our attention will be focused on homogeneous equations.
Remarks on “Linear.”
Intuitively, an
n
th
order diﬀerential equation is linear if
y
and
its derivatives appear in the equation with exponent 1 only, and there are no socalled
”crossproduct” terms,
yy
±
,yy
±±
,y
±
y
±±
, etc.
If we set
L
[
y
]=
y
(
n
)
+
p
n

1
y
(
n

1)
+
+
p
1
(
x
)
y
±
+
p
0
(
x
)
y
, then we can view
L
as an
“operator” that transforms an
n
times diﬀerentiable function
y
=
y
(
x
) into the continuous
function
L
[
y
(
x
)] =
y
(
n
)
(
x
)+
p
n

1
(
x
)
y
(
n

1)
(
x
+
p
1
(
x
)
y
±
(
x
p
0
(
x
)
y
(
x
)
.
251
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View Full DocumentIt is easy to check that, for any two
n
times diﬀerentiable functions
y
1
(
x
) and
y
2
(
x
),
L
[
y
1
(
x
)+
y
2
(
x
)] =
L
[
y
1
(
x
)] +
L
[
y
2
(
x
)]
and, for any
n
times diﬀerentiable function
y
and any constant
c
,
L
[
cy
(
x
)] =
cL
[
y
(
x
)]
.
Therefore, as introduced in Section 2.1,
L
is a
linear diﬀerential operator.
This is the real
reason that equation (L) is said to be a
linear
diﬀerential equation.
±
THEOREM 1. (Existence and Uniqueness Theorem)
Given the
n
th
 order linear
equation (L). Let
a
be any point on the interval
I
, and let
α
0
,α
1
,.
.
.
n

1
be any
n
real numbers. Then the initialvalue problem
y
(
n
)
+
p
n

1
(
x
)
y
(
n

1)
+
p
n

2
(
x
)
y
(
n

2)
+
···
+
p
1
(
x
)
y
±
+
p
0
(
x
)
y
=
f
(
x
);
y
(
a
)=
α
0
,y
±
(
a
α
1
.
.
(
n

1)
(
a
α
n

1
has a unique solution.
Remark:
We can solve any Frst order linear diﬀerential equation, see Section 2.1. In
contrast,
there is no general method for solving second or higher order linear diﬀerential
equations
. However, as we saw in our study of second order equations, there are methods
for solving certain special types of higher order linear equations and we shall look at these
later in this section.
±
Homogeneous Equations
y
(
n
)
+
p
n

1
(
x
)
y
(
n

1)
+
p
n

2
(
x
)
y
(
n

2)
+
+
p
1
(
x
)
y
±
+
p
0
(
x
)
y
=0
.
(H)
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 Fall '08
 morgan
 Math, Equations

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