## 8.4. Linear Differential Equations Of Order *N*

Discussions of LDEs are more easily carried
in the operator notations.

Let
C(
J
)
be the linear space of all real-valued
functions continuous on an interval *J*,
which can be unbounded. Let
C
(
n
)
(
J
)
be the subspace consisting of all real
functions whose first *n* derivatives
are also continuous on *J*. Let
P
1
,⋯,
P
n
be *n*
given functions in
C(
J
)
. We define an operator *L* as the transformation

L:
C
(
n
)
(
J
)→C(
J
)
by
f↦L(
f
)=
f
(
n
)
+
∑
k=1
n
P
k
f
(
n−k
)
=
∑
k=0
n
P
k
f
(
n−k
)

where
P
0
=1
and
f
(
0
)
=f
. Introducing the derivation operators
D
k
,
we have

L=
D
n
+
∑
k=1
n
P
k
D
n−k
=
∑
k=0
n
P
k
D
n−k

where
D
0
=1
. A LDE of order *n* has the form

L(
y
)=R
(8.6)

where *R*
is some known function on *J*. A solution of the LDE is any function in
C
(
n
)
(
J
)
that satisfies (8.6) on *J*.

Using the linearity of *D*, it is easy to show that every *D*^{k}, and hence *L*,
are also linear. Hence, *L* defined above is called a **differential operator of order ***n*.

With each **nonhomogenous equation**
L(
y
)=R
is associated a **homogeneous equation**
L(
y
)=0
. Let *h*
be the most general solution of the homogeneous equation and *p* be any particular solution of the
nonhomogeneous equation. Using the
linearity of *L*, it is easily shown
that the most general solution of the nonhomogeneous equation is given by
h+p
.

The set of all solutions of the homogeneous
equation is the null space
N(
L
)
,
also called the **solution space** of
L(
y
)=0
. The solution space is a subspace of
C
(
n
)
(
J
)
. Although
dim[
C
(
n
)
(
J
)
]
may be infinite,
dim[
N(
L
)
]
is always finite. In fact,
dim[
N(
L
)
]=n
,
the order of *L* [see §8.5].