## 8.6. The Dimension Of The Solution Space Of A Homogeneous Linear
Differential Equation

### Theorem 8.4:
Dimensionality Theorem.

Let
L:
C
(
n
)
(
J
)→C(
J
)
be a linear differential operator of order *n* given by

L=
D
n
+
∑
k=1
n
P
k
D
n−k
(8.7)

Then the solution space of the homogeneous
equation
L(
y
)=0
has dimension *n*.

#### Proof

Let

T:N(
L
)→
R
n
by
f↦T(
f
)=(
f(
x
0
),
f
′
(
x
0
),⋯,
f
(
n−1
)
(
x
0
)
)

be the linear transformation that maps each
function *f* in the solution space
N(
L
)
to the initial- value vector of *f* at
x
0
∈J
. By the uniqueness theorem,
T(
f
)=O
implies
f=0
. Hence, by Theorem 4.10, *T* is 1-1 on
N(
L
)
so that
T
−1
exists and is also 1-1. Furthermore, since every element in **R**^{n}
can be an initial-value vector, the map is onto. By Theorem 4.11,
dimN(
L
)=
R
n
=n
.

### Theorem 8.5.

Let
L:
C
(
n
)
(
J
)→C(
J
)
be a linear differential operator of order *n*.
Let
u
1
,⋯,
u
n
be *n*
independent
solutions of the homogeneous equation
L(
y
)=0
on *J*. Then every solution
y=f(
x
)
on *J*
can be expressed as a linear combination

f(
x
)=
∑
k=1
n
c
k
u
k
(
x
)
(8.8)

where *c*_{k}
are constants.

#### Proof

This is a corollary of Theorem 8.4.

### Comment

Eq(8.8) is called a **general solution** of
L(
y
)=0
.

The dimensionality theorem asserts that an *n*th order LDE always has *n* independent
solutions. There is however no general
method for obtaining these solutions, except for a few special types of
equations, e.g., those with constant coefficients.