## 8.5. The Existence-Uniqueness Theorem

### Theorem 8.3:
Existence-Uniqueness Theorem.

Let
P
1
,⋯,
P
n
be continuous functions on an open interval *J* and let *L* be the linear differential operator

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

where
D
(
0
)
=1
and
P
0
=1
. Let
x
0
∈J
and let
k
0
,
k
1
,⋯,
k
n−1
be *n*
given real numbers. Then there exists 1
and only 1 function
y=f(
x
)
that satisfies both the homogeneous equation
L(
y
)=0
on *J*
and the initial conditions

f
(
j
)
(
x
0
)=
k
j
for
j=0,1,⋯,n−1

#### Proof

Proof of this will be obtained as a corollary of a more
general theorem for systems of differential equations to be discussed in
Chapter 10. A specialized version for
LDEs with constant coefficients is given in §9.9.

### Comment

The initial conditions can be represented
by a vector in **R**^{n} with components

(
f(
x
0
),
f
′
(
x
0
),⋯,
f
(
n−1
)
(
x
0
)
)

This is also called the **initial- value vector** of *f* at *x*_{0}. The uniqueness theorem then implies only
f(
x
)=0
can satisfy an initial-value vector that
equals to *O*.