Consider an ordinary linear differential operator
L
L
, of order
n
⩾
1
n \geqslant 1
, represented by
L
u
≡
a
n
(
t
)
u
(
n
)
+
⋯
+
a
0
(
t
)
u
∀
u
∈
C
n
(
a
,
b
)
Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)
, with real-valued coefficients
a
k
∈
C
k
(
a
,
b
)
{a_k} \in {C^k}(a,b)
,
0
⩽
k
⩽
n
0 \leqslant k \leqslant n
,
a
n
≠
0
{a_n} \ne 0
on
(
a
,
b
)
(a,b)
. According to a classical result, if
L
L
is formally selfadjoint on
(
a
,
b
)
(a,b)
then it has a factorization of the type
L
u
≡
p
n
[
p
n
−
1
(
⋯
(
p
1
(
p
0
u
)
′
)
′
⋯
)
′
]
′
∀
u
∈
C
n
(
a
,
b
)
Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)’)’ \cdots )’]’\forall u \in {C^n}(a,b)
, where the
p
k
{p_k}
’s are sufficiently-smooth and everywhere nonzero functions on
(
a
,
b
)
(a,b)
such that
p
k
=
p
n
−
k
{p_k} = {p_{n - k}}
(
k
=
0
,
…
,
n
)
(k = 0, \ldots ,n)
. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the
p
k
{p_k}
’s at the endpoints of
(
a
,
b
)
(a,b)
which may prove useful in applications.