It is shown that the disconjugate equation (1)
L
x
≡
(
1
/
β
n
)
(
d
/
d
t
)
⋅
(
1
/
β
n
−
1
)
⋯
(
d
/
d
t
)
(
1
/
β
1
)
(
d
/
d
t
)
(
x
/
β
0
)
=
0
Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0
, a
>
t
>
b
> t > b
, where
β
i
>
0
{\beta _i} > 0
, and (2)
β
i
∈
C
(
a
,
b
)
{\beta _i} \in C(a,b)
, can be written in essentially unique canonical forms so that
∫
b
β
i
d
t
=
∞
(
∫
a
β
i
d
t
=
∞
)
{\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )
for
1
≤
i
≤
n
−
1
1 \leq i \leq n - 1
. From this it follows easily that (1) has solutions
x
1
,
…
,
x
n
{x_1}, \ldots ,{x_n}
which are positive in (a, b) near
b
(
a
)
b(a)
and satisfy
lim
t
→
b
−
x
i
(
t
)
/
x
j
(
t
)
=
0
(
lim
t
→
a
+
x
i
(
t
)
/
x
j
(
t
)
=
∞
)
{\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )
for
1
≤
i
>
j
≤
n
1 \leq i > j \leq n
. Necessary and sufficient conditions are given for (1) to have solutions
y
1
,
…
,
y
n
{y_1}, \ldots ,{y_n}
such that
lim
t
→
b
−
y
i
(
t
)
/
y
j
(
t
)
=
lim
t
→
a
+
y
j
(
t
)
/
y
i
(
t
)
=
0
{\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0
for
1
≤
i
>
j
≤
n
1 \leq i > j \leq n
. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition
(
2
′
)
β
i
∈
C
(
n
−
i
)
(
a
,
b
)
(2’){\beta _i} \in {C^{(n - i)}}(a,b)
.