It has been shown that the equation (*)
\[
y
′
+
a
(
t
)
y
+
b
(
t
)
y
′
+
c
(
t
)
y
=
0
,
y’ + a(t)y + b(t)y’ + c(t)y = 0,
\]
where
a
,
b
a,b
, and
c
c
are real-valued continuous functions on
[
α
,
∞
)
[\alpha ,\infty )
such that
a
(
t
)
≥
0
,
b
(
t
)
≤
0
a(t) \geq 0,b(t) \leq 0
, and
c
(
t
)
>
0
c(t) > 0
, admits at most one solution
y
(
t
)
y(t)
(neglecting linear dependence) with the property
y
(
t
)
y
′
(
t
)
>
0
,
y
(
t
)
y
(
t
)
>
0
y(t)y’(t) > 0,y(t)y(t) > 0
for
t
∈
[
α
,
∞
)
t \in [\alpha ,\infty )
and
lim
t
→
∞
y
(
t
)
=
0
{\lim _{t \to \infty }}y(t) = 0
, if (*) has an oscillatory solution. Further, sufficient conditions have been obtained so that (*) admits an oscillatory solution.