The linear differential equation (1)
(
r
(
t
)
x
′
)
′
+
(
f
(
t
)
+
q
(
t
)
)
x
=
0
(r(t)x’)’ + (f(t) + q(t))x= 0
is viewed as a perturbation of the equation (2)
(
r
(
t
)
y
′
)
′
+
(
f
(
t
)
y
=
0
(r(t)y’)’ + (f(t)y = 0
, where
r
>
0
r > 0
,
f
f
and
q
q
are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and
y
1
{y_1}
,
y
2
{y_2}
are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution
x
1
{x_1}
of (1). Under some mild assumptions, we characterize that (1) has a principal solution
x
1
{x_1}
satisfying
x
1
=
y
1
(
1
+
o
(
1
)
)
{x_1}= {y_1}(1 + o(1))
. Sufficient (sometimes necessary) conditions under which the nonprincipal solution
x
2
{x_2}
of (1) behaves, in three different degrees, like
y
2
{y_2}
as
t
→
∞
t \to \infty
are also established.