Let
y
1
y_1
and
y
2
y_2
be principal and nonprincipal solutions of the nonoscillatory differential equation
(
r
(
t
)
y
′
)
′
+
f
(
t
)
y
=
0
(r(t)y’)’+f(t)y=0
. In an earlier paper we showed that if
∫
∞
(
f
−
g
)
y
1
y
2
d
t
\int ^\infty (f-g)y_1y_2\,dt
converges (perhaps conditionally), and a related improper integral converges absolutely and sufficently rapidly, then the differential equation
(
r
(
t
)
x
′
)
′
+
g
(
t
)
x
=
0
(r(t)x’)’+g(t)x=0
has solutions
x
1
x_1
and
x
2
x_2
that behave asymptotically like
y
1
y_1
and
y
2
y_2
. Here we consider the case where
∫
∞
(
f
−
g
)
y
2
2
d
t
\int ^\infty (f-g)y_2^2\,dt
converges (perhaps conditionally) without any additional assumption requiring absolute convergence.