The Sturm-Liouville equation
−
(
p
y
′
)
′
+
q
y
=
λ
r
y
on
[
0
,
l
]
\begin{equation*} -(py’)’ + qy =\lambda ry \;\; \text {on}\;\; [0,l] \end{equation*}
is considered subject to the boundary conditions
y
(
0
)
cos
α
a
m
p
;
=
(
p
y
′
)
(
0
)
sin
α
,
y
(
l
)
cos
β
a
m
p
;
=
(
p
y
′
)
(
l
)
sin
β
.
\begin{align*} y(0)\cos \alpha &= (py’)(0)\sin \alpha ,\\ y(l)\cos \beta &= (py’)(l)\sin \beta . \end{align*}
We assume that
p
p
is positive and that
p
r
pr
is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to
O
(
1
/
n
)
O(1/\sqrt {n})
for
λ
n
\sqrt {\lambda _n}
, or equivalently up to
O
(
n
)
O(\sqrt {n})
for
λ
n
\lambda _n
, the eigenvalues of the above boundary value problem.