This brief report describes some new finite difference methods of order
2
and
4
for computing eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation
y
(
4
)
+
(
p
(
x
)
−
λ
q
(
x
)
)
y
=
0
. These methods are derived from the formula
h
4
y
1
(
4
)
=
(
δ
4
−
1
6
δ
6
+
7
240
δ
8
−
…
)
y
i
.
Numerical results are included to demonstrate practical usefulness of our methods.