This paper 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 differential equation of the form
(
p
y
″
)
′
′
+
(
q
−
λ
r
)
y
=
0
. Numerical results for two typical eigenvalue problems are tabulated to demonstrate practical usefulness of our methods.