This paper describes some new finite difference methods for the approximation of eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation of the type
(
p
y
″
)
′
′
−
(
q
y
′
)
′
+
(
r
−
λ
s
)
y
=
0
. The smallest positive eigenvalue of some typical eigensystems is computed to demonstrate the practical usefulness of the numerical techniques developed.