In this paper it is proved that the two-point boundary value problem, namely
(
d
(
4
)
/
d
x
4
+
f
)
y
=
g
,
y
(
0
)
−
A
1
=
y
(
1
)
−
A
2
=
y
(
0
)
−
B
1
=
y
(
1
)
−
B
2
=
0
({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y(0) - {B_1} = y(1) - {B_2} = 0
, has a unique solution provided
inf
x
f
(
x
)
=
−
η
>
−
π
4
{\inf _x}f(x) = - \eta > - {\pi ^4}
. The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the
L
2
{L_2}
-norm of a vector.