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.