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)−λq(x))y=0. These methods are derived from the formulah4y1(4)=(δ4−16δ6+7240δ8−…)yi.Numerical results are included to demonstrate practical usefulness of our methods.