Abstract: A set of new linear multistep method of order three and four with extra derivatives are developed for solving special second order ordinary differential equations. The extra derivatives are incorporated into the methods, so that, a more accurate numerical results can be obtained. The methods are developed using the sequence of Chebyshev polynomials as the basis function. The methods are then trigonometrically-fitted, so that, they are suitable for solving highly oscillatory problems arise from the special second order ordinary differential equations. Numerical experiments are carried out to show the efficiency and accuracy of the new methods in comparison with methods in the literature.