The objective of this thesis is a study of Delay Differential Equations as well as, their numerical solution. The modification was made by introducing the linear multistep methods, which is for solving numerically ordinary Differential Equations, and Delay Differential Equations. Also, one of themes of this thesis is to reduce the error propagation which is accumulated by using and applying the successive application of the linear multistep methods, this had been done by using one of the popular methods in reducing error constant which is the variable step-size method, which is new in the present subject up to our knowledge. Finally, the suggested approach of the numerical solution of the delay differential equation by multistep method is applied to the population growth model.