The differential equations with a deviating argument are differential equations in which the unknown function and its derivative enter, generally speaking, under different values of the argument, [El'sgol'ts L. and Norkin S., 1973].These equations appeared in the literature in the second half of the eighteenth century by Kondorse in 1771, but a systematic study of equations with a deviating argument began only in the twentieth century (especially in the last forty years by Myshkis A. in the Soviet Union, Wright E. and Bellman R. in other countries) in connection with the requirements of applied science, [El'sgol'ts L. and Norkin S., 1973].A topic of differential equations with deviating arguments which is in a rapid state of development. It was the Russian mathematician Krasovskii who found an accommodation for differential equations with deviating arguments as operators in function spaces. It is worth noting that the theory of differential equations with deviating arguments is not just a simple extension of the theory of ordinary differential equations, [Saaty T., 1967].The differential equations with deviating arguments are integrable in closed form only under very specialized circumstances, and therefore qualitative and approximate methods are of the utmost importance in studying them, [El'sgol'ts L., 1964].Many researchers study the delay differential equations:
Al-Saady A., 2000, gave a new approach for solving the delay differential equations. This approach depends mainly on the Gaussian quadrature numerical integration method and cubic spline interpolator functions for the unknown exact solution, Narie N., 2001, introduced the variational formulations of the delay differential equations and solved them by using the direct Ritz method,Al. Daynee K., 2002, evaluated the variational formulation of the delay BVPs, using two approaches (variational problem with constraint and variational problem using Rayleigh quotient formula), as well as, the equivalence between the solution of the original problem in operator form 
and the variational problem have been proved.Salih S., 2004, studied and modified some numerical and approximate methods for solving the nth order linear delay differential equations with constant coefficients, and Al-Kubeisy S., 2004, solved the delay differential equations numerically by using the linear multistep methods.The main purpose of this work is to derive an estimate of the magnitude of the solutions for special types of ordinary and partial delay differential equations in order to find them by any suitable methods, say the Laplace transform method.
