The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

This work presents the geometrical formulation of variational principle and uses exterior calculus characterized by differential forms, smooth manifolds, and the theory of fiber bundle to deduce variational principle in particle field, and we show the difference between classical form and new form.