Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution

This paper solves the pantograph delay equation which includes inhomogeneous term. The inhomogeneous term is in the form of a class of exponential functions. An efficient transformation is introduced to reduce the inhomogeneous pantograph delay differential equation (IPDDE) to the homogeneous pantograph delay differential equation (HPDDE), which is a homogeneous model. It is found that the solution of the IPDDE depends mainly on the solution of HPDDE. It is well-known in the literature that the analytical solution of the HPDDE is already obtained in a closed series form. Such ready solution of the HPDDE is invested in this paper and accordingly, the solution of the IPDDE under consideration is established. Also, several exact solutions of the present model are determined at specific conditions and values of the parameters. The solutions of some examples in the literature are obtained in exact forms as special cases of the current results. Moreover, the properties of the obtained solutions are theoretically and graphically addressed.