Insight on the Nonhomogeneous Pantograph Equation with an Arbitrary Polynomial of Degree n: Explicit Solution

This paper solves a nonhomogeneous version of the pantograph equation. The nonhomogeneous term is taken as a polynomial of degree n with arbitrary coefficients. The nonhomogeneous pantograph equation is successfully converted to the standard homogeneous version by means of a simple transformation. An explicit formula is derived for the coefficients of the assumed transformation. Accordingly, the solution of the nonhomogeneous version is obtained in different forms in terms of power series, in addition to exponential functions. The obtained solution in power-series form is investigated to produce exact solutions for several examples under specific relationships between the involved parameters. In addition, exact solutions in terms of trigonometric and hyperbolic functions are determined at a certain value of the proportional delay parameter. The obtained results may be reported for the first time for the present nonhomogeneous version of the pantograph equation and can be further applied to include other versions with different nonhomogeneous terms.