Abstract
In this paper, the pantograph delay differential equation y'(t)=ay(t)+byct subject to the condition y(0)=lambda is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for lambda=1, is well-posed if c < 1, but not if c > 1. In addition, the solution is available in the form of a standard power series when lambda=1. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c is an element of(-1,1) such that |b/a| < 1 and it also converges for c > 1 when a < 0. For c=-1, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2 pi/root b(2)-a(2) when b > a. The current results are introduced for the first time and have not been reported in the relevant literature.