Advanced Study on the Delay Differential Equation y ′( t ) = ay ( t ) + by ( ct )

Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation y ′ ( t ) = a y ( t ) + b y c t belongs to such a set of delay differential equations. To the authors’ knowledge, there are no standard methods to solve the delay differential equations, i.e., unlike the ordinary differential equations, for which numerous and standard methods are well-known. In this paper, the Adomian decomposition method is suggested to analyze the pantograph delay differential equation utilizing two different canonical forms. A power series solution is obtained through the first canonical form, while the second canonical form leads to the exponential function solution. The obtained power series solution coincides with the corresponding ones in the literature for special cases. Moreover, several exact solutions are derived from the present power series solution at a specific restriction of the proportional delay parameter c in terms of the parameters a and b . The exponential function solution is successfully obtained in a closed form and then compared with the available exact solutions (derived from the power series solution). The obtained results reveal that the present analysis is efficient and effective in dealing with pantograph delay differential equations.