A novel analytical treatment for the Ambartsumian delay differential equation with a variable coefficient

The Ambartsumian delay differential equation with a variable coefficient is considered in this paper. An effective transformation is produced to convert the extended Ambartsumian equation to the pantograph model. Two kinds of analytical solutions are determined. The first solution is expressed as an exponential function multiplied by an infinite power series. The second solution is obtained as an infinite series in terms of exponential functions. Several exact solutions are established for different forms of the extended Ambartsumian equation under specific relations. In addition, the convergence analysis is addressed theoretically. Moreover, numeric calculations are conducted to estimate the accuracy. The results reveal that the present analysis is efficient and accurate and can be further applied to similar delay models in a straightforward manner.