Initial value problem for differential equation of fractional order with variable coefficients and with variable delay

In this paper, for a linear ordinary differential equation with variable coefficients, with a Dzhrbashyan - Nersesyan fractional differentiation operator of the first order and with variable delay, the method of steps for solving the initial problem is implemented. Fractional operators take into account the history of the process under consideration. There is also a time delay during the processes. The delay occurs because there is always a time duration for some processes. Therefore, differential equations containing both a fractional derivative and an delay argument are more realistic when describing mathematical models of various processes. The equation under study is equivalently reduced to a Volterra integral equation of the second kind. The general representation of the solution is explicitly written out.