Solving Delay Differential Equations Using the Method of Steps

Ahmad M. D. Al-Eybani

Abstract: Delay differential equations (DDEs) are a class of differential equations where the derivative of the unknown function at a given time depends not only on the current state but also on its values at previous times. These equations arise in numerous fields, including biology, engineering, and economics. Unlike ordinary differential equations (ODEs), DDEs incorporate time delays, making their solution more complex due to the need for a history function. A common form of a first-order linear DDE with a constant delay is:

dy(t)dt=-ay(t)+by(t-τ),

where y(t) is the unknown function, a and b are constants, and τ>0 is the time delay. Solving DDEs analytically or numerically requires specialized techniques, and one of the most straightforward and intuitive methods for constant-delay DDEs is the method of steps. This approach breaks the problem into sequential intervals, solving an ODE in each interval using the solution from the previous interval as a history function. In this article, we explore the method of steps in detail, including its mathematical derivation, assumptions, practical implementation, and an illustrative example.