Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay

We randomize the following class of linear differential equations with delay, x(tau)' (t) = ax(tau) (t) bx(tau) (t -tau), t> 0, and initial condition, x(tau )(t) = g(t), -tau <= t <= 0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic process g(t), and then we solve, from a probabilistic point of view, both random initial value problems by determining explicit expressions to the first probability density function, f(x, t; tau), of the corresponding solution stochastic processes. Afterwards, we establish sufficient conditions on the involved random input parameters in order to guarantee that f(x, t; tau) con- verges, as tau -> 0(+), to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (tau = 0). The paper concludes with several numerical experiments illustrating our theoretical findings. (C) 2019 Published by Elsevier Inc.