Solving a random differential equation means to obtain an exact or
approximate expression for the solution stochastic process, and to compute its
statistical properties, mainly the mean and the variance functions. However, a
major challenge is the computation of the probability density function of the
solution. In this article we construct reliable approximations of the probability
density function to the randomized non-autonomous complete linear differential equation by assuming that the diffusion coefficient and the source term are
stochastic processes and the initial condition is a random variable. The key
tools to construct these approximations are the random variable transformation technique and Karhunen-Lo`eve expansions. The study is divided into a
large number of cases with a double aim: firstly, to extend the available results
in the extant literature and, secondly, to embrace as many practical situations
as possible. Finally, a wide variety of numerical experiments illustrate the
potentiality of our findings
