The expectation of a solution of a linear system of differential equations with random coefficients

We consider a linear inhomogeneous system of differential equations of special form with three random coefficients defined by characteristic functionals. Operator functions generated by the functionals are introduced. The problem of finding the expectation of a solution of the Cauchy problem is reduced to the study of an auxiliary deterministic system of differential equations involving ordinary and variational derivatives. The solution of the resulting equation is written in terms of operator functions generated by the functionals. We derive explicit formulas for the expectation of the solution with uniformly distributed random coefficients, random Laplace coefficients, and Gaussian random coefficients.