A note on the mean-square solution of the hypergeometric differential equation with uncertainties

The Fröbenius method of power series has been applied to several linear random differential equations. The interest relies on the derivation of a closed-form mean-square solution and on the possibility of approximating statistical measures at exponential convergence rate. In this paper, we deal with the hypergeometric differential equation with random coefficients and initial conditions. On the interval (0, 1), random power series centered at the regular singular point 0 are employed, which are given in terms of the hypergeometric function. We find the stochastic basis of mean-square solutions and solve random initial-value problems. The approximation of the expectation and the variance is studied and illustrated computationally.