On the mean‐square solution to the Legendre differential equation with random input data

Abstract

In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conjecture posed in a recent paper, concerning the power‐series solution in a Lebesgue sense. By using the finiteness of the moment‐generating function of the equation's coefficient, mean‐square convergence is derived.