Abstract In this work, we study the full randomized versions of Airy, Hermite and Laguerre differential equations, which depend on a random variable appearing as an equation coefficient as well as two random initial conditions. In previous contributions, the mean square stochastic solutions to the aforementioned random differential equations were constructed via the Fröbenius method, under the assumption of exponential growth of the absolute moments of the equation coefficient, which is equivalent to its essential boundedness. In this paper we aim at relaxing the boundedness hypothesis to allow more general probability distributions for the equation coefficient. We prove that the equations are solvable in the mean square sense when the equation coefficient has finite moment-generating function in a neighborhood of the origin. A thorough discussion of the new hypotheses is included.

中文翻译：

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具有不确定性的 Airy、Hermite 和 Laguerre 微分方程的随机系数有界假设之外

摘要 在这项工作中，我们研究了 Airy、Hermite 和 Laguerre 微分方程的完全随机版本，它们依赖于作为方程系数出现的随机变量以及两个随机初始条件。在之前的贡献中，上述随机微分方程的均方随机解是通过Fröbenius方法构建的，假设方程系数的绝对矩呈指数增长，这相当于其本质有界。在本文中，我们旨在放宽有界假设，以允许方程系数的更一般的概率分布。我们证明当方程系数在原点邻域内具有有限矩生成函数时，方程在均方意义上是可解的。