  In this paper, we consider a stochastic system described by a differential
equation admitting a spatially varying random coefficient.
  The differential equation has been employed to model various static physics
systems such as elastic deformation, water flow, electric-magnetic fields,
temperature distribution, etc.
  A random coefficient is introduced to account for the system's uncertainty
and/or imperfect measurements.
  This random coefficient is described by a Gaussian process (the input
process) and thus the solution to the differential equation (under certain
boundary conditions) is a complexed functional of the input Gaussian process.
  In this paper, we focus the analysis on the one-dimensional case and derive
asymptotic approximations of the tail probabilities of the solution to the
equation that has various physics interpretations under different contexts.
  This analysis rests on the literature of the extreme analysis of Gaussian
processes (such as the tail approximations of the supremum) and extends the
analysis to more complexed functionals.
