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This paper concerns the analysis of random second order linear differentialequations. Usually, solving these equations consists of computing the firststatistics of the response process, and that task has been an essential goal inthe literature. A more ambitious objective is the computation of the solutionprobability density function. We present advances on these two aspects in thecase of general random non-autonomous second order linear differentialequations with analytic data processes. The Fr\"obenius method is employed toobtain the stochastic solution in the form of a mean square convergent powerseries. We demonstrate that the convergence requires the boundedness of therandom input coefficients. Further, the mean square error of the Fr\"obeniusmethod is proved to decrease exponentially with the number of terms in theseries, although not uniformly in time. Regarding the probability densityfunction of the solution at a given time, we rely on the law of totalprobability to express it in closed-form as an expectation. For the computationof this expectation, a sequence of approximating density functions isconstructed by reducing the dimensionality of the problem using the truncatedpower series of the fundamental set. We prove several theoretical resultsregarding the pointwise convergence of the sequence of density functions andthe convergence in total variation. The pointwise convergence turns out to beexponential under a Lipschitz hypothesis. As the density functions areexpressed in terms of expectations, we propose a symbolic Monte Carlo samplingalgorithm for their estimation. This algorithm is implemented and applied onseveral numerical examples designed to illustrate the theoretical findings ofthe paper.

Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function