Differential equations of second order with impulses at random moments
are set up and investigated in this paper. The main characteristic of
the studied equations is that the impulses occur at random moments which
are exponentially distributed random variables. The presence of random
variables in the ordinary differential equation leads to a total change
of the behavior of the solution. It is not a function as in the case of
deterministic equations, it is a stochastic process. It requires
combining of the results in Theory of Differential Equations and
Probability Theory. The initial value problem is set up in appropriate
way. Sample path solutions are defined as a solutions of ordinary
differential equations with determined fixed moments of impulses.
P-moment generalized exponential stability is defined and some
sufficient conditions for this type of stability are obtained. The study
is based on the application of Lyapunov functions. The results are
illustrated on example.