  The paper considers the mathematical formulation of the problem of transverse
oscillations of a vertical rod under gravity, friction and external pulse
effect, leading to thermal expansion of the rod. The dynamics of the system
under consideration corresponds to the behavior of a fuel element (FE) in a
pulsed reactor and is related to the dynamic stability of the processes
occurring in it. The FE dynamics is described by inhomogeneous linear
differential equation of the fourth order with non-constant coefficients.
Initial boundary conditions are not smooth since they correspond to the instant
heating of the part of the FE surface exposed to neutron pulse radiation. In
the future the obtained exact solutions will be used as input data to simulate
self-consistent system dynamics of more than 50 FE.
