  The approach to the consideration of the ordinary differential equations with
distributions in the classical space $\mathcal D'$ of distributions with
continuous test functions has certain insufficiencies: the notations are
incorrect from the point of view of distribution theory, the right-hand side
has to satisfy the restrictive conditions of equality type. In the present
paper we consider an initial value problem for the ordinary differential
equation with distributions in the space of distributions with dynamic test
functions $\mathcal T'$, where the continuous operation of multiplication of
distributions by discontinuous functions is defined, and show that this
approach does not have the aforementioned insufficiencies.
  We provide the sufficient conditions for viability of solutions of the
ordinary differential equations with distributions (a generalization of the
Nagumo Theorem), and show that the consideration of the distributional
(impulse) controls in the problem for avoidance of encounters with the set (the
maximal viability time problem) allows us to provide the existence of solution,
which may not exist for the ordinary controls.
