  In this paper we consider the linear ordinary equation of the second order $$
L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the
corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0.
\eqno{(2)} $$ Note that $[\alpha ,\beta ]$ is called a nonoscillation interval
if every nontrivial solution has at most one zero on this interval. Many
investigations which seem to have no connection such as differential
inequalities, the Polia-Mammana decomposition (i.e. representation of the
operator $L$ in the form of products of the first order differential
operators), unique solvability of the interpolation problems, kernels
oscillation, separation of zeros, zones of Lyapunov's stability and some others
have a certain common basis - nonoscillation. Presumably Sturm was the first to
consider the two problems which naturally appear here: to develop corollaries
of nonoscillation and to find methods to check nonoscillation. In this paper we
obtain several tests for nonoscillation on the semiaxis and apply them to
propose new results on asymptotic properties and the exponential stability of
the second order equation (2). Using the Floquet representations and upper and
lower estimates of nonoscillation intervals of oscillatory solutions we deduce
results on the exponential and Lyapunov's stability and instability of equation
(2).
