We investigate the Cauchy problem for second order hyperbolic
equations of complete form, and we prove an extension of a
classical result of Oleĭnik [10] concerning the well-posedness
for equations in which are absent the terms with mixed time-space
derivatives. Then, in space dimension $n=1$, we compare our
results with those in [8] for equations with analytic coefficients,
and those of [7] and [11] for homogeneous equations with
coefficients depending only either on $t$ or on $x$. Moreover
we exhibit, in space dimension $n\ge 2$, an equation of the form
\begin{equation*}
u_{tt} - \sum_{i,j=1}^{n} (a_{ij}(t,x)u_{x_{j}})_{x_{i}}
= 0{,}
\quad\text{with}\quad
\sum a_{ij} \xi_{i}\xi_{j} \ge 0,
\end{equation*}
where the coefficients are analytic
functions, for which the Cauchy problem is ill-posed. Finally,
we present a sufficient condition for the well-posedness of
$2 \times 2$ systems.