  We prove the strong completeness for a class of non-degenerate SDEs, whose
coefficients are not necessarily uniformly elliptic nor locally Lipschitz
continuous nor bounded. Moreover, for each $t$, the solution flow $F_t$ is
weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such
that for all $t<T(p)$, the solution flow $F_t(\cdot)$ belongs to the Sobolev
space $W_{\loc}^{1,p}$. The main tool for this is the approximation of the
associated derivative flow equations. As an application a differential formula
is also obtained.
