  Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of
$\mathbb P^{n+1}$. In this paper we find an effective lower bound for the
degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an
algebraic differential equation of order $k=n=\dim X$, and also similar bounds
for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are
no such algebraic differential equations of order $k<n$ for a smooth
hypersurface in $\mathbb P^{n+1}$.
