In this paper, we investigate the growth of meromorphic solutions of the differential equations
$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 $$
and
$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$
where
$A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z)$
and
$F(z)\not \equiv0$
are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaïdi, Chen, *etc.*