  We study a binary spin-mixture of a zero-temperature repulsively interacting
$^6$Li atoms using both the atomic-orbital and the density functional
approaches. The gas is initially prepared in a configuration of two magnetic
domains and we determine the frequency of the spin-dipole oscillations which
are emerging after the repulsive barrier, initially separating the domains, is
removed. We find, in agreement with recent experiment (G. Valtolina et al.,
arXiv:1605.07850 (2016)), the occurrence of a ferromagnetic instability in an
atomic gas while the interaction strength between different spin states is
increased, after which the system becomes ferromagnetic. The ferromagnetic
instability is preceded by the softening of the spin-dipole mode.
