  We study cosmological solutions in the effective heterotic string theory with
$\alpha'$-correction terms in string frame. It is pointed out that the
effective theory has an ambiguity via field redefinition and we analyze
generalized effective theories due to this ambiguity. We restrict our analysis
to the effective theories which give equations of motion of second order in the
derivatives, just as "Galileon" field theory. This class of effective actions
contains two free coupling constants. We find de Sitter solutions as well as
the power-law expanding universes in our four-dimensional Einstein frame. The
accelerated expanding universes are always the attractors in the present
dynamical system.
