  In a class of non-singular cosmologies derived from higher-order corrections
to the low-energy bosonic string action, we derive evolution equations for the
most general cosmological scalar, vector and tensor perturbations. In the large
scale limit, the evolutions of both scalar and tensor perturbations are
characterised by conserved quantities, the usual curvature perturbation in the
uniform-field gauge and the tensor-type perturbed metric. The vector
perturbation is not affected, being described by the conservation of the
angular momentum of the fluid component in the absence of any additional
dissipative process. For the scalar- and tensor-type perturbations, we show
how, given a background evolution during kinetic driven inflation of the
dilaton field, we can obtain the final power spectra generated from the vacuum
quantum fluctuations of the metric and the dilaton field during the inflation.
