  While the classification of $\alpha'$ corrections of string inspired
effective theories remains an unsolved problem, we show how to classify duality
invariant $\alpha'$ corrections for purely time-dependent (cosmological)
backgrounds. We determine the most general duality invariant theory to all
orders in $\alpha'$ for the metric, $b$-field, and dilaton. The resulting
Friedmann equations are studied when the spatial metric is a time-dependent
scale factor times the Euclidean metric and the $b$-field vanishes. These
equations can be integrated perturbatively to any order in $\alpha'$. We
construct non-perturbative solutions and display duality invariant theories
featuring string-frame de Sitter vacua.
