  Demanding $O(d,d)$-duality covariance, Hohm and Zwiebach have written down
the action for the most general cosmology involving the metric, $b$-field and
dilaton, to all orders in $\alpha'$ in the string frame. Remarkably, for an FRW
metric-dilaton ansatz the equations of motion turn out to be quite simple,
except for the presence of an unknown function of a single variable. If this
unknown function satisfies some simple properties, it allows de Sitter
solutions in the string frame. In this note, we write down the Einstein frame
analogues of these equations, and make some observations that make the system
tractable. Perhaps surprisingly, we find that a necessary condition for de
Sitter solutions to exist is that the unknown function must satisfy a certain
second order non-linear ODE. The solutions of the ODE do not have a simple
power series expansion compatible with the leading supergravity expectation. We
discuss possible interpretations of this fact. After emphasizing that all
(potential) string and Einstein frame de Sitter solutions have a running
dilaton, we write down the most general cosmologies with a constant dilaton in
string/Einstein frame: these have power law scale factors.
