We apply the long-wavelength approximation to the low-energy effective string action in the context of the Hamilton-Jacobi theory. The Hamilton-Jacobi equation for the effective string action is explicitly invariant under scale factor duality. We present the leading-order, general solution of the Hamilton-Jacobi equation. The Hamilton-Jacobi approach yields a solution consistent with the Lagrange formalism. The momentum constraints take an elegant, simple form. Furthermore, this general solution reduces to the quasi-isotropic one, if the evolution of the gravitational radiation is neglected. Duality transformation for the general solution is written as a coordinate transformation in an abstract held space.