def METHOD_NAME(x: np.ndarray, y: np.ndarray) -> np.ndarray:
    """return the exact fully developed solution at specified points"""
    return np.where(y > -halfheight,
                    - (5 - y**2) * (1 - y**2) / 16 - y / 2,
                    1 / 2 - (1 + y) / kratio) + longitudinal_gradient * x