In this paper, by using the Nevanlinna value distribution theory of meromorphic functions on an annulus, we deal with the growth properties of solutions of the linear differential equation f(k)+Bk−1 (z) f(k−1)+· · ·+B1 (z) f0+B0 (z) f = 0, where k ≥ 2 is an integer and Bk−1 (z) , ...,B1 (z) ,B0 (z) are analytic on an annulus. Under some conditions on the coefficients, we obtain some results concerning the estimates of the order and the hyper-order of solutions of the above equation. The results obtained extend and improve those of Wu and Xuan in [16].