Recent studies have shown that the short time solution of the equation of motion for the rolling of ships is important in deciding the possibility of capsize of ships due to the excessive heel. Since most of known solutions for nonlinear equations of motion are long time or steady periodic solutions, here a simple way is described to get the short time solutions of the Duffing equation, which was chosen for deriving a criterion for the capsize of the ship. With the small external rolling moment, we first assume the state of the small damping and near resonance. Then, for cases when the frequency of the external moment is higher than the resonant one, an inequality was derived as a criterion for the capsize. This gives the range of the initial condition and the magnitude of the external moment which should be avoided for a ship to be safe from capsize. Furthermore, from the linearized equation, it is also shown that a simple and self-explanatory solution can be obtained consistent with that for the case of no damping, which yields the well-known linear growth with time.