This study introduces an analytically tractable and computationally efficient model of the legged robot dynamics associated with locomotion on a dynamic rigid surface (DRS), and develops a real-time motion planner based on the proposed model and its analytical solution. This study first theoretically extends the classical linear inverted pendulum (LIP) model from legged locomotion on a static surface to DRS locomotion, by relaxing the LIP's underlying assumption that the surface is static. The resulting model, which we call "DRS-LIP", is explicitly time-varying. After converting the DRS-LIP into Mathieu's equation, an approximate analytical solution of the DRS-LIP is obtained, which is reasonably accurate with a low computational cost. Furthermore, to illustrate the practical uses of the analytical results, they are exploited to develop a hierarchical motion planner that efficiently generates physically feasible trajectories for DRS locomotion. Finally, the effectiveness of the proposed theoretical results and motion planner is demonstrated both through PyBullet simulations and experimentally on a Laikago quadrupedal robot that walks on a rocking treadmill. The videos of simulations and hardware experiments are available at https://youtu.be/u2Q_u2pR99c.