In this paper the analytical solutions of the differential equations are presented. These equations describe the pulsatory liposome dynamics. We consider a unilamellar liposome filled with an aqueous solution of osmotic solute inserted in a hypotonic aqueous medium. Due to the osmosis process the liposome has a cyclic evolution. The lipid vesicle swells to a critical size, when a transbilayer pore suddenly appears. Part of the internal solution leaks through this pore. The liposome relaxes and returns to the initial size. The swelling starts again and the liposome goes through a periodical process. The swelling of the liposome is described by a differential equation. All the processes which contribute to the liposome relaxing and its coming back to the initial size are described by three differential equations. Based on some analytical methods, we solve these equations and their explicit solutions are validated by comparing with previous study numerical results.