Dye-sensitized solar cells (DSSCs) are a novel approach to the renewable energy problem, allowing sunlight conversion while reducing production costs. Electricity generation is achieved through a series of chemical reactions designed to transport electrons as a means of creating a circuit. Current challenges facing their development is to reduce the impact of recombination reactions, whilst enhancing their efficiency. In this paper, we investigate the behaviour of DSSCs by solving a set of partial differential equations which are used to model the density of electrons in the conduction band of a DSSC's semiconductor and in the electrolyte couple. Previously, approaches to solve these equations are by assuming steady-state models and then making use of numerical methods. In this paper, we obtain full analytical solutions to these equations and based on values of parameters associated with DSSCs available in the literature, we derive results which can be used to determine the performance of DSSCs.