The oscillation model, based on the wave equation on networks, can describe user dynamics in online social networks. The fundamental equation of user dynamics can be introduced into the oscillation model to explicitly describe the causal relation of user dynamics yielded by certain specific network structures. Moreover, by considering the sparseness of online social networks, a novel fundamental equation of different form has been devised. In this paper, we derive a closed-form solution of the new fundamental equation. Also, we find the closed-form solution of the new fundamental equation can generate the general solution of the original wave equation.