The theory of stochastic dynamic equations extends and unifies the theories of stochastic difference and differential equations. In this paper, we prove the existence and uniqueness of the strong solution of a certain class of stochastic dynamic equations. As a principal tool in the proof, we define and develop the properties of stochastic dynamic integrals with respect to a Brownian motion indexed by a time scale. Finally we illustrate our theory with the examples of stochastic exponential and geometric Brownian motion.