Dynamical sampling is a recent topic in sampling theory that focuses on sampling and reconstructions of functions that evolve in time. In this talk, we will review several results that illustrate how the ideas of dynamical sampling can be used in the context of initial value problems in differential equations. We will focus on two types of results. The first type deals with necessary and sufficient conditions for reconstructing time-evolving band-limited functions that may arise as solutions to a PDE. The second type focuses on an inverse problem of recovering burst-like forcing terms in an abstract initial value problem.