Differential equation descriptions for continuous-time linear time-invariant systems are unique in that they allow analysis of the effect of stored energy on the system output. You will learn how to represent the output of such a system as a sum of a steady-state and a transient component. The steady-state component is of the same form as the input, while the transient component reflects the effects of stored energy. You will learn how both of these components depend on the parameters of the differential equation that describes the system.
Differential equation descriptions for systems are widely used to represent continuous-time systems. Such systems occur in signal processing as prototypes for discrete-time filters and as models for continuous-time systems that interact with signals. Understanding how the output depends on the input and stored energy will provide you with very helpful insight for working with such systems.