Note
The boundary conditions gives us , such that when we utilize this we have ,

hence

(2.1)

Now let us consider , such that the following one can easily visualize the following

Table 2.1: Breakup of occurrence of events in the time intervals

Which implies the following holds where  

Using the assumptions under definition # 1 and definition # 2 we have:

Hence:

This is a differential equation in time, but a difference equation in state. As usual we are interested to know about the initial conditions and they are