Word equation

Many modellers suggest that as a preliminary to writing down a mathematical model, that is, a mathematical equation, the modeller should always try to write down a word equation. This is probably a good idea since modellers should always be able to communicate their ideas both to fellow modellers, often mathematicians, and also to non-mathematicians who may need to use the model predictions ultimately obtained.

A word equation describing the model that we now have could be:

The difference model

If we now take the time period to be one year we can write the difference equation

This equation can be presented as

where k=b-d.

If we apply the same ideas to a time period of six months, the basic equation is

and so

In a similar fashion, we can obtain the equation

corresponding to a monthly application, and

corresponding to a daily application.

You should be able to deduce the answer corresponding to an infinitely small time interval from the above information.