SECTION 10.1: Mathematical Modeling: Setting up a differential equation

A differential equation is one that is formed based on the information on rates of changes of available values in the equation. This equation can then be analyzed in order to obtain the original function.

From the examples given in the textbook, it was observed that the rate of change of a certain quantity being study was effectively adjustable to the continuous rate of change of an increase or decrease of that quantity and the constant rate at which it is being increased or decreased. Assuming the Quantity being studied is Q,

dQ/dt = aX-Y where a determined the continuous rate of change and Y determines the constant rate of change. X would refer to the original value of the quantity being studied.

The Logistic Model:

The magnitude of a certain quantity being studied (e.g., population) is proportional to its initial value times the difference between its carrying capacity(the limit to which population in a town can increase) and the current value.

In other words if carrying capacity is L, the original and th original value is P then:

dP/dt = kP(L-P) where k is the constant of proportionality. This is known as a 'logistic differential equation'.

I found it difficult to understand exactly how the carrying capacity of a quantity as large as population can really be determined.

It is apparent that known how to use models with differntial equations assessing rates of changes can be important in assessing a company's revenues in terms of the changing wages it has to pay its workers. This section of mathematics is again directly relatable to economics.