The concept of the differential equation

Section III. Differential equations.

This branch of mathematics appeared immediately after the discovery of differential and integral calculus in the late 17-th century. Many processes in the economy, chemistry, biology are described by the differential equations and their systems.

Let consider the following example. From statistic we know that the number of newborns at one unit of time, for example per year, is proportional to the population of this region with coefficient of proportionality, and the number of deaths is proportional to the population with coefficient. Required to establish the law that determines the population of the region which depending on the time. Denote by the number of people living in the region. According to the condition of the problem, the number of newborns per unit of time is equal to, and number of deaths for the same unit of time will be . Then the increase in population we can find as such difference

were is called the coefficient of natural population growth. During the time the population size will growth and function will get an increment

.

Dividing the last equation by and taking limit as we obtain

.

Using the definition of the derivative we can write

.

The last equality is a mathematical model of the demographical process and we see that the population size obeys the equation which we will call differential equation

.

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