Many students who study advanced mathematics in advanced courses have probably wondered: where are differential equations (DEs) used in practice? As a rule, this issue is not discussed at lectures, and teachers immediately proceed to the solution of the control theory without explaining to students the use of differential equations in real life. We will try to fill this gap.
We start by defining a differential equation. So, a differential equation is an equation that relates the value of a derivative function to the function itself, the values of an independent variable and some numbers (parameters).
The most common area in which differential equations are applied is the mathematical description of natural phenomena. They are also used in solving problems where it is impossible to establish a direct relationship between some values that describe a process. Such tasks arise in biology, physics, and economics.
In biology:
The first substantial mathematical model describing biological communities was the Lotka-Volterra model. It describes a population of two interacting species. The first of them, called predators, dies according to the law x '= –ax (a> 0) in the absence of the second, and the second, victims, in the absence of predators multiplies unlimitedly in accordance with the Malthus law. The interaction of these two species is modeled as follows. Victims die out at a rate equal to the number of encounters of predators and victims, which in this model is assumed to be proportional to the number of both populations, i.e. equal to dxy (d> 0). Therefore, y '= by - dxy. Predators reproduce at a rate proportional to the number of prey eaten: x '= –ax + cxy (c> 0). System of equations
x '= –ax + cxy, (1)
y '= by - dxy, (2)
describing such a population, a predator is a prey and is called the Trays - Volterra system (or model).
In physics:
Newton’s second law can be written in the form of a differential equation
m ((d ^ 2) x) / (dt ^ 2) = F (x, t), where m is the mass of the body, x is its coordinate, F (x, t) is the force acting on the body with the coordinate x at time t. His solution is the trajectory of the body under the action of the indicated force.
