Differential Equations - General Information and Scope

Studying the phenomena of nature, solving all sorts of problems in economics, biology, physics, technology, it is not always possible to directly establish a direct connection between certain values that describe one or another evolutionary process. As a rule, it is possible to determine the relationship between these quantities (functions) and their rate of change with respect to other (independent) variables. This raises The equations in which the unknown functions stand under the sign of the derivative are the differential equations. A lot of well-known scientists spent time on their research: Newton, Bernoulli, Laplace and others. The application of differential equations is quite broad: in models of economic dynamics, which show not only the dependence of the variables in time, but also their relationship with time, in the tasks of micro- and macroeconomics; With their help describe the propagation of electromagnetic and thermal waves and various evolutionary phenomena that occur in animate and inanimate nature.

With the help of electromagnetic waves , information is transmitted at a distance (television, telephone, radio, and the like). Modern macroeconomics widely uses differential and difference equations. For example, in macroeconomics, the so-called basic DN of the neoclassical theory of economic growth is used. Differential equations are also used in biology, chemistry, automation and other special disciplines. The figure shows a graph of the function that is used when considering the increase in population growth. This task is solved with the help of remote control.

So, now there is more theory. The usual differential equation is the non-identical relation between the unknown function Y with one independent argument X, the most independent variable X, and the derivatives of the desired function of some order. There are many types of differential equations, more about which later in the article.

Differential equations are:

1) The usual equations of the ith order, which are integrated in squares. These, in turn, are divided into: differential equations with separated variables; DU with separated variables; Homogeneous remote control; Linear remote control; Equations in total differentials.

2) ДУ of higher orders.

3) Linear DMs of the second order, which are linear homogeneous second-order ДУ with constant coefficients and linear inhomogeneous ДУ with constant coefficients.

DU is also solved in several ways, the most common of which are the Cauchy problem, the Euler and Bernoulli methods, and others.

In many problems of the economy, mathematics, technology, it is necessary to calculate a certain number of functions connected by a certain number of DMs. Then we come to the aid of systems of differential equations: a set of equations, each of which includes an independent variable, the functions of this independent and their derivatives.

If the system is linear with respect to unknown functions, then it is called a linear system of differential equations. A normal system of differential equations can be replaced by one ДУ, the order of which is equal to the number of equations of the system.

The transformation of the DU system to a single equation in some cases is accomplished using the elimination method.

In addition to all of the above, there are linear systems with constant coefficients, which are easily solved by the Euler method.