Differential Equations - General Information and Scope

Studying the phenomena of nature, solving all kinds of problemson economics, biology, physics, engineering, 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

Equations in which unknown functions standunder the sign of the derivative are 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, where not only the dependence of the variables in time is displayed, but also their interrelation with time, in problems 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 is transmittedinformation 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 ordinary equations of the ith order, whichintegrate 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 ДІІ-th order with constant coefficients and linear inhomogeneous ДУ with constant coefficients.

The DM are 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 economics, mathematics, technologyit is necessary to calculate a certain number of functions, connected by some quantity of remote control. 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 unknownsfunctions, then it is called a linear system of differential equations. A normal system of differential equations can be replaced by a single DE whose order 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.

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