studying the phenomena of nature, solving various tasks in economics, biology, physics, engineering, not always possible to immediately establish a direct link between by some values that describe a particular evolutionary process.As a rule, you can determine the relationship between these values (functions) and their rate of change with respect to other (independent) variables.This gives rise to the equation in which the unknown functions are under the sign of the derivative - this differential equation.In their study they spent a lot of time, a lot of famous scientists: Newton, Bernoulli, Laplace and others.Application of differential equations quite widely: in models of economic dynamics, displaying not only the dependent variable in time, and their relationship with the times, in the problems of micro- and macroeconomics;use them to describe the propagation of electromagnetic and thermal waves and different evolutionary phenomena that occur in the animate and inanimate nature.
Using electromagnetic waves to transmit information at a distance (television, telephone, radio, etc.).Modern macroeconomics extensive use of differential and difference equations.For example, in macroeconomics is used so-called primary control of the neoclassical theory of economic growth.Differential equations are also used in biology, chemistry, automation and other specialized disciplines.The figure shows the graph of the function, which is used when considering the increasing population growth.This problem is solved with the help of remote control.
So now more theory.Ordinary differential equation called nonidentical relation between the unknown function Y with a single independent argument X, most of the independent variable X and the derivatives of the unknown function of some order.There are many types of differential equations, more of which later in this article.
Differential equations are:
1) Conventional equation of I-th order, are integrated in the squares.These, in turn, are divided into: differential equations with separable variables;Control with separated variables;uniform control;linear control;Exact differential equations.
2) higher-order control.
3) Linear Control II-th order, which are homogeneous linear control II-th order with constant coefficients and inhomogeneous linear control with constant coefficients.
control also solved in several ways, the most common of which - Cauchy problem, methods of Euler and Bernoulli, and others.
In many problems of economics, mathematics, technology necessary to calculate a certain number of functions associated with each other a certain amount of control.Then we come to the aid of the system of differential equations set of equations, each of which includes an independent variable, the function of this independent and their derivatives.
If the system is linear in the unknown functions, it is called a linear system of differential equations.The normal system of differential equations can be replaced by a single controller, the order is equal to the number of equations in the system.
Conversion control system to one equation in some cases is done using the method of exclusion.
addition to all of the above, there are linear systems with constant coefficients which are easily solved by Euler's method.
