Today, one of the most important skills for any specialist is the ability to solve differential equations. The solution of differential equations - not a single applied task can do without this, whether it is the calculation of any physical parameter or the modeling of changes as a result of the adopted macroeconomic policy. These equations are also important for a number of other sciences such as chemistry, biology, medicine, etc. Below we will give an example of the use of differential equations in economics, but before that we will briefly talk about the main types of equations.

Differential equations - the simplest types

The sages said that the laws of our universe are written in mathematical language. Of course, there are many examples of various equations in algebra, but these are mostly educational examples that are not applicable in practice. The really interesting mathematics begins when we want to describe the processes that take place in real life. But how to reflect the time factor, which is subject to real processes - inflation, output or demographic indicators?

Recall one important definition from a mathematics course regarding the derivative of a function. The derivative is the rate of change of the function, so it can help us reflect the time factor in the equation.

That is, we compose an equation with a function that describes the indicator of interest to us and add the derivative of this function to the equation. This is the differential equation. Now let's move on to the simplest types of differential equations for dummies.

The simplest differential equation has the form $y'(x)=f(x)$, where $f(x)$ is some function, and $y'(x)$ is the derivative or rate of change of the desired function. It is solved by ordinary integration: $$y(x)=\int f(x)dx.$$

The second simplest type is called a separable differential equation. Such an equation looks like this $y’(x)=f(x)\cdot g(y)$. It can be seen that the dependent variable $y$ is also part of the constructed function. The equation is solved very simply - you need to "separate the variables", that is, bring it to the form $y’(x)/g(y)=f(x)$ or $dy/g(y)=f(x)dx$. It remains to integrate both parts $$\int \frac(dy)(g(y))=\int f(x)dx$$ - this is the solution of a separable type differential equation.

The last simple type is the first order linear differential equation. It has the form $y'+p(x)y=q(x)$. Here $p(x)$ and $q(x)$ are some functions, and $y=y(x)$ is the desired function. To solve such an equation, special methods are already used (the Lagrange method of variation of an arbitrary constant, the Bernoulli substitution method).

There are more complex types of equations - equations of the second, third and generally arbitrary order, homogeneous and inhomogeneous equations, as well as systems of differential equations. To solve them, you need preliminary preparation and experience in solving simpler problems.

Of great importance for physics and, surprisingly, finance are the so-called partial differential equations. This means that the desired function depends on several variables at the same time. For example, the Black-Scholes equation from the field of financial engineering describes the value of an option (type of security) depending on its yield, the amount of payments, as well as the timing of the start and end of payments. Solving a partial differential equation is quite complicated, usually you need to use special programs such as Matlab or Maple.

An example of applying a differential equation in economics

We give, as promised, a simple example of solving a differential equation. Let's set the task first.

For some firm, the function of marginal revenue from the sale of its products has the form $MR=10-0.2q$. Here $MR$ is the firm's marginal revenue and $q$ is the output. We need to find the total income.

As can be seen from the problem, this is an applied example from microeconomics. Many firms and enterprises are constantly faced with such calculations in the course of their activities.

Let's get to the decision. As is known from microeconomics, marginal revenue is a derivative of total revenue, and revenue is zero at zero sales.

From a mathematical point of view, the problem was reduced to solving the differential equation $R’=10-0.2q$ under the condition $R(0)=0$.

We integrate the equation, taking the antiderivative function of both parts, we get the general solution: $$R(q) = \int (10-0,2q)dq = 10 q-0,1q^2+C. $$

To find the constant $C$, recall the condition $R(0)=0$. Substitute: $$R(0) =0-0+C = 0. $$ So C=0 and our total revenue function becomes $R(q)=10q-0.1q^2$. Problem solved.

Other examples for different types of remote control are collected on the page:

Appendix

Solving differential equations online on the site for students to consolidate the material they have studied. And practice your practical skills. Differential equations online. Difuras online, online math solution. Step by step solution of mathematical problems online. The order, or degree, of a differential equation is the highest order of the derivatives included in it. Differential equations online. The process of solving a differential equation is called integration. The problem of integrating a differential equation is considered solved if the finding of the unknown function can be brought to quadrature, regardless of whether the resulting integral is expressed in the final form in terms of known functions or not. Step by step solution of differential equations online. All differential equations can be divided into ordinary differential equations (ODE), which include only functions (and their derivatives) of one argument, and partial differential equations (PDE), in which the input functions depend on many variables. Differential equations online. There are also stochastic differential equations (SDEs) involving random processes. Step by step solution of differential equations online. Depending on the combinations of derivatives, functions, independent variables, differential equations are divided into linear and non-linear, with constant or variable coefficients, homogeneous or non-homogeneous. Due to the importance of applications, quasilinear (linear with respect to higher derivatives) partial differential equations are singled out in a separate class. Solutions of differential equations are divided into general and particular solutions. Differential equations online. General solutions include undefined constants, and for partial differential equations, arbitrary functions of independent variables that can be refined from additional integration conditions (initial conditions for ordinary differential equations, initial and boundary conditions for partial differential equations). Step by step solution of differential equations online. After determining the form of these constant and indefinite functions, the solutions become particular. The search for solutions to ordinary differential equations led to the establishment of a class of special functions - functions often encountered in applications that are not expressed in terms of known elementary functions. Differential equations online. Their properties were studied in detail, tables of values ​​were compiled, interconnections were determined, etc. . The set of enumerated numbers can be explored. The best answer to the given problem. How to find in the first approximation the outgoing vector to the region of convergence about Differential equations without clarifying the found upper limit. The choice is obvious for increasing mathematical functions. There is a progressive method above the level of research. To align with the initial condition of the problem, the solution of the differential will help to find a single-valued chosen value. It may be that he can determine the unknown immediately. As in the previous example of indicating a solution to a mathematical problem, linear differential equations are the answer to a specific problem in a specified time frame. Maintenance of the study procedure is not locally defined. It will be so that there is an example for each student and the solution of differential equations will be determined by the person assigned to the responsible executor from at least two values. Take a function of general value on a certain segment and warn along which axis there will be a gap. Having studied the differential equations online, it is possible to unambiguously show how important the result is, if one is provided from the initial conditions. Cutting a region out of a function definition is impossible, since there is no task definition locally. Being found from a system of equations, the answer contains a variable that can be calculated in the general sense, but it will naturally be possible to solve a differential equation online without this action to determine the said condition. Near the interval of the segment, it can be seen how the solution of differential equations online is able to advance the result of research in a positive direction at the time of the cutoff of students' knowledge. The best is not always obtained by the general accepted approach to business. At the 2x level, one can usefully view all the necessary natural linear differential equations, but the ability to calculate a numerical value will lead to an increase in knowledge. According to any technique in mathematics, there are differential equations that are presented in essentially different expressions, such as homogeneous or complex. Having carried out a general analysis of the study of the function, it will become clear that the solution of differential as a set of possibilities represents a clear error in the values. The truth in it lies in the space above the abscissa lines. Somewhere in the domain of a complex function, at some point in its definition, linear differential equations will be able to represent the answer in an analytical form. that is, in general terms, as the essence. Nothing will change when changing the variable. However, it is necessary to peer into the answer with special interest. As a matter of fact, the calculator changes the ratio in the end, that is, how the solution of differential equations is proportional to the global value is indicated within the desired solution. In some cases, a mass error warning is unavoidable. 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However, due to circumstances, in view of the complexity of the setting subproblem, differential equations simplify the process of calculated numerical spaces at the level of three-dimensional space. It is easy to prove otherwise, but it is possible to avoid it, as in the example above. In higher mathematics, the following points are provided: when a problem is reduced to a simplified form, the greatest possible effort on the part of students should be extended to it. Lines superimposed on each other fall into the offset. Pro differential solution still resumes the advantage of the said method on a curved line. If you recognize at first not what you need, then the mathematical formula will make a new value of the expression. The goal is the optimal approach to solving the tasks set by the professor. You should not assume that linear differential equations in a simplified form will exceed the expected result. We place three vectors on a finitely composed surface. orthogonal to each other. Let's calculate the product. Let's carry out the addition of a larger number of symbols and write out all the variables of the function from the resulting expression. There is a proportion. Several actions preceding the end of the calculation will not give an unambiguous answer to the solution of differential equations immediately, but only after the allotted time has elapsed along the y-axis. To the left of the discontinuity point, given implicitly from the function, we draw an axis orthogonal to the best increasing vector and place the online differential equations along the least boundary value of the lower bound of the mathematical object. Let's add an extra argument in the break area of ​​the function. To the right of the points of the curved line, the formulas written by us for reducing to a common denominator will help to solve the differential equation online. The only correct approach is the one that will shed light on unsolved problems from theory to practice, in the general case unequivocally. The lines in the direction of the coordinates of the given points have never closed the extreme position of the square, however, the solution of differential equations online will help both students and us, and just beginners in this field, to study mathematics. We are talking about the possibility of substituting the value argument into all significant sublines of one field. In principle, as one would expect, our linear differential equations are something isolated in a single concept of the reduced meaning. To help students, one of the best among similar services is a calculator. Go through all the courses and choose the best one for you.

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Often just a mention differential equations makes students uncomfortable. Why is this happening? Most often, because when studying the basics of the material, a gap in knowledge arises, due to which the further study of difurs becomes simply torture. Nothing is clear what to do, how to decide where to start?

However, we will try to show you that difurs is not as difficult as it seems.

Basic concepts of the theory of differential equations

From school, we know the simplest equations in which we need to find the unknown x. In fact differential equations only slightly different from them - instead of a variable X they need to find a function y(x) , which will turn the equation into an identity.

D differential equations are of great practical importance. This is not abstract mathematics that has nothing to do with the world around us. With the help of differential equations, many real natural processes are described. For example, string vibrations, the movement of a harmonic oscillator, by means of differential equations in problems of mechanics, find the speed and acceleration of a body. Also DU are widely used in biology, chemistry, economics and many other sciences.

Differential equation (DU) is an equation containing the derivatives of the function y(x), the function itself, independent variables and other parameters in various combinations.

There are many types of differential equations: ordinary differential equations, linear and non-linear, homogeneous and non-homogeneous, differential equations of the first and higher orders, partial differential equations, and so on.

The solution to a differential equation is a function that turns it into an identity. There are general and particular solutions of remote control.

The general solution of the differential equation is the general set of solutions that turn the equation into an identity. A particular solution of a differential equation is a solution that satisfies additional conditions specified initially.

The order of a differential equation is determined by the highest order of the derivatives included in it.

Ordinary differential equations

Ordinary differential equations are equations containing one independent variable.

Consider the simplest ordinary differential equation of the first order. It looks like:

This equation can be solved by simply integrating its right side.

Examples of such equations:

Separable Variable Equations

In general, this type of equation looks like this:

Here's an example:

Solving such an equation, you need to separate the variables, bringing it to the form:

After that, it remains to integrate both parts and get a solution.

Linear differential equations of the first order

Such equations take the form:

Here p(x) and q(x) are some functions of the independent variable, and y=y(x) is the desired function. Here is an example of such an equation:

Solving such an equation, most often they use the method of variation of an arbitrary constant or represent the desired function as a product of two other functions y(x)=u(x)v(x).

To solve such equations, a certain preparation is required, and it will be quite difficult to take them “on a whim”.

An example of solving a DE with separable variables

So we have considered the simplest types of remote control. Now let's take a look at one of them. Let it be an equation with separable variables.

First, we rewrite the derivative in a more familiar form:

Then we will separate the variables, that is, in one part of the equation we will collect all the “games”, and in the other - the “xes”:

Now it remains to integrate both parts:

We integrate and obtain the general solution of this equation:

Of course, solving differential equations is a kind of art. You need to be able to understand what type an equation belongs to, and also learn to see what transformations you need to make with it in order to bring it to one form or another, not to mention just the ability to differentiate and integrate. And it takes practice (as with everything) to succeed in solving DE. And if at the moment you don’t have time to figure out how differential equations are solved or the Cauchy problem has risen like a bone in your throat or you don’t know, contact our authors. In a short time, we will provide you with a ready-made and detailed solution, the details of which you can understand at any time convenient for you. In the meantime, we suggest watching a video on the topic "How to solve differential equations":

This article is a starting point in the study of the theory of differential equations. Here are collected the main definitions and concepts that will constantly appear in the text. For better assimilation and understanding, the definitions are provided with examples.

Differential Equation (DE)- this is an equation that includes an unknown function under the sign of the derivative or differential.

If the unknown function is a function of one variable, then the differential equation is called ordinary(abbreviated ODE - ordinary differential equation). If the unknown function is a function of many variables, then the differential equation is called partial differential equation.

The maximum order of the derivative of an unknown function included in a differential equation is called the order of the differential equation.

Here are examples of ODEs of the first, second and fifth orders, respectively

As examples of partial differential equations of the second order, we present

Further, we will consider only ordinary differential equations of the nth order of the form or , where Ф(x, y) = 0 is an unknown function defined implicitly (when possible, we will write it in explicit representation y = f(x) ).

The process of finding solutions to a differential equation is called integration of the differential equation.

Solving a Differential Equation is an implicitly given function Ф(x, y) = 0 (in some cases, the function y can be expressed explicitly in terms of the argument x), which turns the differential equation into an identity.

NOTE.

The solution of a differential equation is always sought on a predetermined interval X .

Why are we talking about this separately? Yes, because in the conditions of many problems the interval X is not mentioned. That is, the condition of the problems is usually formulated as follows: “find a solution to the ordinary differential equation ". In this case, it is understood that the solution should be sought for all x for which both the desired function y and the original equation make sense.

The solution of a differential equation is often referred to as differential equation integral.

Functions or can be called a solution to a differential equation.

One of the solutions of the differential equation is the function . Indeed, substituting this function into the original equation, we obtain the identity . It is easy to see that another solution to this ODE is, for example, . Thus, differential equations can have many solutions.

General solution of differential equation is the set of solutions containing all solutions of this differential equation without exception.

The general solution of a differential equation is also called general integral of the differential equation.

Let's go back to the example. The general solution of the differential equation has the form or , where C is an arbitrary constant. Above, we indicated two solutions to this ODE, which are obtained from the general integral of the differential equation by substituting C = 0 and C = 1, respectively.

If the solution of a differential equation satisfies the initially given additional conditions, then it is called a particular solution of the differential equation.

A particular solution of the differential equation that satisfies the condition y(1)=1 is . Really, and .

The main problems of the theory of differential equations are Cauchy problems, boundary value problems and problems of finding a general solution of a differential equation on any given interval X .

Cauchy problem is the problem of finding a particular solution of a differential equation that satisfies the given initial conditions, where are numbers.

Boundary problem is the problem of finding a particular solution to a second-order differential equation that satisfies additional conditions at the boundary points x 0 and x 1:
f (x 0) \u003d f 0, f (x 1) \u003d f 1, where f 0 and f 1 are given numbers.

The boundary value problem is often called boundary value problem.

An ordinary differential equation of the nth order is called linear, if it has the form , and the coefficients are continuous functions of the argument x on the integration interval.

Either already solved with respect to the derivative, or they can be solved with respect to the derivative .

General solution of differential equations of the type on the interval X, which is given, can be found by taking the integral of both sides of this equality.

Get .

If we look at the properties of the indefinite integral, we find the desired general solution:

y = F(x) + C,

where F(x)- one of the antiderivatives of the function f(x) in between X, a With is an arbitrary constant.

Please note that in most tasks the interval X do not indicate. This means that a solution must be found for everyone. x, for which and the desired function y, and the original equation make sense.

If you need to calculate a particular solution of a differential equation that satisfies the initial condition y(x0) = y0, then after calculating the general integral y = F(x) + C, it is still necessary to determine the value of the constant C=C0 using the initial condition. That is, a constant C=C0 determined from the equation F(x 0) + C = y 0, and the desired particular solution of the differential equation will take the form:

y = F(x) + C0.

Consider an example:

Find the general solution of the differential equation , check the correctness of the result. Let's find a particular solution of this equation that would satisfy the initial condition .

Decision:

After we have integrated the given differential equation, we get:

.

We take this integral by the method of integration by parts:

That., is a general solution of the differential equation.

Let's check to make sure the result is correct. To do this, we substitute the solution that we found into the given equation:

.

That is, at the original equation turns into an identity:

therefore, the general solution of the differential equation was determined correctly.

The solution we found is the general solution of the differential equation for each real value of the argument x.

It remains to calculate a particular solution of the ODE that would satisfy the initial condition . In other words, it is necessary to calculate the value of the constant With, at which the equality will be true:

.

.

Then, substituting C = 2 into the general solution of the ODE, we obtain a particular solution of the differential equation that satisfies the initial condition:

.

Ordinary differential equation can be solved with respect to the derivative by dividing the 2 parts of the equation by f(x). This transformation will be equivalent if f(x) does not go to zero for any x from the interval of integration of the differential equation X.

Situations are likely when, for some values ​​of the argument x ∈ X functions f(x) and g(x) turn to zero at the same time. For similar values x the general solution of the differential equation is any function y, which is defined in them, because .

If for some values ​​of the argument x ∈ X the condition is satisfied, which means that in this case the ODE has no solutions.

For all others x from interval X the general solution of the differential equation is determined from the transformed equation.

Let's look at examples:

Example 1

Let us find the general solution of the ODE: .

Decision.

From the properties of the basic elementary functions, it is clear that the natural logarithm function is defined for non-negative values ​​of the argument, therefore, the domain of the expression log(x+3) there is an interval x > -3 . Hence, the given differential equation makes sense for x > -3 . With these values ​​of the argument, the expression x + 3 does not vanish, so one can solve the ODE with respect to the derivative by dividing the 2 parts by x + 3.

We get .

Next, we integrate the resulting differential equation, solved with respect to the derivative: . To take this integral, we use the method of subsuming under the sign of the differential.