6.1. BASIC CONCEPTS AND DEFINITIONS

When solving various problems in mathematics and physics, biology and medicine, quite often it is not possible to immediately establish a functional relationship in the form of a formula connecting variables, which describe the process under study. Usually you have to use equations that contain, in addition to the independent variable and the unknown function, also its derivatives.

Definition. An equation connecting an independent variable, an unknown function and its derivatives of various orders is called differential.

An unknown function is usually denoted y(x) or simply y, and its derivatives - y", y" etc.

Other designations are also possible, for example: if y= x(t), then x"(t), x""(t)- its derivatives, and t- independent variable.

Definition. If a function depends on one variable, then the differential equation is called ordinary. General form ordinary differential equation:

or

Functions F And f may not contain some arguments, but for the equations to be differential, the presence of a derivative is essential.

Definition.The order of the differential equation is called the order of the highest derivative included in it.

For example, x 2 y"- y= 0, y" + sin x= 0 are first order equations, and y"+ 2 y"+ 5 y= x- second order equation.

When solving differential equations, the integration operation is used, which is associated with the appearance of an arbitrary constant. If the integration action is applied n times, then, obviously, the solution will contain n arbitrary constants.

6.2. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

General form first order differential equation is determined by the expression

The equation may not explicitly contain x And y, but necessarily contains y".

If the equation can be written as

then we obtain a first-order differential equation resolved with respect to the derivative.

Definition. The general solution of the first order differential equation (6.3) (or (6.4)) is the set of solutions , Where WITH- arbitrary constant.

The graph of the solution to a differential equation is called integral curve.

Giving an arbitrary constant WITH different values, partial solutions can be obtained. On surface xOy the general solution is a family of integral curves corresponding to each particular solution.

If you set a point A (x 0 , y 0), through which the integral curve must pass, then, as a rule, from a set of functions One can single out one - a private solution.

Definition.Private decision of a differential equation is its solution that does not contain arbitrary constants.

If is a general solution, then from the condition

you can find a constant WITH. The condition is called initial condition.

The problem of finding a particular solution to the differential equation (6.3) or (6.4) satisfying the initial condition at called Cauchy problem. Does this problem always have a solution? The answer is contained in the following theorem.

Cauchy's theorem(theorem of existence and uniqueness of a solution). Let in the differential equation y"= f(x,y) function f(x,y) and her

partial derivative defined and continuous in some

region D, containing a point Then in the area D exists

the only solution to the equation that satisfies the initial condition at

Cauchy's theorem states that under certain conditions there is a unique integral curve y= f(x), passing through a point Points at which the conditions of the theorem are not met

Cauchies are called special. At these points it breaks f(x, y) or.

Either several integral curves or none pass through a singular point.

Definition. If the solution (6.3), (6.4) is found in the form f(x, y, C)= 0, not allowed relative to y, then it is called general integral differential equation.

Cauchy's theorem only guarantees that a solution exists. Since there is no single method for finding a solution, we will consider only some types of first-order differential equations that can be integrated into quadratures

Definition. The differential equation is called integrable in quadratures, if finding its solution comes down to integrating functions.

6.2.1. Differential equations first order with separable variables

Definition. A first order differential equation is called an equation with separable variables,

The right side of equation (6.5) is the product of two functions, each of which depends on only one variable.

For example, the equation is an equation with separating

mixed with variables
and the equation

cannot be represented in the form (6.5).

Considering that , we rewrite (6.5) in the form

From this equation we obtain a differential equation with separated variables, in which the differentials are functions that depend only on the corresponding variable:

Integrating term by term, we have

where C = C 2 - C 1 - arbitrary constant. Expression (6.6) is the general integral of equation (6.5).

By dividing both sides of equation (6.5) by, we can lose those solutions for which, Indeed, if at

That obviously is a solution to equation (6.5).

Example 1. Find a solution to the equation that satisfies

condition: y= 6 at x= 2 (y(2) = 6).

Solution. We will replace y" then . Multiply both sides by

dx, since during further integration it is impossible to leave dx in the denominator:

and then dividing both parts by we get the equation,

which can be integrated. Let's integrate:

Then ; potentiating, we get y = C. (x + 1) - ob-

general solution.

Using the initial data, we determine an arbitrary constant, substituting them into the general solution

Finally we get y= 2(x + 1) is a particular solution. Let's look at a few more examples of solving equations with separable variables.

Example 2. Find the solution to the equation

Solution. Considering that , we get .

Integrating both sides of the equation, we have

where

Example 3. Find the solution to the equation Solution. We divide both sides of the equation into those factors that depend on a variable that does not coincide with the variable under the differential sign, i.e. and integrate. Then we get

and finally

Example 4. Find the solution to the equation

Solution. Knowing what we will get. Section

lim variables. Then

Integrating, we get

Comment. In examples 1 and 2, the required function is y expressed explicitly (general solution). In examples 3 and 4 - implicitly (general integral). In the future, the form of the decision will not be specified.

Example 5. Find the solution to the equation Solution.

Example 6. Find the solution to the equation , satisfying

condition y(e)= 1.

Solution. Let's write the equation in the form

Multiplying both sides of the equation by dx and on, we get

Integrating both sides of the equation (the integral on the right side is taken by parts), we obtain

But according to the condition y= 1 at x= e. Then

Let's substitute the found values WITH to the general solution:

The resulting expression is called a partial solution of the differential equation.

6.2.2. Homogeneous differential equations of the first order

Definition. The first order differential equation is called homogeneous, if it can be represented in the form

Let us present an algorithm for solving a homogeneous equation.

1.Instead y let's introduce a new functionThen and therefore

2.In terms of function u equation (6.7) takes the form

that is, the replacement reduces a homogeneous equation to an equation with separable variables.

3. Solving equation (6.8), we first find u and then y= ux.

Example 1. Solve the equation Solution. Let's write the equation in the form

We make the substitution:
Then

We will replace

Multiply by dx: Divide by x and on Then

Having integrated both sides of the equation over the corresponding variables, we have

or, returning to the old variables, we finally get

Example 2.Solve the equation Solution.Let Then

Let's divide both sides of the equation by x2: Let's open the brackets and rearrange the terms:

Moving on to the old variables, we arrive at the final result:

Example 3.Find the solution to the equation given that

Solution.Performing a standard replacement we get

or

or

This means that the particular solution has the form Example 4. Find the solution to the equation

Solution.

Example 5.Find the solution to the equation Solution.

Independent work

Find solutions to differential equations with separable variables (1-9).

Find a solution to homogeneous differential equations (9-18).

6.2.3. Some applications of first order differential equations

Radioactive decay problem

The rate of decay of Ra (radium) at each moment of time is proportional to its available mass. Find the law of radioactive decay of Ra if it is known that at the initial moment there was Ra and the half-life of Ra is 1590 years.

Solution. Let at the instant the mass Ra be x= x(t) g, and Then the decay rate Ra is equal to

According to the conditions of the problem

Where k

Separating the variables in the last equation and integrating, we get

where

For determining C we use the initial condition: when .

Then and, therefore,

Proportionality factor k determined from the additional condition:

We have

From here and the required formula

Bacterial reproduction rate problem

The rate of reproduction of bacteria is proportional to their number. At the beginning there were 100 bacteria. Within 3 hours their number doubled. Find the dependence of the number of bacteria on time. How many times will the number of bacteria increase within 9 hours?

Solution. Let x- number of bacteria at a time t. Then, according to the condition,

Where k- proportionality coefficient.

From here From the condition it is known that . Means,

From the additional condition . Then

The function you are looking for:

So, when t= 9 x= 800, i.e. within 9 hours the number of bacteria increased 8 times.

The problem of increasing the amount of enzyme

In a brewer's yeast culture, the rate of growth of the active enzyme is proportional to its initial amount x. Initial amount of enzyme a doubled within an hour. Find dependency

x(t).

Solution. By condition, the differential equation of the process has the form

from here

But . Means, C= a and then

It is also known that

Hence,

6.3. SECOND ORDER DIFFERENTIAL EQUATIONS

6.3.1. Basic Concepts

Definition.Second order differential equation is called a relation connecting the independent variable, the desired function and its first and second derivatives.

In special cases, x may be missing from the equation, at or y". However, a second-order equation must necessarily contain y." IN general case the second order differential equation is written as:

or, if possible, in the form resolved with respect to the second derivative:

As in the case of a first-order equation, for a second-order equation there can be general and particular solutions. The general solution is:

Finding a Particular Solution

under initial conditions - given

numbers) is called Cauchy problem. Geometrically, this means that we need to find the integral curve at= y(x), passing through a given point and having a tangent at this point which is

aligns with the positive axis direction Ox specified angle. e. (Fig. 6.1). The Cauchy problem has a unique solution if the right-hand side of equation (6.10), incessant

is discontinuous and has continuous partial derivatives with respect to uh, uh" in some neighborhood of the starting point

To find constants included in a private solution, the system must be resolved

Rice. 6.1. Integral curve

Instructions

If the equation is presented in the form: dy/dx = q(x)/n(y), classify them as differential equations with separable variables. They can be solved by writing the condition in differentials as follows: n(y)dy = q(x)dx. Then integrate both sides. In some cases, the solution is written in the form of integrals taken from known functions. For example, in the case of dy/dx = x/y, we get q(x) = x, n(y) = y. Write it in the form ydy = xdx and integrate. It should be y^2 = x^2 + c.

To linear equations relate the equations to “first”. An unknown function with its derivatives enters such an equation only to the first degree. Linear has the form dy/dx + f(x) = j(x), where f(x) and g(x) are functions depending on x. The solution is written using integrals taken from known functions.

Please note that many differential equations are second order equations (containing second derivatives). For example, the equation of simple harmonic motion is written in general form: md 2x/dt 2 = –kx. Such equations have, in , particular solutions. The equation of simple harmonic motion is an example of a rather important one: linear differential equations that have constant coefficient.

If in the conditions of the task there is only one linear equation, which means that you have been given additional conditions through which you can find a solution. Read the problem carefully to find these conditions. If variables x and y indicate distance, speed, weight - feel free to set the limit x≥0 and y≥0. It is quite possible that x or y hides the number of apples, etc. – then the values ​​can only be . If x is the son’s age, it is clear that he cannot be older than his father, so indicate this in the conditions of the problem.

Sources:

• how to solve an equation with one variable

Differential and integral calculus problems are important elements consolidation of the theory mathematical analysis, a branch of higher mathematics studied in universities. Differential the equation solved by the integration method.

Instructions

Differential calculus explores the properties of . And vice versa, integrating a function allows for given properties, i.e. derivatives or differentials of a function to find it itself. This is the solution to the differential equation.

Anything is a relationship between an unknown quantity and known data. In the case of a differential equation, the role of the unknown is played by a function, and the role of known quantities is played by its derivatives. In addition, the relation may contain an independent variable: F(x, y(x), y'(x), y''(x),…, y^n(x)) = 0, where x is an unknown variable, y (x) is the function to be determined, the order of the equation is the maximum order of the derivative (n).

Such an equation is called an ordinary differential equation. If the relationship contains several independent variables and partial derivatives (differentials) of the function with respect to these variables, then the equation is called a partial differential equation and has the form: x∂z/∂y - ∂z/∂x = 0, where z(x, y) is the required function.

So, in order to learn how to solve differential equations, you need to be able to find antiderivatives, i.e. solve the problem inverse to differentiation. For example: Solve the first order equation y’ = -y/x.

SolutionReplace y’ with dy/dx: dy/dx = -y/x.

Reduce the equation to a form convenient for integration. To do this, multiply both sides by dx and divide by y:dy/y = -dx/x.

Integrate: ∫dy/y = - ∫dx/x + Сln |y| = - ln |x| +C.

This solution is called the general differential equation. C is a constant whose set of values ​​determines the set of solutions to the equation. For any specific meaning C will be the only solution. This solution is a partial solution of the differential equation.

Solving most higher-order equations degrees does not have a clear formula for finding square roots equations. However, there are several reduction methods that allow you to transform a higher degree equation into a more visual form.

Instructions

The most common method for solving higher degree equations is expansion. This approach is a combination of selecting integer roots, divisors of the free term, and subsequent division of the general polynomial into the form (x – x0).

For example, solve the equation x^4 + x³ + 2 x² – x – 3 = 0. Solution: The free term of this polynomial is -3, therefore, its integer divisors can be the numbers ±1 and ±3. Substitute them one by one into the equation and find out whether you get the identity: 1: 1 + 1 + 2 – 1 – 3 = 0.

Second root x = -1. Divide by the expression (x + 1). Write down the resulting equation (x - 1)·(x + 1)·(x² + x + 3) = 0. The degree has been reduced to the second, therefore, the equation can have two more roots. To find them, solve the quadratic equation: x² + x + 3 = 0D = 1 – 12 = -11

The discriminant is a negative value, which means that the equation no longer has real roots. Find the complex roots of the equation: x = (-2 + i·√11)/2 and x = (-2 – i·√11)/2.

Another method for solving a higher degree equation is to change variables to make it quadratic. This approach is used when all powers of the equation are even, for example: x^4 – 13 x² + 36 = 0

Now find the roots of the original equation: x1 = √9 = ±3; x2 = √4 = ±2.

Tip 10: How to Determine Redox Equations

A chemical reaction is a process of transformation of substances that occurs with a change in their composition. Those substances that react are called initial substances, and those that are formed as a result of this process are called products. It happens that during a chemical reaction, the elements that make up the starting substances change their oxidation state. That is, they can accept someone else's electrons and give away their own. In both cases, their charge changes. Such reactions are called redox reactions.

Today, one of the most important skills for any specialist is the ability to solve differential equations. Solving differential equations - not a single applied task can do without this, be it calculating any physical parameter or modeling changes as a result of adopted macroeconomic policies. 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, in algebra there are many examples of various equations, but these are, for the most part, educational examples that are not applicable in practice. For real interesting mathematics begins when we want to describe the processes occurring in real life. But how can we reflect the time factor that governs real processes—inflation, output, or demographic indicators?

Let us recall one important definition from a mathematics course concerning the derivative of a function. The derivative is the rate of change of a function, hence it can help us reflect the time factor in the equation.

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

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

The second simplest type is called a differential equation with separable variables. 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 can be 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 sides $$\int \frac(dy)(g(y))=\int f(x)dx$$ - this is the solution to the differential equation of separable type.

The last simple type is a 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 required function. To solve such an equation, special methods are used (Lagrange’s method of variation of an arbitrary constant, Bernoulli’s 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. Solving them requires preliminary preparation and experience in solving simpler problems.

The so-called partial differential equations are of great importance for physics and, unexpectedly, finance. 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 profitability, the size of payments, and the start and end dates of payments. Solving a partial differential equation is quite complex, usually you need to use special programs, such as Matlab or Maple.

An example of the application of a differential equation in economics

Let us give, as promised, a simple example of solving a differential equation. First, let's set the task.

For some company, 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 volume of production. We need to find the total revenue.

As you can see from the problem, this is an applied example from microeconomics. Many firms and enterprises constantly face such calculations in the course of their activities.

Let's start with the solution. 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 sides, and obtain the general solution: $$R(q) = \int (10-0.2q)dq = 10 q-0.1q^2+C. $$

To find the constant $C$, recall the condition $R(0)=0$. Let's substitute: $$R(0) =0-0+C = 0. $$ So C=0 and our total revenue function takes the form $R(q)=10q-0.1q^2$. The problem is solved.

Other examples by different types Remote controls are collected on the page:

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

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

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 entering a differential equation is called order of the differential equation.

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

As examples of second order partial differential equations, we give

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 specified implicitly (when possible, we will write it in explicit representation y = f(x) ).

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

Solving a differential equation- it's implicit given functionФ(x, y) = 0 (in some cases the function y can be expressed explicitly through the argument x), which turns the differential equation into an identity.

NOTE.

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

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

The solution to a differential equation is often called integral of the differential equation.

Functions or can be called the solution of a differential equation.

One of the solutions to 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 a differential equation is a set of solutions containing all, without exception, solutions to this differential equation.

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 specified additional conditions, then it is called partial solution of the differential equation.

A partial solution of the differential equation satisfying 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 general solution differential equation on any given interval X.

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

Boundary value 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) = f 0, f (x 1) = f 1, where f 0 and f 1 are given numbers.

The boundary value problem is often called boundary problem.

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