I think we should start with the history of such a glorious mathematical tool as differential equations. Like all differential and integral calculus, these equations were invented by Newton in the late 17th century. He considered this particular discovery of his to be so important that he even encrypted a message, which today can be translated something like this: “All laws of nature are described by differential equations.” This may seem like an exaggeration, but it is true. Any law of physics, chemistry, biology can be described by these equations.

Mathematicians Euler and Lagrange made a huge contribution to the development and creation of the theory of differential equations. Already in the 18th century, they discovered and developed what they now study in senior university courses.

A new milestone in the study of differential equations began thanks to Henri Poincaré. He created the “qualitative theory of differential equations”, which, combined with the theory of functions of a complex variable, made a significant contribution to the foundation of topology - the science of space and its properties.

What are differential equations?

Many people are afraid of one phrase. However, in this article we will outline in detail the whole essence of this very useful mathematical apparatus, which is actually not as complicated as it seems from the name. In order to start talking about first-order differential equations, you should first become familiar with the basic concepts that are inherently associated with this definition. And we'll start with the differential.

Differential

Many people have known this concept since school. However, let’s take a closer look at it. Imagine the graph of a function. We can increase it to such an extent that any segment of it will take the form of a straight line. Let’s take two points on it that are infinitely close to each other. The difference between their coordinates (x or y) will be infinitesimal. It is called the differential and is denoted by the signs dy (differential of y) and dx (differential of x). It is very important to understand that the differential is not a finite quantity, and this is its meaning and main function.

Now we need to consider the next element, which will be useful to us in explaining the concept of a differential equation. This is a derivative.

Derivative

We all probably heard this concept at school. The derivative is said to be the rate at which a function increases or decreases. However, from this definition much becomes unclear. Let's try to explain the derivative through differentials. Let's return to the infinitesimal segment of the function with two points that are on minimum distance from each other. But even over this distance the function manages to change by some amount. And to describe this change they came up with a derivative, which can otherwise be written as a ratio of differentials: f(x)"=df/dx.

Now it’s worth considering the basic properties of the derivative. There are only three of them:

1. The derivative of a sum or difference can be represented as a sum or difference of derivatives: (a+b)"=a"+b" and (a-b)"=a"-b".
2. The second property is related to multiplication. The derivative of a product is the sum of the products of one function and the derivative of another: (a*b)"=a"*b+a*b".
3. The derivative of the difference can be written as the following equality: (a/b)"=(a"*b-a*b")/b 2 .

All these properties will be useful to us for finding solutions to first-order differential equations.

There are also partial derivatives. Let's say we have a function z that depends on the variables x and y. To calculate the partial derivative of this function, say, with respect to x, we need to take the variable y as a constant and simply differentiate.

Integral

Another important concept is integral. In fact, this is the exact opposite of a derivative. There are several types of integrals, but to solve the simplest differential equations we need the most trivial ones

So, let's say we have some dependence of f on x. We take the integral from it and get the function F(x) (often called the antiderivative), the derivative of which is equal to the original function. Thus F(x)"=f(x). It also follows that the integral of the derivative is equal to the original function.

When solving differential equations, it is very important to understand the meaning and function of the integral, since you will have to take them very often to find the solution.

Equations vary depending on their nature. In the next section, we will look at the types of first-order differential equations, and then learn how to solve them.

Classes of differential equations

"Diffurs" are divided according to the order of the derivatives involved in them. Thus there is first, second, third and more order. They can also be divided into several classes: ordinary and partial derivatives.

In this article we will look at first order ordinary differential equations. We will also discuss examples and ways to solve them in the following sections. We will consider only ODEs, because these are the most common types of equations. Ordinary ones are divided into subspecies: with separable variables, homogeneous and heterogeneous. Next, you will learn how they differ from each other and learn how to solve them.

In addition, these equations can be combined so that we end up with a system of first-order differential equations. We will also consider such systems and learn how to solve them.

Why are we only considering first order? Because you need to start with something simple, and it is simply impossible to describe everything related to differential equations in one article.

Separable equations

These are perhaps the simplest first order differential equations. These include examples that can be written as follows: y"=f(x)*f(y). To solve this equation, we need a formula for representing the derivative as a ratio of differentials: y"=dy/dx. Using it we get the following equation: dy/dx=f(x)*f(y). Now we can turn to the solution method standard examples: let's divide the variables into parts, i.e., move everything with the y variable to the part where dy is located, and do the same with the x variable. We obtain an equation of the form: dy/f(y)=f(x)dx, which is solved by taking integrals of both sides. Don’t forget about the constant that needs to be set after taking the integral.

The solution to any “diffure” is a function of the dependence of x on y (in our case) or, if a numerical condition is present, then the answer in the form of a number. Let's look at the whole solution process using a specific example:

Let's move the variables in different directions:

Now let's take the integrals. All of them can be found in a special table of integrals. And we get:

ln(y) = -2*cos(x) + C

If required, we can express "y" as a function of "x". Now we can say that our differential equation is solved if the condition is not specified. A condition can be specified, for example, y(n/2)=e. Then we simply substitute the values ​​of these variables into the solution and find the value of the constant. In our example it is 1.

Homogeneous differential equations of the first order

Now let's move on to the more difficult part. Homogeneous first order differential equations can be written in general view like this: y"=z(x,y). It should be noted that right function on two variables is homogeneous, and it cannot be divided into two dependences: z on x and z on y. Checking whether an equation is homogeneous or not is quite simple: we make the replacement x=k*x and y=k*y. Now we reduce all k. If all these letters are reduced, then the equation is homogeneous and you can safely begin to solve it. Looking ahead, let's say: the principle of solving these examples is also very simple.

We need to make a replacement: y=t(x)*x, where t is a certain function that also depends on x. Then we can express the derivative: y"=t"(x)*x+t. Substituting all this into our original equation and simplifying it, we get an example with separable variables t and x. We solve it and get the dependence t(x). When we received it, we simply substitute y=t(x)*x into our previous replacement. Then we get the dependence of y on x.

To make it clearer, let's look at an example: x*y"=y-x*e y/x .

When checking with replacement, everything is reduced. This means that the equation is truly homogeneous. Now we make another replacement that we talked about: y=t(x)*x and y"=t"(x)*x+t(x). After simplification, we obtain the following equation: t"(x)*x=-e t. We solve the resulting example with separated variables and get: e -t =ln(C*x). All we have to do is replace t with y/x (after all, if y =t*x, then t=y/x), and we get the answer: e -y/x =ln(x*C).

Linear differential equations of the first order

It's time to look at another broad topic. We will analyze first-order inhomogeneous differential equations. How are they different from the previous two? Let's figure it out. Linear differential equations of the first order in general form can be written as follows: y" + g(x)*y=z(x). It is worth clarifying that z(x) and g(x) can be constant quantities.

And now an example: y" - y*x=x 2 .

There are two solutions, and we will look at both in order. The first is the method of varying arbitrary constants.

In order to solve the equation this way, you must first equate right side to zero and solve the resulting equation, which after transferring the parts will take the form:

ln|y|=x 2 /2 + C;

y=e x2/2 *y C =C 1 *e x2/2 .

Now we need to replace the constant C 1 with the function v(x), which we have to find.

Let's replace the derivative:

y"=v"*e x2/2 -x*v*e x2/2 .

And substitute these expressions into the original equation:

v"*e x2/2 - x*v*e x2/2 + x*v*e x2/2 = x 2 .

You can see that on the left side two terms cancel. If in some example this did not happen, then you did something wrong. Let's continue:

v"*e x2/2 = x 2 .

Now we solve the usual equation in which we need to separate the variables:

dv/dx=x 2 /e x2/2 ;

dv = x 2 *e - x2/2 dx.

To extract the integral, we will have to apply integration by parts here. However, this is not the topic of our article. If you are interested, you can learn how to perform such actions yourself. It is not difficult, and with sufficient skill and care it does not take much time.

Let's turn to the second method of solving inhomogeneous equations: Bernoulli's method. Which approach is faster and easier is up to you to decide.

So, when solving an equation using this method, we need to make a substitution: y=k*n. Here k and n are some x-dependent functions. Then the derivative will look like this: y"=k"*n+k*n". We substitute both replacements into the equation:

k"*n+k*n"+x*k*n=x 2 .

Grouping:

k"*n+k*(n"+x*n)=x 2 .

Now we need to equate to zero what is in parentheses. Now, if we combine the two resulting equations, we get a system of first-order differential equations that needs to be solved:

We solve the first equality as an ordinary equation. To do this you need to separate the variables:

We take the integral and get: ln(n)=x 2 /2. Then, if we express n:

Now we substitute the resulting equality into the second equation of the system:

k"*e x2/2 =x 2 .

And transforming, we get the same equality as in the first method:

dk=x 2 /e x2/2 .

We will also not disassemble further actions. It is worth saying that at first solving first-order differential equations causes significant difficulties. However, with a deeper immersion in the topic, it begins to work out better and better.

Where are differential equations used?

Differential equations are used very actively in physics, since almost all basic laws are written in differential form, and the formulas that we see are the solution to these equations. In chemistry they are used for the same reason: fundamental laws are derived with their help. In biology, differential equations are used to model the behavior of systems, such as predator and prey. They can also be used to create reproduction models of, say, a colony of microorganisms.

How can differential equations help you in life?

The answer to this question is simple: not at all. If you are not a scientist or engineer, then they are unlikely to be useful to you. However for general development It doesn't hurt to know what a differential equation is and how it is solved. And then the son or daughter’s question is “what is a differential equation?” won't confuse you. Well, if you are a scientist or engineer, then you yourself understand the importance of this topic in any science. But the most important thing is that now the question “how to solve a first-order differential equation?” you can always give an answer. Agree, it’s always nice when you understand something that people are even afraid to understand.

Main problems in studying

The main problem in understanding this topic is poor skill in integrating and differentiating functions. If you are bad at taking derivatives and integrals, then it’s probably worth studying and mastering different methods integration and differentiation, and only then begin to study the material that was described in the article.

Some people are surprised when they learn that dx can be carried over, because previously (at school) it was stated that the fraction dy/dx is indivisible. Here you need to read the literature on the derivative and understand that it is a ratio of infinitesimal quantities that can be manipulated when solving equations.

Many people do not immediately realize that solving first-order differential equations is often a function or an integral that cannot be taken, and this misconception gives them a lot of trouble.

What else can you study for a better understanding?

It is best to start further immersion into the world of differential calculus with specialized textbooks, for example, on mathematical analysis for students of non-mathematical specialties. Then you can move on to more specialized literature.

It is worth saying that, in addition to differential equations, there are also integral equations, so you will always have something to strive for and something to study.

Conclusion

We hope that after reading this article you have an idea of ​​what differential equations are and how to solve them correctly.

In any case, mathematics will be useful to us in life in some way. It develops logic and attention, without which every person is without hands.

Lecture notes on

differential equations

Differential equations

Introduction

When studying certain phenomena, a situation often arises when the process cannot be described using the equation y=f(x) or F(x;y)=0. In addition to the variable x and the unknown function, the derivative of this function enters the equation.

Definition: The equation connecting the variable x, the unknown function y(x) and its derivatives is called differential equation. In general, the differential equation looks like this:

F(x;y(x); ;;...;y (n))=0

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

–differential equation of 1st order

–differential equation of 3rd order

Definition: The solution to a differential equation is a function that, when substituted into the equation, turns it into an identity.

Differential equations 1st order

Definition: Equation of the form =f(x;y) or F(x;y; )=0is called a 1st order differential equation.

Definition: The general solution of a 1st order differential equation is the function y=γ(x;c), where (c –const), which, when substituted into the equation, turns it into an identity. Geometrically, on the plane, the general solution corresponds to a family of integral curves depending on the parameter c.

Definition: The integral curve passing through a point in the plane with coordinates (x 0 ; y 0) corresponds to a particular solution of the differential equation satisfying the initial condition:

Theorem on the existence of uniqueness of a solution to a 1st order differential equation

Given a 1st order differential equation
and the function f(x;y) is continuous along with partial derivatives in some region D of the XOY plane, then through the point M 0 (x 0 ;y 0) D passes through the only curve corresponding to a particular solution of the differential equation corresponding to the initial condition y(x 0)=y 0

One integral curve passes through a point in the plane with given coordinates.

If you can't get common decision differential equation of 1st order in explicit form, i.e.
, then it can be obtained implicitly:

F(x; y; c) =0 – implicit form

The general solution in this form is called general integral differential equation.

In relation to the 1st order differential equation, 2 problems are posed:

1) Find the general solution (general integral)

2) Find a particular solution (partial integral) that satisfies the given initial condition. This problem is called the Cauchy problem for a differential equation.

Differential equations with separable variables

Equations of the form:
is called a differential equation with separable variables.

Let's substitute

multiply by dx

let's separate the variables

divide by

Note: it is necessary to consider the special case when

variables are separated

let's integrate both sides of the equation

- common decision

A differential equation with separable variables can be written as:

An isolated case
!

Let's integrate both sides of the equation:

1)

2)
beginning conditions:

Homogeneous differential equations of 1st order

Definition: Function
is called homogeneous of order n if

Example: - homogeneous function of ordern=2

Definition: A homogeneous function of order 0 is called homogeneous.

Definition: Differential equation
is called homogeneous if
- homogeneous function, i.e.

Thus, the homogeneous differential equation can be written as:

Using replacement , where t is a function of the variable x, the homogeneous differential equation is reduced to an equation with separable variables.

- substitute into the equation

The variables are separated, let's integrate both sides of the equation

Let's make the reverse substitution by substituting , we obtain a general solution in implicit form.

A homogeneous differential equation can be written in differential form.

M(x;y)dx+N(x;y)dy=0, where M(x;y) and N(x;y) are homogeneous functions of the same order.

Divide by dx and express

1)

A first-order equation of the form a 1 (x)y" + a 0 (x)y = b(x) is called a linear differential equation. If b(x) ≡ 0 then the equation is called homogeneous, otherwise - heterogeneous. For a linear differential equation, the existence and uniqueness theorem has a more specific form.

Purpose of the service. An online calculator can be used to check the solution homogeneous and inhomogeneous linear differential equations of the form y"+y=b(x) .

=

Use variable substitution y=u*v
Use the method of variation of an arbitrary constant
Find a particular solution for y( ) = .

To obtain a solution, the original expression must be reduced to the form: a 1 (x)y" + a 0 (x)y = b(x). For example, for y"-exp(x)=2*y it will be y"-2 *y=exp(x) .

Theorem. Let a 1 (x) , a 0 (x) , b(x) be continuous on the interval [α,β], a 1 ≠0 for ∀x∈[α,β]. Then for any point (x 0 , y 0), x 0 ∈[α,β], there is a unique solution to the equation that satisfies the condition y(x 0) = y 0 and is defined on the entire interval [α,β].
Consider the homogeneous linear differential equation a 1 (x)y"+a 0 (x)y=0.
Separating the variables, we get , or, integrating both parts, The last relation, taking into account the notation exp(x) = e x , is written in the form

Let us now try to find a solution to the equation in the indicated form, in which instead of the constant C the function C(x) is substituted, that is, in the form

Substituting this solution into the original one, after the necessary transformations we obtain Integrating the latter, we have

where C 1 is some new constant. Substituting the resulting expression for C(x), we finally obtain the solution to the original linear equation
.

Example. Solve the equation y" + 2y = 4x. Consider the corresponding homogeneous equation y" + 2y = 0. Solving it, we get y = Ce -2 x. We are now looking for a solution to the original equation in the form y = C(x)e -2 x. Substituting y and y" = C"(x)e -2 x - 2C(x)e -2 x into the original equation, we have C"(x) = 4xe 2 x, whence C(x) = 2xe 2 x - e 2 x + C 1 and y(x) = (2xe 2 x - e 2 x + C 1)e -2 x = 2x - 1 + C 1 e -2 x is the general solution to the original equation. In this solution, y 1 (. x) = 2x-1 - motion of the object under the influence of force b(x) = 4x, y 2 (x) = C 1 e -2 x - proper motion of the object.

Example No. 2. Find the general solution to the first order differential equation y"+3 y tan(3x)=2 cos(3x)/sin 2 2x.
This is not a homogeneous equation. Let's make a change of variables: y=u v, y" = u"v + uv".
3u v tg(3x)+u v"+u" v = 2cos(3x)/sin 2 2x or u(3v tg(3x)+v") + u" v= 2cos(3x)/sin 2 2x
The solution consists of two stages:
1. u(3v tan(3x)+v") = 0
2. u"v = 2cos(3x)/sin 2 2x
1. Equate u=0, find a solution for 3v tan(3x)+v" = 0
Let's present it in the form: v" = -3v tg(3x)

Integrating, we get:

ln(v) = ln(cos(3x))
v = cos(3x)
2. Knowing v, Find u from the condition: u"v = 2cos(3x)/sin 2 2x
u" cos(3x) = 2cos(3x)/sin 2 2x
u" = 2/sin 2 2x
Integrating, we get:
From the condition y=u v, we get:
y = u v = (C-cos(2x)/sin(2x)) cos(3x) or y = C cos(3x)-cos(2x) cot(3x)

Educational institution "Belarusian State

agricultural Academy"

Department of Higher Mathematics

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

Lecture notes for accounting students

correspondence form of education (NISPO)

Gorki, 2013

First order differential equations

The concept of a differential equation. General and particular solutions

When studying various phenomena, it is often not possible to find a law that directly connects the independent variable and the desired function, but it is possible to establish a connection between the desired function and its derivatives.

The relationship connecting the independent variable, the desired function and its derivatives is called differential equation :

Here x– independent variable, y– the required function,
- derivatives of the desired function. In this case, relation (1) must have at least one derivative.

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

Consider the differential equation

. (2)

Since this equation includes only a first-order derivative, it is called is a first order differential equation.

If equation (2) can be resolved with respect to the derivative and written in the form

, (3)

then such an equation is called a first order differential equation in normal form.

In many cases it is advisable to consider an equation of the form

which is called a first order differential equation written in differential form.

Because
, then equation (3) can be written in the form
or
, where we can count
And
. This means that equation (3) is converted to equation (4).

Let us write equation (4) in the form
. Then
,
,
, where we can count
, i.e. an equation of the form (3) is obtained. Thus, equations (3) and (4) are equivalent.

Solving a differential equation (2) or (3) is called any function
, which, when substituting it into equation (2) or (3), turns it into an identity:

or
.

The process of finding all solutions to a differential equation is called its integration , and the solution graph
differential equation is called integral curve this equation.

If the solution to the differential equation is obtained in implicit form
, then it is called integral of this differential equation.

General solution of a first order differential equation is a family of functions of the form
, depending on an arbitrary constant WITH, each of which is a solution to a given differential equation for any admissible value of an arbitrary constant WITH. Thus, the differential equation has an infinite number of solutions.

Private decision differential equation is a solution obtained from the general solution formula for a specific value of an arbitrary constant WITH, including
.

Cauchy problem and its geometric interpretation

Equation (2) has an infinite number of solutions. In order to select one solution from this set, which is called a private one, you need to set some additional conditions.

The problem of finding a particular solution to equation (2) under given conditions is called Cauchy problem . This problem is one of the most important in the theory of differential equations.

The Cauchy problem is formulated as follows: among all solutions of equation (2) find such a solution
, in which the function
takes the given numeric value , if the independent variable x takes the given numeric value , i.e.

,
, (5)

Where D– domain of definition of the function
.

Meaning called the initial value of the function , A – initial value of the independent variable . Condition (5) is called initial condition or Cauchy condition .

From a geometric point of view, the Cauchy problem for differential equation (2) can be formulated as follows: from the set of integral curves of equation (2), select the one that passes through a given point
.

Differential equations with separable variables

One of the simplest types of differential equations is a first-order differential equation that does not contain the desired function:

. (6)

Considering that
, we write the equation in the form
or
. Integrating both sides of the last equation, we get:
or

. (7)

Thus, (7) is a general solution to equation (6).

Example 1 . Find the general solution to the differential equation
.

Solution . Let's write the equation in the form
or
. Let's integrate both sides of the resulting equation:
,
. We'll finally write it down
.

Example 2 . Find the solution to the equation
given that
.

Solution . Let's find a general solution to the equation:
,
,
,
. By condition
,
. Let's substitute into the general solution:
or
. We substitute the found value of an arbitrary constant into the formula for the general solution:
. This is a particular solution of the differential equation that satisfies the given condition.

The equation

(8)

Called a first order differential equation that does not contain an independent variable . Let's write it in the form
or
. Let's integrate both sides of the last equation:
or
- general solution of equation (8).

Example . Find the general solution to the equation
.

Solution . Let's write this equation in the form:
or
. Then
,
,
,
. Thus,
is the general solution of this equation.

Equation of the form

(9)

integrates using separation of variables. To do this, we write the equation in the form
, and then using the operations of multiplication and division we bring it to such a form that one part includes only the function of X and differential dx, and in the second part – the function of at and differential dy. To do this, both sides of the equation need to be multiplied by dx and divide by
. As a result, we obtain the equation

, (10)

in which the variables X And at separated. Let's integrate both sides of equation (10):
. The resulting relation is the general integral of equation (9).

Example 3 . Integrate Equation
.

Solution . Let's transform the equation and separate the variables:
,
. Let's integrate:
,
or is the general integral of this equation.
.

Let the equation be given in the form

This equation is called first order differential equation with separable variables in a symmetrical form.

To separate the variables, you need to divide both sides of the equation by
:

. (12)

The resulting equation is called separated differential equation . Let's integrate equation (12):

.(13)

Relation (13) is the general integral of differential equation (11).

Example 4 . Integrate a differential equation.

Solution . Let's write the equation in the form

and divide both parts by
,
. The resulting equation:
is a separated variable equation. Let's integrate it:

,
,

,
. The last equality is the general integral of this differential equation.

Example 5 . Find a particular solution to the differential equation
, satisfying the condition
.

Solution . Considering that
, we write the equation in the form
or
. Let's separate the variables:
. Let's integrate this equation:
,
,
. The resulting relation is the general integral of this equation. By condition
. Let's substitute it into the general integral and find WITH:
,WITH=1. Then the expression
is a partial solution of a given differential equation, written as a partial integral.

Linear differential equations of the first order

The equation

(14)

called linear differential equation of the first order . Unknown function
and its derivative enter into this equation linearly, and the functions
And
continuous.

If
, then the equation

(15)

called linear homogeneous . If
, then equation (14) is called linear inhomogeneous .

To find a solution to equation (14) one usually uses substitution method (Bernoulli) , the essence of which is as follows.

We will look for a solution to equation (14) in the form of a product of two functions

, (16)

Where
And
- some continuous functions. Let's substitute
and derivative
into equation (14):

Function v we will select in such a way that the condition is satisfied
.
Then

. Thus, to find a solution to equation (14), it is necessary to solve the system of differential equations
,
,
,
,
The first equation of the system is a linear homogeneous equation and can be solved by the method of separation of variables:
. As a function WITH=1:
you can take one of the partial solutions of the homogeneous equation, i.e. at
or
. Let's substitute into the second equation of the system:
.Then
.

. Thus, the general solution to a first order linear differential equation has the form . Solve the equation
.

Solution . We will look for a solution to the equation in the form
. Then
. Let's substitute into the equation:

or
. Function v choose in such a way that the equality holds
. Then
. Let's solve the first of these equations using the method of separation of variables:
,
,
,
,. Function v Let's substitute into the second equation:
,
,
,
. The general solution to this equation is
.

Questions for self-control of knowledge

What is a differential equation?

What is the order of a differential equation?

Which differential equation is called a first order differential equation?

How is a first order differential equation written in differential form?

What is the solution to a differential equation?

What is an integral curve?

What is the general solution of a first order differential equation?

What is called a partial solution of a differential equation?

How is the Cauchy problem formulated for a first order differential equation?

What is the geometric interpretation of the Cauchy problem?

How to write a differential equation with separable variables in symmetric form?

Which equation is called a first order linear differential equation?

What method can be used to solve a first-order linear differential equation and what is the essence of this method?

Tasks for independent work

Solve differential equations with separable variables:

A)
;
;

b)
V)
.

;

A)
;
G)
;

2. Solve first order linear differential equations:
;
.

V)
G)

;

d) First order differential equations. Examples of solutions. Differential equations with separable variables Differential equations (DE). These two words usually terrify the average person. Differential equations seem to be something prohibitive and difficult to master for many students. Uuuuuu... differential equations, how can I survive all this?! This opinion and this attitude is fundamentally wrong, because in fact DIFFERENTIAL EQUATIONS - IT'S SIMPLE AND EVEN FUN. What do you need to know and be able to do in order to learn how to solve differential equations? To successfully study diffuses, you must be good at integrating and differentiating. The better the topics are studied Derivative of a function of one variable you know how to decide - so much the better. Why? You'll have to integrate a lot. And differentiate. Also highly recommend learn to find.

In 95% of cases in tests There are 3 types of first order differential equations: separable equations which we will look at in this lesson; homogeneous equations This opinion and this attitude is fundamentally wrong, because in fact linear inhomogeneous equations. For those starting to study diffusers, I advise you to read the lessons in exactly this order, and after studying the first two articles, it won’t hurt to consolidate your skills in an additional workshop - equations reducing to homogeneous.

There are even rarer types of differential equations: total differential equations, Bernoulli equations and some others. The most important of the last two types are the equations in full differentials, since in addition to this remote control I am considering new material – partial integration.

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So, the landmarks are set - let's go:

First, let's remember the usual algebraic equations. They contain variables and numbers. The simplest example: . What does it mean to solve an ordinary equation? This means finding set of numbers, which satisfy this equation. It is easy to notice that the children's equation has a single root: . Just for fun, let’s check and substitute the found root into our equation:

– the correct equality is obtained, which means that the solution was found correctly.

The diffusers are designed in much the same way!

Differential equation first order V general case contains:
1) independent variable;
2) dependent variable (function);
3) the first derivative of the function: .

In some 1st order equations there may be no “x” and/or “y”, but this is not significant - important to go to the control room was first derivative, and did not have derivatives of higher orders – , etc.

What means ? Solving a differential equation means finding set of all functions, which satisfy this equation. Such a set of functions often has the form (– an arbitrary constant), which is called general solution of the differential equation.

Example 1

Solve differential equation

Full ammunition. Where to begin solution?

First of all, you need to rewrite the derivative in a slightly different form. We recall the cumbersome designation, which many of you probably seemed ridiculous and unnecessary. This is what rules in diffusers!

In the second step, let's see if it's possible separate variables? What does it mean to separate variables? Roughly speaking, on the left side we need to leave only "Greeks", A on the right side organize only "X's". The division of variables is carried out using “school” manipulations: putting them out of brackets, transferring terms from part to part with a change of sign, transferring factors from part to part according to the rule of proportion, etc.

Differentials and are full multipliers and active participants in hostilities. In the example under consideration, the variables are easily separated by tossing the factors according to the rule of proportion:

Variables are separated. On the left side there are only “Y’s”, on the right side – only “X’s”.

Next stage - integration of differential equation. It’s simple, we put integrals on both sides:

Of course, we need to take integrals. IN in this case they are tabular:

As we remember, a constant is assigned to any antiderivative. There are two integrals here, but it is enough to write the constant once (since constant + constant is still equal to another constant). In most cases it is placed on the right side.

Strictly speaking, after the integrals are taken, the differential equation is considered solved. The only thing is that our “y” is not expressed through “x”, that is, the solution is presented in an implicit form. The solution to a differential equation in implicit form is called general integral of the differential equation. That is, this is a general integral.

The answer in this form is quite acceptable, but is there a better option? Let's try to get common decision.

Please, remember the first technique, it is very common and is often used in practical tasks: if a logarithm appears on the right side after integration, then in many cases (but not always!) it is also advisable to write the constant under the logarithm.

That is, INSTEAD OF entries are usually written .

Why is this necessary? And in order to make it easier to express “game”. Using the property of logarithms . In this case:

Now logarithms and modules can be removed:

The function is presented explicitly. This is the general solution.

Answer: common decision: .

The answers to many differential equations are fairly easy to check. In our case, this is done quite simply, we take the solution found and differentiate it:

Then we substitute the derivative into the original equation:

– the correct equality is obtained, which means that the general solution satisfies the equation, which is what needed to be checked.

By giving a constant different values, you can get an infinite number of private solutions differential equation. It is clear that any of the functions , , etc. satisfies the differential equation.

Sometimes the general solution is called family of functions. In this example, the general solution - this is a family linear functions, or rather, a family of direct proportionality.

After a thorough review of the first example, it is appropriate to answer several naive questions about differential equations:

1)In this example, we were able to separate the variables. Can this always be done? No not always. And even more often, variables cannot be separated. For example, in homogeneous first order equations, you must first replace it. In other types of equations, for example, in a first order linear inhomogeneous equation, you need to use various techniques and methods for finding a general solution. Equations with separable variables, which we consider in the first lesson - simplest type differential equations.

2) Is it always possible to integrate a differential equation? No not always. It is very easy to come up with a “fancy” equation that cannot be integrated; in addition, there are integrals that cannot be taken. But similar DEs can be solved approximately using special methods. D’Alembert and Cauchy guarantee... ...ugh, lurkmore.to read a lot just now, I almost added “from the other world.”

3) In this example, we obtained a solution in the form of a general integral . Is it always possible to find a general solution from a general integral, that is, to express the “y” explicitly? No not always. For example: . Well, how can you express “Greek” here?! In such cases, the answer should be written as a general integral. In addition, sometimes it is possible to find a general solution, but it is written so cumbersome and clumsily that it is better to leave the answer in the form of a general integral

4) ...perhaps that’s enough for now. In the first example we encountered Another one important point , but so as not to cover the “dummies” with an avalanche new information, I'll leave it until the next lesson.

Let's not rush. Another simple remote control and another typical solution:

Example 2

Find a particular solution to the differential equation that satisfies the initial condition

Solution: according to the condition, you need to find private solution DE that satisfies a given initial condition. This formulation of the question is also called Cauchy problem.

First we find a general solution. There is no “x” variable in the equation, but this should not confuse, the main thing is that it has the first derivative.

We rewrite the derivative in the required form:

Obviously, the variables can be separated, boys to the left, girls to the right:

Let's integrate the equation:

The general integral is obtained. Here I drew a constant with an asterisk, the fact is that very soon it will turn into another constant.

Now we try to transform the general integral into a general solution (express the “y” explicitly). Let's remember the good old things from school: . In this case:

The constant in the indicator looks somehow unkosher, so it is usually brought down to earth. In detail, this is how it happens. Using the property of degrees, we rewrite the function as follows:

If is a constant, then is also some constant, let’s redesignate it with the letter :

Remember “demolishing” a constant is second technique, which is often used when solving differential equations.

So, the general solution is: . This is a nice family of exponential functions.

At the final stage, you need to find a particular solution that satisfies the given initial condition. This is also simple.

What is the task? Need to pick up such the value of the constant so that the condition is satisfied.

It can be formatted in different ways, but this will probably be the clearest way. In the general solution, instead of the “X” we substitute a zero, and instead of the “Y” we substitute a two:



That is,

Standard design version:

Now we substitute the found value of the constant into the general solution:
– this is the particular solution we need.

Answer: private solution:

Let's check. Checking a private solution includes two stages:

First you need to check whether the particular solution found really satisfies the initial condition? Instead of the “X” we substitute a zero and see what happens:
- yes, indeed, a two was received, which means that the initial condition is met.

The second stage is already familiar. We take the resulting particular solution and find the derivative:

We substitute into the original equation:

– the correct equality is obtained.

Conclusion: the particular solution was found correctly.

Let's move on to more meaningful examples.

Example 3

Solve differential equation

Solution: We rewrite the derivative in the form we need:

We evaluate whether it is possible to separate the variables? Can. We move the second term to the right side with a change of sign:

And we transfer the multipliers according to the rule of proportion:

The variables are separated, let's integrate both parts:

I must warn you, judgment day is approaching. If you haven't studied well indefinite integrals, have solved few examples, then there is nowhere to go - you will have to master them now.

The integral of the left side is easy to find; we deal with the integral of the cotangent using the standard technique that we looked at in the lesson Integrating trigonometric functions last year:

On the right side we have a logarithm, and, according to my first technical recommendation, the constant should also be written under the logarithm.

Now we try to simplify the general integral. Since we only have logarithms, it is quite possible (and necessary) to get rid of them. By using known properties We “pack” the logarithms as much as possible. I'll write it down in great detail:

The packaging is finished to be barbarically tattered:

Is it possible to express “game”? Can. It is necessary to square both parts.

But you don't need to do this.

Third technical tip: if to obtain a general solution it is necessary to raise to a power or take roots, then In most cases you should refrain from these actions and leave the answer in the form of a general integral. The fact is that the general solution will look simply terrible - with large roots, signs and other trash.

Therefore, we write the answer in the form of a general integral. It is considered good practice to present it in the form , that is, on the right side, if possible, leave only a constant. It is not necessary to do this, but it is always beneficial to please the professor ;-)

Answer: general integral:

! Note: The general integral of any equation can be written in more than one way. Thus, if your result does not coincide with the previously known answer, this does not mean that you solved the equation incorrectly.

The general integral is also quite easy to check, the main thing is to be able to find derivative of a function specified implicitly. Let's differentiate the answer:

We multiply both terms by:

And divide by:

The original differential equation has been obtained exactly, which means that the general integral has been found correctly.

Example 4

Find a particular solution to the differential equation that satisfies the initial condition. Perform check.

This is an example for independent decision.

Let me remind you that the algorithm consists of two stages:
1) finding a general solution;
2) finding the required particular solution.

The check is also carried out in two steps (see sample in Example No. 2), you need to:
1) make sure that the particular solution found satisfies the initial condition;
2) check that a particular solution generally satisfies the differential equation.

Full solution and answer at the end of the lesson.

Example 5

Find a particular solution to the differential equation , satisfying the initial condition. Perform check.

Solution: First, let's find a general solution. This equation already contains ready-made differentials and, which means the solution is simplified. We separate the variables:

Let's integrate the equation:

The integral on the left is tabular, the integral on the right is taken method of subsuming a function under the differential sign:

The general integral has been obtained; is it possible to successfully express the general solution? Can. We hang logarithms on both sides. Since they are positive, the modulus signs are unnecessary:

(I hope everyone understands the transformation, such things should already be known)

So, the general solution is:

Let's find a particular solution corresponding to the given initial condition.
In the general solution, instead of “X” we substitute zero, and instead of “Y” we substitute the logarithm of two:

More familiar design:

We substitute the found value of the constant into the general solution.

Answer: private solution:

Check: First, let's check if the initial condition is met:
- everything is good.

Now let’s check whether the found particular solution satisfies the differential equation at all. Finding the derivative:

Let's look at the original equation: – it is presented in differentials. There are two ways to check. It is possible to express the differential from the found derivative:

Let us substitute the found particular solution and the resulting differential into the original equation :

We use the basic logarithmic identity:

The correct equality is obtained, which means that the particular solution was found correctly.

The second method of checking is mirrored and more familiar: from the equation Let's express the derivative, to do this we divide all the pieces by:

And into the transformed DE we substitute the obtained partial solution and the found derivative. As a result of simplifications, the correct equality should also be obtained.

Example 6

Solve differential equation. Present the answer in the form of a general integral.

This is an example for you to solve on your own, complete solution and answer at the end of the lesson.

What difficulties lie in wait when solving differential equations with separable variables?

1) It is not always obvious (especially to a “teapot”) that variables can be separated. Let's consider conditional example: . Here you need to take the factors out of brackets: and separate the roots: . It’s clear what to do next.

2) Difficulties with the integration itself. Integrals are often not the simplest, and if there are flaws in the skills of finding indefinite integral, then it will be difficult with many diffusers. In addition, the logic “since the differential equation is simple, then at least let the integrals be more complicated” is popular among compilers of collections and training manuals.

3) Transformations with a constant. As everyone has noticed, the constant in differential equations can be handled quite freely, and some transformations are not always clear to a beginner. Let's look at another conditional example: . It is advisable to multiply all terms in it by 2: . The resulting constant is also some kind of constant, which can be denoted by: . Yes, and since there is a logarithm on the right side, then it is advisable to rewrite the constant in the form of another constant: .

The trouble is that they often don’t bother with indexes and use the same letter. As a result, the decision record takes the following form:

What kind of heresy? There are mistakes right there! Strictly speaking, yes. However, from a substantive point of view, there are no errors, because as a result of transforming a variable constant, a variable constant is still obtained.

Or another example, suppose that in the course of solving the equation a general integral is obtained. This answer looks ugly, so it is advisable to change the sign of each term: . Formally, there is another mistake here - it should be written on the right. But informally it is implied that “minus ce” is still a constant ( which can just as easily take any meaning!), so putting a “minus” doesn’t make sense and you can use the same letter.

I will try to avoid a careless approach, and still assign different indices to constants when converting them.

Example 7

Solve differential equation. Perform check.

Solution: This equation allows for separation of variables. We separate the variables:

Let's integrate:

It is not necessary to define the constant here as a logarithm, since nothing useful will come of this.

Answer: general integral:

Check: Differentiate the answer (implicit function):

We get rid of fractions by multiplying both terms by:

The original differential equation has been obtained, which means that the general integral has been found correctly.

Example 8

Find a particular solution of the DE.
,

This is an example for you to solve on your own. The only hint is that here you will get a general integral, and, more correctly speaking, you need to contrive to find not a particular solution, but partial integral. Full solution and answer at the end of the lesson.