A method for solving differential equations with separable variables is considered. An example is given detailed solution differential equation with separable variables.

Content

Definition

Let s (x), q (x)- functions of the variable x ;
p (y), r (y)- functions of the variable y .

A differential equation with separable variables is an equation of the form

Method for solving a differential equation with separable variables

Consider the equation:
(i) .
We express the derivative y in terms of differentials.
;
.
Multiply by dx .
(ii)
Divide the equation by s (x)r(y). This can be done if s (x) r(y) ≠ 0. For s (x) r(y) ≠ 0 we have
.
Integrating, we obtain the general integral in quadratures
(iii) .

Since we divided by s (x)r(y), then we get the integral of the equation for s (x) ≠ 0 and r (y) ≠ 0. Next, you need to solve the equation
r (y) = 0.
If this equation has roots, then they are also solutions of equation (i). Let the equation r (y) = 0. has n roots a i , r (a i ) = 0, i = 1, 2, ... , n. Then the constants y = a i are solutions of equation (i). Some of these solutions may already be contained in the general integral (iii).

Note that if the original equation is given in the form (ii), then the equation should also be solved
s (x) = 0.
Its roots b j , s (b j ) = 0, j = 1, 2, ... , m. give solutions x = b j .

An example of solving a differential equation with separable variables

solve the equation

We express the derivative in terms of differentials:

Multiply by dx and divide by . For y ≠ 0 we have:

Let's integrate.

We calculate the integrals using the formula.

Substituting, we obtain the general integral of the equation
.

Now consider the case, y = 0 .
It is obvious that y = 0 is a solution to the original equation. It is not included in the general integral.
So let's add it to the final result.

; y= 0 .

References:
N.M. Gunther, R.O. Kuzmin, Collection of problems in higher mathematics, Lan, 2003.

A method for solving differential equations reducing to equations with separable variables is considered. An example of a detailed solution of a differential equation that reduces to an equation with separable variables is given.

Content

Formulation of the problem

Consider the differential equation
(i) ,
where f is a function, a, b, c are constants, b ≠ 0 .
This equation is reduced to an equation with separable variables.

Solution method

We make a substitution:
u = ax + by + c
Here y is a function of x . Therefore, u is also a function of x .
Differentiate with respect to x
u′ = (ax + by + c)′ = a + by′
Substitute (i)
u′ = a + by′ = a + b f(ax + by + c) = a + b f (u)
Or:
(ii)
Separate variables. Multiply by dx and divide by a + b f (u). If a + b f (u) ≠ 0, then

By integrating, we obtain the general integral of the original equation (i) in squares:
(iii) .

Finally, consider the case
(iv) a + b f (u) = 0.
Suppose this equation has n roots u = r i , a + b f (r i ) = 0, i = 1, 2, ...n. Since the function u = r i is constant, its derivative with respect to x is equal to zero. Therefore, u = r i is a solution to the equation (ii).
However, the equation (ii) does not match the original equation (i) and, perhaps, not all solutions u = r i , expressed in terms of the variables x and y , satisfy the original equation (i).

Thus, the solution to the original equation is the general integral (iii) and some roots of the equation (iv).

An example of solving a differential equation that reduces to an equation with separable variables

solve the equation
(1)

We make a substitution:
u = x - y
Differentiate with respect to x and perform transformations:
;

Multiply by dx and divide by u 2 .

If u ≠ 0, then we get:

We integrate:

We apply the formula from the table of integrals:

We calculate the integral

Then
;
, or

Common decision:
.

Now consider the case u = 0 , or u = x - y = 0 , or
y=x.
Since y′ = (x)′ = 1, then y = x is a solution to the original equation (1) .

;
.

References:
N.M. Gunther, R.O. Kuzmin, Collection of problems in higher mathematics, Lan, 2003.

Differential Equations first order. Solution examples.
Differential equations with separable variables

Differential Equations (DE). These two words usually terrify the average layman. Differential equations seem to be something outrageous and difficult to master for many students. Uuuuuu… differential equations, how would I survive all this?!

Such an opinion and such an attitude is fundamentally wrong, because in fact DIFFERENTIAL EQUATIONS ARE SIMPLE AND EVEN FUN. What do you need to know and be able to learn to solve differential equations? To successfully study diffures, you must be good at integrating and differentiating. The better the topics are studied Derivative of a function of one variable and Indefinite integral, the easier it will be to understand differential equations. I will say more, if you have more or less decent integration skills, then the topic is practically mastered! The more integrals various types you know how to decide - the better. Why? You have to integrate a lot. And differentiate. Also highly recommend learn to find.

In 95% of cases in control work there are 3 types of first-order differential equations: separable equations, which we will cover in this lesson; homogeneous equations and linear inhomogeneous equations. For beginners to study diffusers, I advise you to read the lessons in this sequence, and after studying the first two articles, it will not hurt to consolidate your skills in an additional workshop - equations that reduce to homogeneous.

There are even rarer types of differential equations: equations in total differentials, Bernoulli's equations, and some others. The most important of the last two types are the equations in total differentials, because in addition to this DE, I am considering a new material - partial integration.

If you only have a day or two left, then for ultra-fast preparation there is blitz course in pdf format.

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

Let us first recall the usual algebraic equations. They contain variables and numbers. The simplest example: . What does it mean to solve an ordinary equation? This means to find set of numbers that satisfy this equation. It is easy to see that the children's equation has a single root: . For fun, let's do a check, substitute the found root into our equation:

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

Diffuras are arranged in much the same way!

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

In some equations of the 1st order, there may be no "x" or (and) "y", but this is not essential - important so that in DU was first derivative, and did not have derivatives of higher orders - , etc.

What means ? To solve a differential equation means to find set of all functions that satisfy this equation. Such a set of functions often has the form ( is an arbitrary constant), which is called general solution of the differential equation.

Example 1

Solve differential equation

Full ammo. Where to begin solution?

First of all, you need to rewrite the derivative in a slightly different form. We recall the cumbersome notation, which many of you probably thought was ridiculous and unnecessary. It is it that rules in diffusers!

In the second step, let's see if it's possible split variables? What does it mean to separate variables? Roughly speaking, on the left side we need to leave only "games", a on the right side organize only x's. Separation of variables is carried out with the help of "school" manipulations: bracketing, transferring terms from part to part with a sign change, transferring factors from part to part according to the rule of proportion, etc.

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

Variables are separated. On the left side - only "Game", on the right side - only "X".

Next stage - differential equation integration. It's simple, we hang integrals on both parts:

Of course, integrals must be taken. 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 (because a constant + a constant is still equal to another constant). In most cases, it is placed in right side.

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

An 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 often used in practical tasks: if a logarithm appears on the right side after integration, then in many cases (but by no means always!) it is advisable to write the constant under the logarithm as well. And write ALWAYS if only logarithms are obtained (as in the example under consideration).

That is, INSTEAD OF records are usually written .

Why is this needed? And in order to make it easier to express "y". We use 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 found solution 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 was required to be checked.

Giving a constant various meanings, you can get infinitely many private decisions 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 is a family of linear functions, or rather, a family of direct proportionalities.

After a detailed discussion of the first example, it is appropriate to answer a few naive questions about differential equations:

1)In this example, we managed to separate the variables. Is it always possible to do this? No not always. And even more often the variables cannot be separated. For example, in homogeneous first order equations must be replaced first. In other types of equations, for example, in a linear non-homogeneous equation of the first order, you need to use various tricks and methods to find a general solution. The separable variable equations that we consider in the first lesson are the simplest type of 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 such DEs can be solved approximately using special methods. D'Alembert and Cauchy guarantee... ...ugh, lurkmore.to I read a lot just now, I almost added "from the other world."

3) In this example, we have obtained a solution in the form of a general integral . Is it always possible to find a general solution from the general integral, that is, to express "y" in an explicit form? No not always. For example: . Well, how can I express "y" 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 cumbersomely and clumsily that it is better to leave the answer in the form of a general integral

4) ...perhaps enough for now. In the first example, we met another important point , but in order not to cover the "dummies" with an avalanche of new information, I will leave it until the next lesson.

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

Example 2

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

Solution: according to the condition it is required to find private decision DE that satisfies a given initial condition. This kind of questioning is also called Cauchy problem.

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

We rewrite the derivative in desired form:

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

We integrate the equation:

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

Now we are trying to convert the general integral into a general solution (express "y" explicitly). We remember the old, good, school: . In this case:

The constant in the indicator looks somehow not kosher, so it is usually lowered from heaven to earth. In detail, it happens like this. Using the property of degrees, we rewrite the function as follows:

If is a constant, then is also some constant, redesignate it with the letter :
- at the same time, we remove the module, after which the constant "ce" can take both positive and negative values

Remember the "demolition" of a constant is second technique, which is often used in the course of solving differential equations. On a clean copy, you can immediately go from to , but always be prepared to explain this transition.

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

At the final stage, you need to find a particular solution that satisfies the given initial condition . It's simple too.

What is the task? Need to pick up such the value of the constant to satisfy the condition .

You can arrange it in different ways, but the most understandable, perhaps, will be like this. In the general solution, instead of “x”, we substitute zero, and instead of “y”, 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 do a check. Verification of a particular solution includes two stages:

First, it is necessary to check whether the found particular solution really satisfies the initial condition ? Instead of "x" we substitute zero and see what happens:
- yes, indeed, a deuce was obtained, which means that the initial condition is satisfied.

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

Substitute in the original equation:

- the correct equality is obtained.

Conclusion: the particular solution is 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:

Assessing whether variables can be separated? Can. We transfer the second term to the right side with a sign change:

And we flip the factors according to the rule of proportion:

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

I must warn you, judgment day is coming. If you have not learned well indefinite integrals, solved few examples, then there is nowhere to go - you have to master them now.

The integral of the left side is easy to find, with the integral of the cotangent we deal with the standard technique that we considered in the lesson Integration of trigonometric functions In the past year:

As a result, we got only logarithms, and, according to my first technical recommendation, we also define the constant under the logarithm.

Now we try to simplify the general integral. Since we have only logarithms, it is quite possible (and necessary) to get rid of them. By using known properties maximally "pack" the logarithms. I will write in great detail:

The packaging is complete to be barbarously tattered:
, and immediately-immediately give common integral to the mind, as soon as possible:

Generally speaking, it is not necessary to do this, but it is always beneficial to please the professor ;-)

In principle, this masterpiece can be written as an answer, but here it is still appropriate to square both parts and redefine the constant:

Answer: general integral:

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

Is it possible to express "y"? Can. Let's express the general solution:

Of course, the result obtained is suitable for an answer, but note that the general integral looks more compact, and the solution turned out to be shorter.

Third tech tip:if a significant number of actions must be performed to obtain a general solution, then in most cases it is better to refrain from these actions and leave the answer in the form of a general integral. The same applies to “bad” actions when it is required to express an inverse function, raise to a power, take a root, etc. The fact is that the general solution will look pretentious and cumbersome - with large roots, signs and other mathematical trash.

How to check? Verification can be done in two ways. Method one: take the general solution , we find the derivative and substitute them into the original equation. Try it yourself!

The second way is to differentiate the general integral. It's pretty easy, the main thing is to be able to find derivative of a function defined implicitly:

divide each term by:

and on :

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

Example 4

Find a particular solution of the differential equation that satisfies the initial condition. Run a check.

This is a do-it-yourself example.

I 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 the sample in Example No. 2), you need:
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 of a differential equation , satisfying the initial condition . Run a check.

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

We integrate the equation:

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

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 modulo signs are redundant:

(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”, the logarithm of two:

More familiar design:

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

Answer: private solution:

Check: First, 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. We find 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:

We 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 is found correctly.

The second way of checking is mirrored and more familiar: from the equation express the derivative, for this we divide all the pieces by:

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

Example 6

Find the general integral of the equation , present the answer as .

This is an example for self-solving, full solution and answer at the end of the lesson.

What difficulties await in solving differential equations with separable variables?

1) It is not always obvious (especially to a teapot) that variables can be separated. Consider a conditional example: . Here you need to take the factors out of brackets: and separate the roots:. How to proceed further is clear.

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

3) Transformations with a constant. As everyone has noticed, a constant in differential equations can be handled quite freely, and some transformations are not always clear to a beginner. Let's look at another hypothetical example: . In it, it is advisable to multiply all the terms by 2: . The resulting constant is also some kind of constant, which can be denoted by: . Yes, and since we have the same logarithms, it is advisable to rewrite the constant as another constant: .

The trouble is that they often do not bother with indices and use the same letter. As a result, the decision record takes the following form:

What the heck?! Here are the errors! Strictly speaking, yes. However, from a substantive point of view, there are no errors, because as a result of the transformation of a variable constant, an equivalent variable constant is 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 again an error - on the right, it should be written . But it is informally implied that “minus ce” is still a constant that just as well takes on the same set of values, and therefore putting “minus” does not make sense.

I will try to avoid a careless approach, and still put down different indexes for constants when converting them. Which is what I advise you to do.

Example 7

Solve the differential equation. Run a check.

Solution: This equation admits separation of variables. Separating variables:

We integrate:

The constant here does not have to be defined under the logarithm, since nothing good will come of it.

Answer: general integral:

And, of course, here it is NOT NECESSARY to express “y” explicitly, because it will turn out to be trash (remember the third technical tip).

Examination: Differentiate the answer (implicit function):

We get rid of fractions, for this we multiply 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 DE.
,

Definition 7. An equation of the form is called an equation with separable variables.

This equation can be reduced to the form by dividing all the terms of the equation by the product.

For example, solve the equation

Solution. The derivative is equal to

Separating the variables, we get:

Now let's integrate:

Solve Differential Equation

Solution. This is a first order equation with separable variables. To separate the variables of this equation in the form and divide it term by term into the product . As a result, we get or

integrating both parts of the last equation, we obtain the general solution

arcsin y = arcsin x + C

Let us now find a particular solution that satisfies the initial conditions . Substituting the initial conditions into the general solution, we obtain

; whence C=0

Therefore, a particular solution has the form arc sin y \u003d arc sin x, but the sines of equal arcs are equal to each other

sin (arcsin y) = sin (arcsin x).

Whence, by definition of the arcsine, it follows that y = x.

Homogeneous differential equations

Definition 8. A differential equation of the form that can be reduced to the form is called homogeneous.

To integrate such equations, a change of variables is made, assuming . This substitution results in a differential equation for x and t in which the variables are separated, after which the equation can be integrated. To get the final answer, you need to replace the variable t with .

For example, solve the equation

Solution. Let's rewrite the equation like this:

we get:

After reduction by x 2 we have:

Let's replace t with:

Review questions

1 What is a differential equation?

2 Name the types of differential equations.

3 Tell the algorithms for solving all these equations.

Example 3

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

Assessing whether variables can be separated? Can. We transfer the second term to the right side with a sign change:

And we flip the factors according to the rule of proportion:

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

I must warn you, judgment day is coming. If you have not learned well indefinite integrals, solved few examples, then there is nowhere to go - you have to master them now.

On the right side, we got a logarithm, according to my first technical recommendation, in this case, the constant should also be written under the logarithm.

Now we try to simplify the general integral. Since we have only logarithms, it is quite possible (and necessary) to get rid of them. We “pack” the logarithms as much as possible. Packaging is carried out using three properties:

Please rewrite these three formulas to yourself in workbook, they are used very often when solving diffuses.

I will write the solution in great detail:

Packaging is complete, remove the logarithms:

Is it possible to express "y"? Can. Both parts must be squared. But you don't have to.

Third tech tip: If, to obtain a general solution, you need 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 pretentious and terrible - with large roots, signs.

Therefore, we write the answer as a general integral. It is considered good form to present the general integral 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 did not coincide with a previously known answer, then this does not mean that you have solved the equation incorrectly.

The general integral is also checked quite easily, the main thing is to be able to find derivatives of a function defined implicitly. Let's differentiate the answer:

We multiply both terms by:

And we divide by:

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

Example 4

Find a particular solution of the differential equation that satisfies the initial condition. Run a check.

This is a do-it-yourself example. I remind you that the Cauchy problem consists of two stages:
1) Finding a general solution.
2) Finding a particular solution.

The check is also carried out in two stages (see also the sample of Example 2), you need:
1) Make sure that the particular solution found really satisfies the initial condition.
2) Check that the particular solution generally satisfies the differential equation.

Full solution and answer at the end of the lesson.

Example 5

Find a particular solution of a differential equation , satisfying the initial condition . Run a check.

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

We integrate the equation:

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

The general integral has been obtained, is it possible to successfully express the general solution? Can. We hang logarithms:

(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”, the logarithm of two:

More familiar design:

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

Answer: private solution:

Check: First, 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. We find the derivative:

The second way of checking is mirrored and more familiar: from the equation express the derivative, for this we divide all the pieces by:

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

Example 6

Solve the differential equation. Express the answer as a general integral.

This is an example for self-solving, full solution and answer at the end of the lesson.

What difficulties await in solving differential equations with separable variables?

3) Transformations with a constant. As everyone has noticed, with a constant in differential equations, you can do almost anything. And not always such transformations are clear to a beginner. Consider another conditional example: . In it, it is advisable to multiply all the terms 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, it is advisable to rewrite the constant as another constant: .

The trouble is that they often do not bother with indices, and use the same letter . And as a result, the decision record takes the following form:

What the hell? Here are the errors. Formally, yes. And informally - there is no error, it is understood that when converting a constant, some other constant is still obtained.

Or such an 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 signs of all multipliers: . Formally, according to the record, there is again an error, it should have been written. But it is informally implied that - it's still some other constant (all the more it can take any value), so changing the sign of the constant does not make any sense and you can use the same letter .

I will try to avoid a careless approach, and still put down different indexes for constants when converting them.

Example 7

Solve the differential equation. Run a check.

Solution: This equation admits separation of variables. Separating variables:

We integrate:

The constant here does not have to be defined under the logarithm, since nothing good will come of it.

Answer: general integral:

Check: Differentiate the answer (implicit function):

We get rid of fractions, for this we multiply 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 DE.
,

This is a do-it-yourself example. The only comment, here you get a general integral, and, more correctly, you need to contrive to find not a particular solution, but private integral. Full solution and answer at the end of the lesson.

As already noted, in diffuras with separable variables, not the simplest integrals often appear. And here are a couple of such examples for an independent decision. I recommend everyone to solve examples No. 9-10, regardless of the level of training, this will update the skills of finding integrals or fill in knowledge gaps.

Example 9

Solve differential equation

Example 10

Solve differential equation

Remember that the general integral can be written in more than one way, and the appearance of your answers may differ from appearance my answers. Brief solution and answers at the end of the lesson.

Successful promotion!

Solutions and answers:

Example 4:Solution: Let's find a general solution. Separating variables:

We integrate:

The general integral has been obtained, we are trying to simplify it. We pack the logarithms and get rid of them:

We express the function explicitly using .
Common decision:

Find a particular solution that satisfies the initial condition .
Method one, instead of "x" we substitute 1, instead of "y" - "e":
.
Method two:

We substitute the found value of the constant into a general solution.
Answer: private solution:

Check: Check if the initial condition is indeed true:
, yes, initial condition performed.
We check whether the particular solution satisfies at all differential equation. First we find the derivative:

We substitute the obtained particular solution and found derivative into the original equation :

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

Example 6:Solution: This equation admits separation of variables. We separate the variables and integrate:

Answer: general integral:

Note: here you can get a general solution:

But, according to my third technical tip, it is not desirable to do this, since such an answer looks pretty bad.

Example 8:Solution: This remote control allows the separation of variables. Separating variables:

We integrate:

General integral:
Find a particular solution (partial integral) corresponding to the given initial condition . We substitute into the general solution and :

Answer: Private integral:
In principle, the answer can be combed and get something more compact. .

Differential equations.

Basic concepts about ordinary differential equations.

Definition 1. Ordinary differential equation n-th order for the function y argument x is called a relation of the form

where F is a given function of its arguments. In the name of this class of mathematical equations, the term "differential" emphasizes that they include derivatives (functions formed as a result of differentiation); the term - "ordinary" says that the desired function depends on only one real argument.

An ordinary differential equation may not explicitly contain an argument x, the desired function and any of its derivatives, but the highest derivative must be included in the equation n- order. For example

a) is a first order equation;

b) is a third order equation.

When writing ordinary differential equations, the notation of derivatives through differentials is often used:

in) is a second order equation;

d) is a first order equation,

forming after division by dx equivalent form of the equation: .

A function is called a solution to an ordinary differential equation if, when substituted into it, it becomes an identity.

For example, the 3rd order equation

Has a solution .

Finding by one method or another, for example, selection, one function that satisfies the equation does not mean solving it. To solve an ordinary differential equation means to find all functions that form an identity when substituted into the equation. For equation (1.1), the family of such functions is formed with the help of arbitrary constants and is called the general solution of the ordinary differential equation n th order, and the number of constants coincides with the order of the equation: y(x): In this case, the solution is called the general integral of equation (1.1).

For example, the following expression is a general solution to a differential equation: , and the second term can be written as , since an arbitrary constant divided by 2 can be replaced by a new arbitrary constant .

By setting some admissible values for all arbitrary constants in the general solution or in the general integral, we obtain a certain function that no longer contains arbitrary constants. This function is called a particular solution or a particular integral of equation (1.1). To find the values of arbitrary constants, and hence the particular solution, various additional conditions to equation (1.1) are used. For example, the so-called initial conditions for (1.2) can be given

In the right parts of the initial conditions (1.2), the numerical values of the function and derivatives are given, and the total number of initial conditions is equal to the number of arbitrary constants being determined.

The problem of finding a particular solution to equation (1.1) from initial conditions is called the Cauchy problem.

§ 2. Ordinary differential equations of the 1st order - basic concepts.

Ordinary differential equation of the 1st order ( n=1) has the form: or, if it can be resolved with respect to the derivative: . Common decision y=y(x, C) or the general integral of the 1st order equations contain one arbitrary constant. The only initial condition for the 1st order equation allows you to determine the value of the constant from the general solution or from the general integral. Thus, a particular solution will be found or, which is also the Cauchy problem will be solved. The question of the existence and uniqueness of a solution to the Cauchy problem is one of the central questions in general theory ordinary differential equations. For a first-order equation, in particular, the theorem is valid, which is accepted here without proof.

Theorem 2.1. If in an equation a function and its partial derivative are continuous in some region D plane XOY , and a point is given in this region, then there exists and, moreover, a unique solution that satisfies both the equation and the initial condition.

The geometrically general solution of the 1st order equation is a family of curves in the plane XOY, who do not have common points and differ from each other by one parameter - the value of the constant C. These curves are called integral curves for the given equation. The integral curves of the equation have an obvious geometric property: at each point, the tangent of the slope of the tangent to the curve is equal to the value of the right side of the equation at that point: . In other words, the equation is given in the plane XOY field of directions of tangents to integral curves. Comment: It should be noted that for the equation the equation and the so-called equation in symmetric form are given .

First order differential equations with separable variables.

Definition. A differential equation with separable variables is an equation of the form (3.1)

or an equation of the form (3.2)

In order to separate the variables in equation (3.1), i.e. reduce this equation to the so-called equation with separated variables, perform the following actions:

;