When deciding various tasks physics, chemistry, mathematics and others exact sciences often used mathematical models in the form of equations relating one or more independent variables, an unknown function of these variables, and derivatives (or differentials) of this function. This kind the equations are called differential.
If there is only one independent variable, then the equation is called ordinary; if there are two or more independent variables, then the equation is called partial differential equation. In order to obtain highly qualified specialists in all universities where exact disciplines are studied, a course in differential equations is required. For some students, theory is difficult, practice is a struggle; for others, both theory and practice are difficult. If you analyze differential equations from a practical perspective, then to calculate them you only need to be good at integrating and taking derivatives. All other transformations come down to several schemes that can be understood and studied. Below we will study the basic definitions and method for solving simple DR.

Theory of differential equations

Definition: Ordinary differential equation is an equation that connects the independent variable x, the function y(x), its derivatives y"(x), y n (x) and has general formF(x,y(x),y" (x), …, y n (x))=0
Differential equation(DR) is called either an ordinary differential equation or a partial differential equation. Order of differential equation is determined by the order of the highest derivative (n), which is included in this differential equation.

General solution of the differential equation is a function that contains as many constants as the order of the differential equation, and the substitution of which into a given differential equation turns it into an identity, that is, it has the form y=f(x, C 1, C 2, ..., C n).
A general solution that is not resolved with respect to y(x) and has the form F(x,y,C 1 ,C 2 , …, C n)=0 is called general integral of a differential equation.
The solution found from the general one for fixed values ​​of the constants C 1 , C 2 , …, C n is called private solution of a differential equation.
The simultaneous specification of a differential equation and the corresponding number of initial conditions is called Cauchy problem.
F(x,y,C 1 ,C 2 , …, C n)=0
y(x0)=y0;
….
y n (x0)=y n (0)
Ordinary differential equation of the first order called an equation of the form
F(x, y, y")=0. (1)
Integral of the equation(1) is called a relation of the form Ф (x,y)=0 if each continuously differentiated function implicitly specified by it is a solution to equation (1).
An equation that has the form (1) and cannot be reduced to simple view is called an equation, undecidable with respect to the derivative. If it can be written in the form
y" = f(x,y), then it is called solved equation for the derivative.
Cauchy problem for a first order equation contains only one initial condition and has the form:
F(x,y,y")=0
y(x 0)=y 0 .
Equations of the form
M(x,y)dx+N(x,y)dx=0 (2)
where the variables x i y are "symmetric": we can assume that x is an independent variable and y is a dependent variable, or vice versa, y is an independent variable and x is a dependent variable, called equation in symmetric form.
Geometric meaning of a first order differential equation
y"=f(x,y) (3)
is as follows.
This equation establishes a connection (dependence) between the coordinates of the point (x;y) and the angular coefficient y" of the tangent to the integral curve passing through this point. Thus, the equation y"= f(x,y) is a set directions (directions field) on the Cartesian Oxy plane.
A curve constructed at points at which the direction of the field is the same is called an isocline. Isoclins can be used to approximate the construction of integral curves. The isocline equation can be obtained by putting the derivative equal to the constant y"=C
f(x, y)=C - isocline equation..
Integral line of the equation(3) is called the graph of the solution to this equation.
Ordinary differential equations whose solutions can be specified analytically y=g(x) are called integrable equations.
Equations of the form
M 0 (x)dx+N 0 (y)dy=0 (3)
are called equations with separate interchangeables.
From them we will begin our acquaintance with differential equations. The process of finding solutions to DR is called integration of a differential equation.

Separated Variable Equations

Example 1. Find the solution to the equation y"=x .
Check the solution.
Solution: Write the equation in differentials
dy/dx=x or dy=x*dx.
Let's find the integral of the right and left sides of the equation
int(dy)=int(x*dx);
y=x 2 /2+C.
This is the DR integral.
Let's check its correctness and calculate the derivative of the function
y"=1/2*2x+0=x.
As you can see, we received the original DR, therefore the calculations are correct.
We have just found a solution to a first order differential equation. This is exactly simpler equations, which can be imagined.

Example 2. Find the general integral of a differential equation
(x+1)y"=y+3
Solution: Let's write the original equation in differentials
(x+1)dy=(y+3)dx.
The resulting equation is reduced to DR with separated variables

All that's left is to take the integral of both sides

Using tabular formulas we find
ln|y+3|=ln|x+1|+C.
If we expose both parts, we get
y+3=e ln|x+1|+C or y=e ln|x+1|+C -3.
This notation is correct, but not compact.
In practice, a different technique is used; when calculating the integral, the constant is entered under the logarithm
ln|y+3|=ln|x+1|+ln(C).
According to the properties of the logarithm, this allows you to collapse the last two terms
ln|y+3|=ln(C|x+1|).
Now when exposing solving a differential equation will be compact and easy to read
y=С|x+1|+3
Remember this rule; in practice it is used as a calculation standard.

Example 3. Solve differential equation
y"=-y*sin(x).
Solution: Let's write it down equation in differentials
dy/dx= y*sin(x)
or after rearranging the factors in the form separated equations
dy/ y=-sin(x)dx.
It remains to integrate the equation
int(1/y,y)=-int(sin(x), x);
ln|y|=cos(x)-ln(C).
It is convenient to enter the constant under the logarithm, and even with a negative value, so that it can be transferred to left side get
ln|С*y|=cos(x).
Exposing both parts of the dependence
С*y=exp(cos(x)).
This is what it is. You can leave it as is, or you can permanently transfer it to right side

The calculations are not complicated; in most cases, integrals can also be found using tabular integration formulas.

Example 4. Solve the Cauchy problem
y"=y+x, y(1)=e 3 -2.
Solution: Preliminary transformations will no longer take place here. However, the equation is linear and quite simple. In such cases, you need to introduce a new variable
z=y+x.
Remembering that y=y(x) let's find the derivative of z.
z"= y"+1,
from where we express the old derivative
y"= z"-1.
Let's substitute all this into the original equation
z"-1=z or z"=z+1.
Let's write it down differential equation through differentials
dz=(z+1)dx.
Separating the variables in the equation

All that remains is to calculate simple integrals that anyone can do

We expose the dependence to get rid of the logarithm of the function
z+1=e x+C or z=e x+1 -1
Don't forget to return to the completed replacement.
z=x+y= e x+С -1,
write it out from here common decision differential equation
y= e x+C -x-1.
Find a solution to the Cauchy problem in DR in in this case not difficult. We write out the Cauchy condition
y(1)=e 3 -2
and substitute into the solution we just found
e 1 + C -1-1 = e 3 -2.
From here we obtain the condition for calculating the constant
1+C=3; C=3-1=2.
Now we can write solution of the Cauchy problem (partial solution of DR)
y= e x+2 -x-1.
If you know how to integrate well, and you are also doing well with derivatives, then the topic of differential equations will not be an obstacle in your education.
In further study, you will need to study several important diagrams so that you can distinguish between equations and know which substitution or technique works in each case.
After this, homogeneous and inhomogeneous DR, differential equations of the first and higher orders await you. In order not to burden you with theory, in the following lessons we will give only the type of equations and a brief scheme for their calculations. You can read the whole theory from methodological recommendations to study the course " Differential equations" (2014) authors Bokalo Nikolay Mikhailovich, Domanskaya Elena Viktorovna, Chmyr Oksana Yuryevna. You can use other sources that contain explanations of the theory of differential equations that you understand. Ready-made examples for differential. equations taken from the program for mathematicians of LNU named after. I. Frank.
We know how to solve differential equations and we will try to easy way instill this knowledge in you.

Differential equation (DE) - this is the equation,
where are the independent variables, y is the function and are the partial derivatives.

Ordinary differential equation is a differential equation that has only one independent variable, .

Partial differential equation is a differential equation that has two or more independent variables.

The words “ordinary” and “partial derivatives” may be omitted if it is clear which equation is being considered. In what follows, ordinary differential equations are considered.

Order of differential equation is the order of the highest derivative.

Here is an example of a first order equation:

Here is an example of a fourth order equation:

Sometimes a first order differential equation is written in terms of differentials:

In this case, the variables x and y are equal. That is, the independent variable can be either x or y. In the first case, y is a function of x. In the second case, x is a function of y. If necessary, we can reduce this equation to a form that explicitly includes the derivative y′.
Dividing this equation by dx we get:
.
Since and , it follows that
.

Solving differential equations

Derivatives from elementary functions are expressed through elementary functions. Integrals of elementary functions are often not expressed in terms of elementary functions. With differential equations the situation is even worse. As a result of the solution you can get:

• explicit dependence of a function on a variable;

  Solving a differential equation is the function y = u (x), which is defined, n times differentiable, and .

• implicit dependence in the form of an equation of type Φ (x, y) = 0 or systems of equations;

  Integral of a differential equation is a solution to a differential equation that has an implicit form.

• dependence expressed through elementary functions and integrals from them;

  Solving a differential equation in quadratures - this is finding a solution in the form of a combination of elementary functions and integrals of them.

• the solution may not be expressed through elementary functions.

Since solving differential equations comes down to calculating integrals, the solution includes a set of constants C 1, C 2, C 3, ... C n. The number of constants is equal to the order of the equation. Partial integral of a differential equation is the general integral for given values ​​of the constants C 1, C 2, C 3, ..., C n.

References:
V.V. Stepanov, Course of differential equations, "LKI", 2015.
N.M. Gunter, R.O. Kuzmin, Collection of problems in higher mathematics, “Lan”, 2003.

Ordinary differential equation is an equation that relates an independent variable, an unknown function of this variable and its derivatives (or differentials) of various orders.

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

In addition to ordinary ones, partial differential equations are also studied. These are equations relating independent variables, an unknown function of these variables and its partial derivatives with respect to the same variables. But we will only consider ordinary differential equations and therefore, for the sake of brevity, we will omit the word “ordinary”.

Examples of differential equations:

(1) ;

(3) ;

(4) ;

Equation (1) is fourth order, equation (2) is third order, equations (3) and (4) are second order, equation (5) is first order.

Differential equation n th order does not necessarily have to contain an explicit function, all its derivatives from the first to n-th order and independent variable. It may not explicitly contain derivatives of certain orders, a function, or an independent variable.

For example, in equation (1) there are clearly no third- and second-order derivatives, as well as a function; in equation (2) - the second-order derivative and the function; in equation (4) - the independent variable; in equation (5) - functions. Only equation (3) contains explicitly all the derivatives, the function and the independent variable.

Solving a differential equation every function is called y = f(x), when substituted into the equation it turns into an identity.

The process of finding a solution to a differential equation is called its integration.

Example 1. Find the solution to the differential equation.

Solution. Let's write this equation in the form . The solution is to find the function from its derivative. The original function, as is known from integral calculus, is an antiderivative for, i.e.

That's what it is solution to this differential equation . Changing in it C, we will obtain different solutions. We found out that there is an infinite number of solutions to a first order differential equation.

General solution of the differential equation n th order is its solution, expressed explicitly with respect to the unknown function and containing n independent arbitrary constants, i.e.

The solution to the differential equation in Example 1 is general.

Partial solution of the differential equation a solution in which arbitrary constants are given specific numerical values ​​is called.

Example 2. Find the general solution of the differential equation and a particular solution for .

Solution. Let's integrate both sides of the equation a number of times equal to the order of the differential equation.

,

.

As a result, we received a general solution -

of a given third order differential equation.

Now let's find a particular solution under the specified conditions. To do this, substitute their values ​​instead of arbitrary coefficients and get

.

If, in addition to the differential equation, the initial condition is given in the form , then such a problem is called Cauchy problem . Substitute the values ​​and into the general solution of the equation and find the value of an arbitrary constant C, and then a particular solution of the equation for the found value C. This is the solution to the Cauchy problem.

Example 3. Solve the Cauchy problem for the differential equation from Example 1 subject to .

Solution. Let us substitute the values ​​from the initial condition into the general solution y = 3, x= 1. We get

We write down the solution to the Cauchy problem for this first-order differential equation:

Solving differential equations, even the simplest ones, requires good integration and derivative skills, including complex functions. This can be seen in the following example.

Example 4. Find the general solution to the differential equation.

Solution. The equation is written in such a form that you can immediately integrate both sides.

.

We apply the method of integration by change of variable (substitution). Let it be then.

Required to take dx and now - attention - we do this according to the rules of differentiation of a complex function, since x and there is complex function("apple" - extraction square root or, what is the same thing - raising to the power “one-half”, and “minced meat” is the very expression under the root):

We find the integral:

Returning to the variable x, we get:

.

This is the general solution to this first degree differential equation.

Not only skills from previous sections of higher mathematics will be required in solving differential equations, but also skills from elementary, that is, school mathematics. As already mentioned, in a differential equation of any order there may not be an independent variable, that is, a variable x. Knowledge about proportions from school that has not been forgotten (however, depending on who) from school will help solve this problem. This is the next example.

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 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 has almost been mastered! The more integrals various types 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 And 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 equations in total differentials, since in addition to this differential equation I consider new material – partial integration.

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

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 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.

We won't 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 have drawn 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.

Complete solution and the 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, therefore, 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 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.

In some problems of physics, it is not possible to establish a direct connection between the quantities describing the process. But it is possible to obtain an equality containing the derivatives of the functions under study. This is how differential equations arise and the need to solve them to find an unknown function.

This article is intended for those who are faced with the problem of solving a differential equation in which the unknown function is a function of one variable. The theory is structured in such a way that with zero knowledge of differential equations, you can cope with your task.

Each type of differential equation is associated with a solution method with detailed explanations and solutions to typical examples and problems. All you have to do is determine the type of differential equation of your problem, find a similar analyzed example and carry out similar actions.

To successfully solve differential equations, you will also need the ability to find sets of antiderivatives ( indefinite integrals) various functions. If necessary, we recommend that you refer to the section.

First, we will consider the types of ordinary differential equations of the first order that can be resolved with respect to the derivative, then we will move on to second-order ODEs, then we will dwell on higher-order equations and end with systems of differential equations.

Recall that if y is a function of the argument x.

First order differential equations.

The simplest differential equations of the first order of the form.

Let's write down a few examples of such remote control .

Differential equations can be resolved with respect to the derivative by dividing both sides of the equality by f(x) . In this case, we arrive at an equation that will be equivalent to the original one for f(x) ≠ 0. Examples of such ODEs are .

If there are values ​​of the argument x at which the functions f(x) and g(x) simultaneously vanish, then additional solutions appear. Additional solutions to the equation given x are any functions defined for these argument values. Examples of such differential equations include:

Second order differential equations.

Linear homogeneous differential equations of the second order with constant coefficients.

LDE with constant coefficients is a very common type of differential equation. Their solution is not particularly difficult. First the roots are found characteristic equation . For different p and q, three cases are possible: the roots of the characteristic equation can be real and different, real and coinciding or complex conjugates. Depending on the values ​​of the roots of the characteristic equation, the general solution of the differential equation is written as , or , or respectively.

For example, consider a linear homogeneous second-order differential equation with constant coefficients. The roots of its characteristic equation are k 1 = -3 and k 2 = 0. The roots are real and different, therefore, the general solution of the LODE with constant coefficients has the form


Linear inhomogeneous differential equations of the second order with constant coefficients.

The general solution of a second-order LDDE with constant coefficients y is sought in the form of the sum of the general solution of the corresponding LDDE and a particular solution to the original inhomogeneous equation, that is, . The previous paragraph is devoted to finding a general solution to a homogeneous differential equation with constant coefficients. And a particular solution is determined either by the method uncertain coefficients for a certain form of the function f(x) on the right side original equation, or by the method of varying arbitrary constants.

As examples of second-order LDDEs with constant coefficients, we give


To understand the theory and get acquainted with detailed solutions of examples, we offer you on the page linear inhomogeneous second-order differential equations with constant coefficients.

Linear homogeneous differential equations (LODE) and linear inhomogeneous differential equations (LNDEs) of the second order.

A special case of differential equations of this type are LODE and LDDE with constant coefficients.

The general solution of the LODE on a certain segment is represented by a linear combination of two linearly independent partial solutions y 1 and y 2 of this equation, that is, .

The main difficulty lies precisely in finding linearly independent partial solutions to a differential equation of this type. Typically, particular solutions are selected from the following systems linear independent functions:


However, particular solutions are not always presented in this form.

An example of a LOD is .

The general solution of the LDDE is sought in the form , where is the general solution of the corresponding LDDE, and is the particular solution of the original differential equation. We just talked about finding it, but it can be determined using the method of varying arbitrary constants.

An example of LNDU can be given .

Differential equations of higher orders.

Differential equations that allow a reduction in order.

Order of differential equation , which does not contain the desired function and its derivatives up to k-1 order, can be reduced to n-k by replacing .

In this case, the original differential equation will be reduced to . After finding its solution p(x), it remains to return to the replacement and determine the unknown function y.

For example, the differential equation after the replacement, it will become an equation with separable variables, and its order will be reduced from third to first.