The solution of various geometric, physical and engineering problems often leads to equations that relate the independent variables characterizing a particular problem with some function of these variables and derivatives of this function of various orders.

As an example, we can consider the simplest case of uniformly accelerated motion of a material point.

It is known that the displacement of a material point during uniformly accelerated motion is a function of time and is expressed by the formula:

In turn, acceleration a is derivative with respect to time t from speed V, which is also time derivative t from moving S. Those.

Then we get:
- the equation connects the function f(t) with the independent variable t and the second-order derivative of the function f(t).

Definition. Differential equation is an equation that relates independent variables, their functions, and derivatives (or differentials) of this function.

Definition. If a differential equation has one independent variable, then it is called ordinary differential equation , if there are two or more independent variables, then such a differential equation is called partial differential equation.

Definition. The highest order of derivatives appearing in an equation is called order of the differential equation .

Example.

- ordinary differential equation of the 1st order. IN general view is recorded
.

- ordinary differential equation of the 2nd order. In general it is written

- first order partial differential equation.

Definition. General solution differential equation is such a differentiable function y = (x, C), which, when substituted into the original equation instead of an unknown function, turns the equation into the identity

Properties of the general solution.

1) Because constant C is an arbitrary value, then generally speaking a differential equation has an infinite number of solutions.

2) Under any initial conditions x = x 0, y(x 0) = y 0, there is a value C = C 0 at which the solution to the differential equation is the function y = (x, C 0).

Definition. A solution of the form y = (x, C 0) is called private solution differential equation.

Definition. Cauchy problem (Augustin Louis Cauchy (1789-1857) - French mathematician) is the finding of any particular solution to a differential equation of the form y = (x, C 0), satisfying the initial conditions y(x 0) = y 0.

Cauchy's theorem. (theorem on the existence and uniqueness of a solution to a 1st order differential equation)

If the functionf(x, y) is continuous in some regionDin the planeXOYand has a continuous partial derivative in this region
, then whatever the point (x 0 , y 0 ) in areaD, there is only one solution
equations
, defined in some interval containing point x 0 , taking at x = x 0 meaning (X 0 ) = y 0 , i.e. there is a unique solution to the differential equation.

Definition. Integral A differential equation is any equation that does not contain derivatives and for which the given differential equation is a consequence.

Example. Find common decision differential equation
.

The general solution of the differential equation is sought by integrating the left and right sides of the equation, which is previously transformed as follows:

Now let's integrate:

is the general solution of the original differential equation.

Let's say some initial conditions are given: x 0 = 1; y 0 = 2, then we have

By substituting the obtained value of the constant into the general solution, we obtain a particular solution for the given initial conditions (solution to the Cauchy problem).

Definition. Integral curve is called the graph y = (x) of the solution to a differential equation on the XOY plane.

Definition. By special decision of a differential equation is such a solution at all points of which the Cauchy uniqueness condition is called (see. Cauchy's theorem.) is not fulfilled, i.e. in the neighborhood of some point (x, y) there are at least two integral curves.

Special solutions do not depend on the constant C.

Special solutions cannot be obtained from the general solution for any value of the constant C. If we construct a family of integral curves of a differential equation, then the special solution will be represented by a line that touches at least one integral curve at each point.

Note that not every differential equation has special solutions.

Example. Find the general solution to the differential equation:
Find a special solution if it exists.

This differential equation also has a special solution at= 0. This solution cannot be obtained from the general one, but when substituting into the original equation we obtain an identity. The opinion that the solution y = 0 can be obtained from the general solution with WITH 1 = 0 wrong, because C 1 = e C  0.

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.

Huge contribution to the development and creation of theory differential equations contributed by mathematicians Euler and Lagrange. 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 an infinitesimal segment of a function with two points that are at a 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 method for solving standard examples: we will divide the variables into parts, that is, we will move everything with the variable y to the part where dy is located, and do the same with the variable x. We obtain an equation of the form: dy/f(y)=f(x)dx, which is solved by taking integrals from 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 specific example the whole solution:

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 differential equations of the first order can be written in general form as follows: y"=z(x,y). It should be noted that the right-hand function of two variables is homogeneous, and it cannot be divided into two dependences: z on x and z on y. Check , whether the equation is homogeneous or not is quite simple: we make the replacement x=k*x and y=k*y. Now we cancel all k. If all these letters are canceled, then the equation is homogeneous and you can safely start solving 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 in this way, you must first equate the 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 discuss further actions. It is worth saying that at first solving first-order differential equations causes significant difficulties. However, as you delve deeper into the topic, it starts to work out better and better.

Where are differential equations used?

Differential equations are used very actively in physics, since almost all the basic laws are written in differential form, and the formulas that we see are solutions 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 begin further immersion in 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.

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 on 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.

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 in general 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.

Giving a constant different meanings, you can get infinitely many 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 a few 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 to find 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 advice, 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 not the only 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 a 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 a 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.

The content of the article

DIFFERENTIAL EQUATIONS. Many physical laws that govern certain phenomena are written in the form of a mathematical equation that expresses a certain relationship between certain quantities. Often we're talking about about the relationship between quantities that change over time, for example, engine efficiency, measured by the distance that a car can travel on one liter of fuel, depends on the speed of the car. The corresponding equation contains one or more functions and their derivatives and is called a differential equation. (The rate of change of distance over time is determined by speed; therefore, speed is a derivative of distance; similarly, acceleration is a derivative of speed, since acceleration determines the rate of change of speed with time.) Great importance, which differential equations have for mathematics and especially for its applications, are explained by the fact that the study of many physical and technical problems comes down to solving such equations. Differential equations also play a significant role in other sciences, such as biology, economics and electrical engineering; in fact, they arise wherever there is a need for a quantitative (numerical) description of phenomena (since the world changes over time and conditions change from one place to another).

Examples.

The following examples provide a better understanding of how various problems are formulated in the language of differential equations.

1) The law of decay of some radioactive substances is that the decay rate is proportional to the available amount of this substance. If x– the amount of substance at a certain point in time t, then this law can be written as follows:

Where dx/dt is the decay rate, and k– some positive constant characterizing a given substance. (The minus sign on the right side indicates that x decreases over time; a plus sign, always implied when the sign is not explicitly stated, would mean that x increases over time.)

2) The container initially contains 10 kg of salt dissolved in 100 m 3 of water. If pure water pours into the container at a speed of 1 m 3 per minute and mixes evenly with the solution, and the resulting solution flows out of the container at the same speed, then how much salt will be in the container at any subsequent point in time? If x– amount of salt (in kg) in the container at a time t, then at any time t 1 m 3 of solution in the container contains x/100 kg salt; therefore the amount of salt decreases at a rate x/100 kg/min, or

3) Let there be masses on the body m suspended from the end of the spring, a restoring force acts proportional to the amount of tension in the spring. Let x– the amount of deviation of the body from the equilibrium position. Then, according to Newton's second law, which states that acceleration (the second derivative of x by time, designated d 2 x/dt 2) proportional to force:

The right side has a minus sign because the restoring force reduces the stretch of the spring.

4) The law of body cooling states that the amount of heat in a body decreases in proportion to the difference in body temperature and environment. If a cup of coffee heated to a temperature of 90°C is in a room where the temperature is 20°C, then

Where T– coffee temperature at time t.

5) The Foreign Minister of the State of Blefuscu claims that the arms program adopted by Lilliput forces his country to increase military spending as much as possible. The Minister of Foreign Affairs of Lilliput makes similar statements. The resulting situation (in its simplest interpretation) can be accurately described by two differential equations. Let x And y- expenses for armament of Lilliput and Blefuscu. Assuming that Lilliput increases its expenditures on armaments at a rate proportional to the rate of increase in expenditures on armaments of Blefuscu, and vice versa, we obtain:

where the members are ax And - by describe the military expenditures of each country, k And l are positive constants. (This problem was first formulated in this way in 1939 by L. Richardson.)

After the problem is written in the language of differential equations, you should try to solve them, i.e. find the quantities whose rates of change are included in the equations. Sometimes solutions are found in the form of explicit formulas, but more often they can only be presented in approximate form or qualitative information can be obtained about them. It can often be difficult to determine whether a solution even exists, let alone find one. An important section of the theory of differential equations consists of the so-called “existence theorems”, in which the existence of a solution for one or another type of differential equation is proved.

Initial mathematical formulation physical problem usually contains simplifying assumptions; the criterion of their reasonableness can be the degree of consistency of the mathematical solution with the available observations.

Solutions of differential equations.

Differential equation, for example dy/dx = x/y, is satisfied not by a number, but by a function, in this particular case such that its graph at any point, for example at a point with coordinates (2,3), has a tangent with an angular coefficient equal to the ratio of the coordinates (in our example, 2/3). It is easy to verify this if you construct a large number of points and plot a short segment from each with a corresponding slope. The solution will be a function whose graph touches each of its points to the corresponding segment. If there are enough points and segments, then we can approximately outline the course of the solution curves (three such curves are shown in Fig. 1). There is exactly one solution curve passing through each point with y No. 0. Each individual solution is called a partial solution of a differential equation; if it is possible to find a formula containing all the particular solutions (with the possible exception of a few special ones), then they say that a general solution has been obtained. A particular solution represents one function, while a general solution represents a whole family of them. Solving a differential equation means finding either its particular or general solution. In the example we are considering, the general solution has the form y 2 – x 2 = c, Where c– any number; a particular solution passing through the point (1,1) has the form y = x and it turns out when c= 0; a particular solution passing through point (2,1) has the form y 2 – x 2 = 3. The condition requiring that the solution curve pass, for example, through the point (2,1), is called the initial condition (since it specifies the starting point on the solution curve).

It can be shown that in example (1) the general solution has the form x = ce –kt, Where c– a constant that can be determined, for example, by indicating the amount of substance at t= 0. Equation from example (2) – special case equation from example (1), corresponding k= 1/100. Initial condition x= 10 at t= 0 gives a particular solution x = 10e –t/100 . The equation from example (4) has a general solution T = 70 + ce –kt and private solution 70 + 130 – kt; to determine the value k, additional data is needed.

Differential equation dy/dx = x/y is called a first-order equation, since it contains the first derivative (the order of a differential equation is usually considered to be the order of the highest derivative included in it). For most (though not all) differential equations of the first kind that arise in practice, only one solution curve passes through each point.

There are several important types of first-order differential equations that can be solved in the form of formulas containing only elementary functions– powers, exponents, logarithms, sines and cosines, etc. Such equations include the following.

Equations with separable variables.

Equations of the form dy/dx = f(x)/g(y) can be solved by writing it in differentials g(y)dy = f(x)dx and integrating both parts. In the worst case, the solution can be represented in the form of integrals of known functions. For example, in the case of the equation dy/dx = x/y we have f(x) = x, g(y) = y. By writing it in the form ydy = xdx and integrating, we get y 2 = x 2 + c. Equations with separable variables include equations from examples (1), (2), (4) (they can be solved in the manner described above).

Equations in total differentials.

If the differential equation has the form dy/dx = M(x,y)/N(x,y), Where M And N are two given functions, then it can be represented as M(x,y)dx – N(x,y)dy= 0. If left side is the differential of some function F(x,y), then the differential equation can be written as dF(x,y) = 0, which is equivalent to the equation F(x,y) = const. Thus, the solution curves of the equation are the “lines of constant levels” of the function, or the locus of points that satisfy the equations F(x,y) = c. The equation ydy = xdx(Fig. 1) - with separable variables, and the same - in total differentials: to make sure of the latter, we write it in the form ydy – xdx= 0, i.e. d(y 2 – x 2) = 0. Function F(x,y) in this case is equal to (1/2)( y 2 – x 2); Some of its constant level lines are shown in Fig. 1.

Linear equations.

Linear equations are equations of “first degree” - the unknown function and its derivatives appear in such equations only to the first degree. Thus, the first order linear differential equation has the form dy/dx + p(x) = q(x), Where p(x) And q(x) – functions that depend only on x. Its solution can always be written using integrals of known functions. Many other types of first-order differential equations are solved using special techniques.

Higher order equations.

Many differential equations that physicists encounter are second-order equations (i.e., equations containing second derivatives). Such, for example, is the equation of simple harmonic motion from example (3), md 2 x/dt 2 = –kx. Generally speaking, we can expect that a second-order equation has partial solutions that satisfy two conditions; for example, one can require that the solution curve pass through a given point at in this direction. In cases where the differential equation contains a certain parameter (a number whose value depends on the circumstances), solutions of the required type exist only for certain values ​​of this parameter. For example, consider the equation md 2 x/dt 2 = –kx and we will demand that y(0) = y(1) = 0. Function yє 0 is obviously a solution, but if it is an integer multiple p, i.e. k = m 2 n 2 p 2, where n is an integer, but in reality only in this case, there are other solutions, namely: y= sin npx. The parameter values ​​for which the equation has special solutions are called characteristic or eigenvalues; they play an important role in many tasks.

The equation of simple harmonic motion is an example of an important class of equations, namely linear differential equations with constant coefficients. More general example(also second order) – equation

Where a And b– given constants, f(x) is a given function. Such equations can be solved different ways, for example, using the integral Laplace transform. The same can be said about linear equations of higher orders with constant coefficients. Not small role they also play linear equations with variable odds.

Nonlinear differential equations.

Equations containing unknown functions and their derivatives to powers higher than the first or in some more complex manner are called nonlinear. IN last years they are attracting more and more attention. The fact is that physical equations are usually linear only to a first approximation; Further and more accurate research, as a rule, requires the use of nonlinear equations. In addition, many problems are nonlinear in nature. Since solutions to nonlinear equations are often very complex and difficult to imagine simple formulas, Substantial part modern theory is devoted to a qualitative analysis of their behavior, i.e. the development of methods that make it possible, without solving the equation, to say something significant about the nature of the solutions as a whole: for example, that they are all limited, or have a periodic nature, or depend in a certain way on the coefficients.

Approximate solutions to differential equations can be found numerically, but this requires a lot of time. With the advent of high-speed computers, this time was greatly reduced, which opened up new possibilities for the numerical solution of many problems that were previously intractable to such a solution.

Existence theorems.

An existence theorem is a theorem that states that, under certain conditions, a given differential equation has a solution. There are differential equations that have no solutions or have more of them than expected. The purpose of an existence theorem is to convince us that a given equation actually has a solution, and most often to assure us that it has exactly one solution of the required type. For example, the equation we have already encountered dy/dx = –2y has exactly one solution passing through each point of the plane ( x,y), and since we have already found one such solution, we have thereby completely solved this equation. On the other hand, the equation ( dy/dx) 2 = 1 – y 2 has many solutions. Among them are straight y = 1, y= –1 and curves y= sin( x + c). The solution may consist of several segments of these straight lines and curves, passing into each other at points of contact (Fig. 2).

Partial differential equations.

An ordinary differential equation is a statement about the derivative of an unknown function of one variable. A partial differential equation contains a function of two or more variables and derivatives of that function with respect to at least two different variables.

In physics, examples of such equations are Laplace's equation

X, y) inside the circle if the values u specified at each point of the bounding circle. Since problems with more than one variable in physics are the rule rather than the exception, it is easy to imagine how vast the subject of the theory of partial differential equations is.

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.