Find the general solution of the first differential equation. Differential Equations

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Either already solved with respect to the derivative, or they can be solved with respect to the derivative .

General solution of differential equations of the type on the interval X, which is given, can be found by taking the integral of both sides of this equality.

Get .

Looking at the properties indefinite integral, then we find the desired common decision:

y = F(x) + C,

where F(x)- one of the antiderivatives of the function f(x) in between X, a FROM is an arbitrary constant.

Please note that in most tasks the interval X do not indicate. This means that a solution must be found for everyone. x, for which and the desired function y, and the original equation make sense.

If you need to calculate a particular solution of a differential equation that satisfies the initial condition y(x0) = y0, then after calculating the general integral y = F(x) + C, it is still necessary to determine the value of the constant C=C0 using the initial condition. That is, a constant C=C0 determined from the equation F(x 0) + C = y 0, and the desired particular solution of the differential equation will take the form:

y = F(x) + C0.

Consider an example:

Find the general solution of the differential equation , check the correctness of the result. Let's find a particular solution of this equation that would satisfy the initial condition .

Solution:

After we have integrated the given differential equation, we get:

.

We take this integral by the method of integration by parts:

That., is a general solution of the differential equation.

Let's check to make sure the result is correct. To do this, we substitute the solution that we found into the given equation:

.

That is, at the original equation turns into an identity:

therefore, the general solution of the differential equation was determined correctly.

The solution we found is the general solution of the differential equation for each real value of the argument x.

It remains to calculate a particular solution of the ODE that would satisfy the initial condition . In other words, it is necessary to calculate the value of the constant FROM, at which the equality will be true:

.

.

Then, substituting C = 2 into the general solution of the ODE, we obtain a particular solution of the differential equation that satisfies the initial condition:

.

Ordinary differential equation can be solved with respect to the derivative by dividing the 2 parts of the equation by f(x). This transformation will be equivalent if f(x) does not go to zero for any x from the interval of integration of the differential equation X.

Situations are likely when, for some values ​​of the argument x ∈ X functions f(x) and g(x) turn to zero at the same time. For similar values x the general solution of the differential equation is any function y, which is defined in them, because .

If for some values ​​of the argument x ∈ X the condition is satisfied, which means that in this case the ODE has no solutions.

For all others x from interval X the general solution of the differential equation is determined from the transformed equation.

Let's look at examples:

Example 1

Let us find the general solution of the ODE: .

Solution.

From the properties of the basic elementary functions, it is clear that the natural logarithm function is defined for non-negative values ​​of the argument, therefore, the domain of the expression log(x+3) there is an interval x > -3 . Hence, the given differential equation makes sense for x > -3 . With these values ​​of the argument, the expression x + 3 does not vanish, so one can solve the ODE with respect to the derivative by dividing the 2 parts by x + 3.

We get .

Next, we integrate the resulting differential equation, solved with respect to the derivative: . To take this integral, we use the method of subsuming under the sign of the differential.

First order differential equations. Solution examples.
Differential equations with separable variables

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

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

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

There are even rarer types of differential equations: equations in total differentials, Bernoulli's equations, and some others. Of the last two types, the most important are equations in total differentials, since, in addition to this DE, I consider new material – partial integration.

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

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

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

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

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

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

Example 1

Solve differential equation

Full ammunition. Where to begin solution?

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

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

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

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

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

Of course, integrals must be taken. In this case, they are tabular:

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

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

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

Please, remember the first technique, it is very common and often used in practical tasks: if a logarithm appears on the right side after integration, then in many cases (but by no means always!) it is also advisable to write the constant under the logarithm.

That is, INSTEAD OF records are usually written .

Why is this needed? And in order to make it easier to express "y". We use the property of logarithms . In this case:

Now logarithms and modules can be removed:

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

Answer: common decision: .

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

Then we substitute the derivative into the original equation:

- the correct equality is obtained, which means that the general solution satisfies the equation , which was required to be checked.

Giving a constant various meanings, you can get infinitely many private decisions differential equation. It is clear that any of the functions , , etc. satisfies the differential equation .

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

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

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

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

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

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

Example 2

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

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

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

We rewrite the derivative in the required form:

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

We integrate the equation:

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

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

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

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

Remember the "demolition" of a constant is second technique, which is often used in the course of solving differential equations.

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

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

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

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



That is,

Standard design version:

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

Answer: private solution:

Let's do a check. Verification of a particular solution includes two stages:

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

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

Substitute in the original equation:

- the correct equality is obtained.

Conclusion: the particular solution is found correctly.

Let's move on to more meaningful examples.

Example 3

Solve differential equation

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

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

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

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

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

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

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

The packaging is complete to be barbarously tattered:

Is it possible to express "y"? Can. Both parts must be squared.

But you don't have to.

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

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

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

We multiply both terms by:

And we divide by:

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

Example 4

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

This is an example for independent decision.

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

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

Full solution and answer at the end of the lesson.

Example 5

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

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

We integrate the equation:

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

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

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

So the general solution is:

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

More familiar design:

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

Answer: private solution:

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

Now let's check whether the found particular solution satisfies the differential equation at all. We find the derivative:

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

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

We use the basic logarithmic identity:

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

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

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

Example 6

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

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

What difficulties await in solving differential equations with separable variables?

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

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

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

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

What heresy? Here are the errors! Strictly speaking, yes. However, from a substantive point of view, there are no errors, because as a result of the transformation of a variable constant, 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 again an error - on the right, it should be written . But it is informally implied that “minus ce” is still a constant ( which just as well takes on any values!), so putting a "minus" does not make sense and you can use the same letter.

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

Example 7

Solve the differential equation. Run a check.

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

We integrate:

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

Answer: general integral:

Check: Differentiate the answer (implicit function):

We get rid of fractions, for this we multiply both terms by:

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

Example 8

Find a particular solution of DE.
,

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

I. Ordinary differential equations

1.1. Basic concepts and definitions

A differential equation is an equation that relates an independent variable x, the desired function y and its derivatives or differentials.

Symbolically, the differential equation is written as follows:

F(x,y,y")=0, F(x,y,y")=0, F(x,y,y",y",.., y(n))=0

A differential equation is called ordinary if the desired function depends on one independent variable.

By solving the differential equation is called such a function that turns this equation into an identity.

The order of the differential equation is the order of the highest derivative in this equation

Examples.

1. Consider the first order differential equation

The solution to this equation is the function y = 5 ln x. Indeed, by substituting y" into the equation, we get - an identity.

And this means that the function y = 5 ln x– is the solution of this differential equation.

2. Consider the second order differential equation y" - 5y" + 6y = 0. The function is the solution to this equation.

Really, .

Substituting these expressions into the equation, we get: , - identity.

And this means that the function is the solution of this differential equation.

Integration of differential equations is the process of finding solutions to differential equations.

General solution of the differential equation is called a function of the form , which includes as many independent arbitrary constants as the order of the equation.

Partial solution of the differential equation is called the solution obtained from the general solution for different numerical values ​​of arbitrary constants. The values ​​of arbitrary constants are found at certain initial values ​​of the argument and function.

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

Examples

1. Find a particular solution to a first-order differential equation

xdx + ydy = 0, if y= 4 at x = 3.

Solution. Integrating both sides of the equation, we get

Comment. An arbitrary constant C obtained as a result of integration can be represented in any form convenient for further transformations. In this case, taking into account the canonical equation of the circle, it is convenient to represent an arbitrary constant С in the form .

is the general solution of the differential equation.

A particular solution of an equation that satisfies the initial conditions y = 4 at x = 3 is found from the general by substituting the initial conditions into the general solution: 3 2 + 4 2 = C 2 ; C=5.

Substituting C=5 into the general solution, we get x2+y2 = 5 2 .

This is a particular solution of the differential equation obtained from the general solution under given initial conditions.

2. Find the general solution of the differential equation

The solution of this equation is any function of the form , where C is an arbitrary constant. Indeed, substituting into the equations, we obtain: , .

Therefore, this differential equation has an infinite number of solutions, since for different values ​​of the constant C, the equality determines different solutions of the equation.

For example, by direct substitution, one can verify that the functions are solutions of the equation .

A problem in which it is required to find a particular solution to the equation y" = f(x, y) satisfying the initial condition y(x0) = y0, is called the Cauchy problem.

Equation solution y" = f(x, y), satisfying the initial condition, y(x0) = y0, is called a solution to the Cauchy problem.

The solution of the Cauchy problem has a simple geometric meaning. Indeed, according to these definitions, to solve the Cauchy problem y" = f(x, y) on condition y(x0) = y0, means to find the integral curve of the equation y" = f(x, y) which goes through given point M0 (x0,y 0).

II. First order differential equations

2.1. Basic concepts

A first-order differential equation is an equation of the form F(x,y,y") = 0.

The first order differential equation includes the first derivative and does not include higher order derivatives.

The equation y" = f(x, y) is called a first-order equation solved with respect to the derivative.

A general solution of a first-order differential equation is a function of the form , which contains one arbitrary constant.

Example. Consider a first order differential equation.

The solution to this equation is the function .

Indeed, replacing in this equation with its value, we obtain

that is 3x=3x

Therefore, the function is a general solution of the equation for any constant C.

Find a particular solution of this equation that satisfies the initial condition y(1)=1 Substituting initial conditions x=1, y=1 into the general solution of the equation , we obtain whence C=0.

Thus, we obtain a particular solution from the general one by substituting into this equation, the resulting value C=0 is a private decision.

2.2. Differential equations with separable variables

A differential equation with separable variables is an equation of the form: y"=f(x)g(y) or through differentials , where f(x) and g(y) are given functions.

For those y, for which , the equation y"=f(x)g(y) is equivalent to the equation in which the variable y is present only on the left side, and the variable x is present only on the right side. They say, "in the equation y"=f(x)g(y separating the variables.

Type equation is called a separated variable equation.

After integrating both parts of the equation on x, we get G(y) = F(x) + C is the general solution of the equation, where G(y) and F(x) are some antiderivatives, respectively, of functions and f(x), C arbitrary constant.

Algorithm for solving a first-order differential equation with separable variables

Example 1

solve the equation y" = xy

Solution. Derivative of a function y" replace with

we separate the variables

Let's integrate both parts of the equality:

Example 2

2yy" = 1- 3x 2, if y 0 = 3 at x0 = 1

This is a separated variable equation. Let's represent it in differentials. To do this, we rewrite this equation in the form From here

Integrating both parts of the last equality, we find

Substituting initial values x 0 = 1, y 0 = 3 find FROM 9=1-1+C, i.e. C = 9.

Therefore, the desired partial integral will be or

Example 3

Write an equation for a curve passing through a point M(2;-3) and having a tangent with a slope

Solution. According to the condition

This is a separable variable equation. Dividing the variables, we get:

Integrating both parts of the equation, we get:

Using the initial conditions, x=2 and y=-3 find C:

Therefore, the desired equation has the form

2.3. Linear differential equations of the first order

A first-order linear differential equation is an equation of the form y" = f(x)y + g(x)

where f(x) and g(x)- some given functions.

If a g(x)=0 then the linear differential equation is called homogeneous and has the form: y" = f(x)y

If then the equation y" = f(x)y + g(x) called heterogeneous.

General solution of a linear homogeneous differential equation y" = f(x)y given by the formula: where FROM is an arbitrary constant.

In particular, if C \u003d 0, then the solution is y=0 If linear homogeneous equation has the form y" = ky where k is some constant, then its general solution has the form: .

General solution of a linear inhomogeneous differential equation y" = f(x)y + g(x) given by the formula ,

those. is equal to the sum of the general solution of the corresponding linear homogeneous equation and the particular solution of this equation.

For a linear inhomogeneous equation of the form y" = kx + b,

where k and b- some numbers and a particular solution will be a constant function . Therefore, the general solution has the form .

Example. solve the equation y" + 2y +3 = 0

Solution. We represent the equation in the form y" = -2y - 3 where k=-2, b=-3 The general solution is given by the formula .

Therefore, where C is an arbitrary constant.

2.4. Solution of linear differential equations of the first order by the Bernoulli method

Finding a General Solution to a First-Order Linear Differential Equation y" = f(x)y + g(x) reduces to solving two differential equations with separated variables using the substitution y=uv, where u and v- unknown functions from x. This solution method is called the Bernoulli method.

Algorithm for solving a first-order linear differential equation

y" = f(x)y + g(x)

1. Enter a substitution y=uv.

2. Differentiate this equality y"=u"v + uv"

3. Substitute y and y" into this equation: u"v + uv" =f(x)uv + g(x) or u"v + uv" + f(x)uv = g(x).

4. Group the terms of the equation so that u take it out of brackets:

5. From the bracket, equating it to zero, find the function

This is a separable equation:

Divide the variables and get:

Where . .

6. Substitute the received value v into the equation (from item 4):

and find the function This is a separable equation:

7. Write the general solution in the form: , i.e. .

Example 1

Find a particular solution to the equation y" = -2y +3 = 0 if y=1 at x=0

Solution. Let's solve it with substitution y=uv,.y"=u"v + uv"

Substituting y and y" into this equation, we get

Grouping the second and third terms on the left side of the equation, we take out the common factor u out of brackets

We equate the expression in brackets to zero and, having solved the resulting equation, we find the function v = v(x)

We got an equation with separated variables. We integrate both parts of this equation: Find the function v:

Substitute the resulting value v into the equation We get:

This is a separated variable equation. We integrate both parts of the equation: Let's find the function u = u(x,c) Let's find a general solution: Let us find a particular solution of the equation that satisfies the initial conditions y=1 at x=0:

III. Higher order differential equations

3.1. Basic concepts and definitions

A second-order differential equation is an equation containing derivatives not higher than the second order. In the general case, the second-order differential equation is written as: F(x,y,y",y") = 0

The general solution of a second-order differential equation is a function of the form , which includes two arbitrary constants C1 and C2.

A particular solution of a second-order differential equation is a solution obtained from the general one for some values ​​of arbitrary constants C1 and C2.

3.2. Linear homogeneous differential equations of the second order with constant ratios.

Linear homogeneous differential equation of the second order with constant coefficients is called an equation of the form y" + py" + qy = 0, where p and q are constant values.

Algorithm for solving second-order homogeneous differential equations with constant coefficients

1. Write the differential equation in the form: y" + py" + qy = 0.

2. Compose its characteristic equation, denoting y" through r2, y" through r, y in 1: r2 + pr +q = 0

The online calculator allows you to solve differential equations online. It is enough to enter your equation in the appropriate field, denoting the "derivative of the function" with an apostrophe and click on the "solve equation" button. And the system implemented on the basis of the popular WolframAlpha website will give a detailed differential equation solution absolutely free. You can also set the Cauchy problem to choose from the entire set of possible solutions a particular one corresponding to given initial conditions. The Cauchy problem is entered in a separate field.

Differential equation

By default, in the equation, the function y is a function of a variable x. However, you can set your own variable notation, if you write, for example, y(t) in an equation, the calculator will automatically recognize that y is a function of a variable t. With the calculator you can solve differential equations of any complexity and type: homogeneous and inhomogeneous, linear or non-linear, first order or second and higher orders, equations with separable or non-separable variables, etc. Solution diff. equation is given in analytical form, has detailed description. Differential equations are very common in physics and mathematics. Without their calculation, it is impossible to solve many problems (especially in mathematical physics).

One of the steps in solving differential equations is the integration of functions. There are standard methods for solving differential equations. It is necessary to bring the equations to the form with separable variables y and x and separately integrate the separated functions. To do this, sometimes you need to make a certain replacement.