Partial solution of differential equation calculator in detail. Solving the simplest differential equations of the first order
I. Ordinary differential equations
1.1. Basic concepts and definitions
A differential equation is an equation that relates an independent variable x, the required 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 required function depends on one independent variable.
Solving a differential equation is called a function that turns this equation into an identity.
The order of the differential equation is the order of the highest derivative included in this equation
Examples.
1. Consider a first order differential equation
The solution to this equation is the function y = 5 ln x. Indeed, substituting y" into the equation, we get the identity.
And this means that the function y = 5 ln x– is a solution to 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 obtain: , – identity.
And this means that the function is the solution to this differential equation.
Integrating differential equations is the process of finding solutions to differential equations.
General solution of the differential equation 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 a solution obtained from a general solution for various 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 to 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 a circle, it is convenient to represent an arbitrary constant C in the form .
- general solution of the differential equation.
Particular solution of the equation satisfying 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 x 2 +y 2 = 5 2 .
This is a particular solution to a differential equation obtained from a general solution under given initial conditions.
2. Find the general solution to the differential equation
The solution to this equation is any function of the form , where C is an arbitrary constant. Indeed, substituting into the equations, we get: , .
Consequently, this differential equation has an infinite number of solutions, since for different values of the constant C, equality determines different solutions to the equation.
For example, by direct substitution you can verify that the functions 
are solutions to the equation.
A problem in which you need to find a particular solution to the equation y" = f(x,y) satisfying the initial condition y(x 0) = y 0, is called the Cauchy problem.
Solving the equation y" = f(x,y), satisfying the initial condition, y(x 0) = y 0, is called a solution to the Cauchy problem.
The solution to the Cauchy problem has a simple geometric meaning. Indeed, according to these definitions, to solve the Cauchy problem y" = f(x,y) given that y(x 0) = y 0, means to find the integral curve of the equation y" = f(x,y) which passes through a given point M 0 (x 0,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.
A 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.
The 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 this equation with its value, we get
that is 3x=3x
Therefore, the function is a general solution to the equation for any constant C.
Find a particular solution to 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 get from where C=0.
Thus, we obtain a particular solution from the general one by substituting into this equation the resulting value C=0– private solution.
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)– specified 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 only on the right side. They say, "in Eq. y"=f(x)g(y Let's separate the variables."
Equation of the form called a separated variable equation.
Integrating both sides of the equation By x, we get G(y) = F(x) + C is the general solution of the equation, where G(y) And F(x)– 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 it with
let's separate the variables
Let's integrate both sides of the equality:
Example 2
2yy" = 1- 3x 2, If y 0 = 3 at x 0 = 1
This is a separated variable equation. Let's imagine it in differentials. To do this, we rewrite this equation in the form 
From here 
Integrating both sides of the last equality, we find
Substituting the initial values x 0 = 1, y 0 = 3 we'll find WITH 9=1-1+C, i.e. C = 9.
Therefore, the required 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 an angular coefficient
Solution. According to the condition
This is an equation with separable variables. Dividing the variables, we get: 
Integrating both sides of the equation, we get: 
Using the initial conditions, x = 2 And y = - 3 we'll find C:
Therefore, the required equation has the form 
2.3. Linear differential equations of the first order
A linear differential equation of the first order is an equation of the form y" = f(x)y + g(x)
Where f(x) And g(x)- some specified functions.
If 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) is called heterogeneous.
General solution of a linear homogeneous differential equation y" = f(x)y is given by the formula: where WITH– arbitrary constant.
In particular, if C =0, then the solution is y = 0 If a 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) is 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. Let's 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. Solving 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 substitution y=uv, Where u And v- unknown functions from x. This solution method is called Bernoulli's method.
Algorithm for solving a first order linear differential equation
y" = f(x)y + g(x)
1. Enter 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: 
Let's divide the variables and get: 
Where 
.
.
6. Substitute the resulting value v into the equation (from step 4):
and find the function This is an equation with separable variables:
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 using substitution y=uv,.y" = u"v + uv"
Substituting y And y" into this equation, we get
By 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 get an equation with separated variables. Let's integrate both sides of this equation: Find the function v:
Let's substitute the resulting value v into the equation we get:
This is a separated variable equation. Let's integrate both sides of the equation: 
Let's find the function u = u(x,c) 
Let's find a general solution: 
Let us find a particular solution to 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 of no higher than second order. In the general case, a 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 C 1 And C 2.
A particular solution to a second-order differential equation is a solution obtained from a general solution for certain values of arbitrary constants C 1 And C 2.
3.2. Linear homogeneous differential equations of the second order with constant coefficients.
Linear homogeneous differential equation of the second order with constant coefficients called an equation of the form y" + py" +qy = 0, Where p And q- constant values.
Algorithm for solving homogeneous second order differential equations with constant coefficients
1. Write the differential equation in the form: y" + py" +qy = 0.
2. Create its characteristic equation, denoting y" through r 2, y" through r, y in 1: r 2 + pr +q = 0
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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 of various types you can solve, the better. Why? You'll have to integrate a lot. And differentiate. Also highly recommend learn to find.
In 95% of cases, test papers contain 3 types of first-order differential equations: separable equations which we will look at in this lesson; homogeneous equations And linear inhomogeneous equations. For those starting to study diffusers, I advise you to read the lessons in exactly this order, and after studying the first two articles, it won’t hurt to consolidate your skills in an additional workshop - equations reducing to homogeneous.
There are even rarer types of differential equations: total differential equations, Bernoulli equations and some others. The most important of the last two types are equations in total differentials, since in addition to this differential equation I 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 this case they are tabular:
As we remember, a constant is assigned to any antiderivative. There are two integrals here, but it is enough to write the constant once (since constant + constant is still equal to another constant). In most cases it is placed on the right side.
Strictly speaking, after the integrals are taken, the differential equation is considered solved. The only thing is that our “y” is not expressed through “x”, that is, the solution is presented in an implicit form. The solution to a differential equation in implicit form is called general integral of the differential equation. That is, this is a general integral.
The answer in this form is quite acceptable, but is there a better option? Let's try to get common decision.
Please, remember the first technique, it is very common and is often used in practical tasks: if a logarithm appears on the right side after integration, then in many cases (but not always!) it is also advisable to write the constant under the logarithm.
That is, INSTEAD OF entries are usually written 
.
Why is this necessary? And in order to make it easier to express “game”. Using the property of logarithms 
. In this case:
Now logarithms and modules can be removed:
The function is presented explicitly. This is the general solution.
Answer: common decision: 
.
The answers to many differential equations are fairly easy to check. In our case, this is done quite simply, we take the solution found and differentiate it:
Then we substitute the derivative into the original equation:
– the correct equality is obtained, which means that the general solution satisfies the equation, which is what needed to be checked.
By giving a constant different values, you can get an infinite number of private solutions differential equation. It is clear that any of the functions , , etc. satisfies the differential equation.
Sometimes the general solution is called family of functions. In this example, the general solution 
is a family of linear functions, or more precisely, a family of direct proportionality.
After a thorough review of the first example, it is appropriate to answer several naive questions about differential equations:
1)In this example, we were able to separate the variables. Can this always be done? No not always. And even more often, variables cannot be separated. For example, in homogeneous first order equations, you must first replace it. In other types of equations, for example, in a first-order linear inhomogeneous equation, you need to use various techniques and methods to find a general solution. Equations with separable variables, which we consider in the first lesson, are the simplest type of differential equations.
2) Is it always possible to integrate a differential equation? No not always. It is very easy to come up with a “fancy” equation that cannot be integrated; in addition, there are integrals that cannot be taken. But such DEs can be solved approximately using special methods. D’Alembert and Cauchy guarantee... ...ugh, lurkmore.to 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 important point, but in order not to cover the “dummies” with an avalanche of new information, I’ll leave it until the next lesson.
We won't rush. Another simple remote control and another typical solution:
Example 2
Find a particular solution to the differential equation that satisfies the initial condition
Solution: according to the condition, you need to find private solution DE that satisfies a given initial condition. This formulation of the question is also called Cauchy problem.
First we find a general solution. There is no “x” variable in the equation, but this should not confuse, the main thing is that it has the first derivative.
We rewrite the derivative in the required form:
Obviously, the variables can be separated, boys to the left, girls to the right:
Let's integrate the equation: 
The general integral is obtained. Here I have drawn a constant with an asterisk, the fact is that very soon it will turn into another constant.
Now we try to transform the general integral into a general solution (express the “y” explicitly). Let's remember the good old things from school: 
. In this case:
The constant in the indicator looks somehow unkosher, so it is usually brought down to earth. In detail, this is how it happens. Using the property of degrees, we rewrite the function as follows:
If is a constant, then is also some constant, let’s redesignate it with the letter :
Remember “demolishing” a constant is second technique, which is often used when solving differential equations.
So, the general solution is: . This is a nice family of exponential functions.
At the final stage, you need to find a particular solution that satisfies the given initial condition. This is also simple.
What is the task? Need to pick up such the value of the constant so that the condition is satisfied.
It can be formatted in different ways, but this will probably be the clearest way. In the general solution, instead of the “X” we substitute a zero, and instead of the “Y” we substitute a two:
That is,
Standard design version:
Now we substitute the found value of the constant into the general solution:
– this is the particular solution we need.
Answer: private solution:
Let's check. Checking a private solution includes two stages:
First you need to check whether the particular solution found really satisfies the initial condition? Instead of the “X” we substitute a zero and see what happens:
- yes, indeed, a two was received, which means that the initial condition is met.
The second stage is already familiar. We take the resulting particular solution and find the derivative:
We substitute into the original equation:
– the correct equality is obtained.
Conclusion: the particular solution was found correctly.
Let's move on to more meaningful examples.
Example 3
Solve differential equation
Solution: We rewrite the derivative in the form we need: 
We evaluate whether it is possible to separate the variables? Can. We move the second term to the right side with a change of sign: 
And we transfer the multipliers according to the rule of proportion: 
The variables are separated, let's integrate both parts: 
I must warn you, judgment day is approaching. If you haven't studied well indefinite integrals, have solved few examples, then there is nowhere to go - you will have to master them now.
The integral of the left side is easy to find; we deal with the integral of the cotangent using the standard technique that we looked at in the lesson Integrating trigonometric functions last year: 
On the right side we have a logarithm, and, according to my first technical recommendation, the constant should also be written under the logarithm.
Now we try to simplify the general integral. Since we only have logarithms, it is quite possible (and necessary) to get rid of them. By using known properties We “pack” the logarithms as much as possible. I'll write it down in great detail: 
The packaging is finished to be barbarically tattered: 
Is it possible to express “game”? Can. It is necessary to square both parts.
But you don't need to do this.
Third technical tip: if to obtain a general solution it is necessary to raise to a power or take roots, then In most cases you should refrain from these actions and leave the answer in the form of a general integral. The fact is that the general solution will look simply terrible - with large roots, signs and other trash.
Therefore, we write the answer in the form of a general integral. It is considered good practice to present it in the form , that is, on the right side, if possible, leave only a constant. It is not necessary to do this, but it is always beneficial to please the professor ;-)
Answer: general integral:
! Note: The general integral of any equation can be written in more than one way. Thus, if your result does not coincide with the previously known answer, this does not mean that you solved the equation incorrectly.
The general integral is also quite easy to check, the main thing is to be able to find derivative of a function specified implicitly. Let's differentiate the answer: 
We multiply both terms by: 
And divide by: 
The original differential equation has been obtained exactly, which means that the general integral has been found correctly.
Example 4
Find a particular solution to the differential equation that satisfies the initial condition. Perform check.
This is an example for you to solve on your own.
Let me remind you that the algorithm consists of two stages:
1) finding a general solution;
2) finding the required particular solution.
The check is also carried out in two steps (see sample in Example No. 2), you need to:
1) make sure that the particular solution found satisfies the initial condition;
2) check that a particular solution generally satisfies the differential equation.
Full solution and answer at the end of the lesson.
Example 5
Find a particular solution to 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.
This 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 through an apostrophe, and click on the “solve equation” button. And the system, implemented on the basis of the popular WolframAlpha website, will give detailed solving a differential equation absolutely free. You can also define a Cauchy problem to select from the entire set of possible solutions the quotient that corresponds to the given initial conditions. The Cauchy problem is entered in a separate field.
Differential equation
By default, the function in the equation y is a function of a variable x. However, you can specify your own designation for the variable; if you write, for example, y(t) in the equation, the calculator will automatically recognize that y there is a function from a variable t. With the help of a calculator you can solve differential equations of any complexity and type: homogeneous and inhomogeneous, linear or nonlinear, first order or second and higher orders, equations with separable or nonseparable variables, etc. Solution diff. the equation is given in analytical form and has a detailed description. Differential equations are very common in physics and mathematics. Without calculating them, it is impossible to solve many problems (especially in mathematical physics).
One of the stages of solving differential equations is integrating functions. There are standard methods for solving differential equations. It is necessary to reduce the equations to a form with separable variables y and x and separately integrate the separated functions. To do this, sometimes a certain replacement must be made.
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 a complex function (“apple” is the extraction of a square root or, which 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.