Differential equations of first order

Features of the first-order differential equations

When solving the first order equations, the function y and the variable x should be considered equal. That is, the solution may be in the form and in the form.

Differential equations of first order allowed relative to the derivative

Equations with separating variables

Equations resulting in equations with dividing variables

Uniform equations

Equations leading to homogeneous

Generalized homogeneous equations

Linear differential equations

• Linear on y.
• Linear software F (Y)
• Linear by X.
• Linear software F (X)

Bernoulli equations

Riccati equations

Jacobi equations

Equations in full differentials

given that

Integrating multiplier

If the first order differential equation is not given any of the listed types, then you should try to find an integrating multiplier to reduce it to the equation in full differentials.

Equations not solved relative to the derivative Y '

Equations that make a decision relative to the derivative y '

First you need to try to resolve the equation relative to the y 'derivative. If possible, the equation can be given to one of the types listed above.

Equations permitting multiplied

Equations that do not contain x and y

Equations that do not contain x or y

Or

Equations permitted relative to y

Equations Clero

Lagrange equations

Equations leading to the Bernoulli equation

Differential equations of higher orders

Differential equations that reduce order

Equations that are resolved direct integration

Equations that do not contain y

Equations that do not contain x

Equations, homogeneous relative to y, y ', y' ', ...

Linear inhomogeneous equations with a special inhomogeneous part

,
where - polynomials degrees and.

Euler equations

References:
V.V. Stepanov, course of differential equations, "LCA", 2015.
N.M. Gunter, R.O. Kuzmin, Collection of tasks higher Mathematics, "Lan", 2003.

Often, only the mention of differential equations causes an unpleasant feeling among students. Why is this happening? Most often because when studying the basics of the material there is a gap in knowledge, because of which the further study of the difuri becomes simply torture. Nothing is clear what to do, how to decide where to start?

However, we will try to show you that Difura is not as difficult as it seems.

The main concepts of the theory of differential equations

From school, we know the simplest equations in which you need to find an unknown X. In fact differential equations Only a bit different from them - instead of a variable h. They need to find a feature. y (x) which will turn the equation into identity.

Differential equations have a huge applied value. This is not an abstract mathematics, which has no relation to the world around us. With the help of differential equations, many real natural processes are described. For example, string fluctuations, the movement of the harmonic oscillator, by means of differential equations in the tasks of mechanics, the speed and acceleration of the body are found. Also D. Focus are widely used in biology, chemistry, economics and many other sciences.

Differential equation (D.) - This is an equation containing derivatives y (x), the function itself, independent variables and other parameters in various combinations.

There are many species of differential equations: ordinary differential equations, linear and nonlinear, homogeneous and inhomogeneous, differential equations of the first and higher orders, difura in private derivatives and so on.

The solution of the differential equation is a function that turns it into identity. There are general and private solutions of the Du.

The general solution of the Du is the total set of solutions that turn the equation into identity. A particular solution of the differential equation is a solution that satisfies additional conditions specified initially.

The order of the differential equation is determined highest order derivatives included in it.

Ordinary differential equations

Ordinary differential equations - These are equations containing one independent variable.

Consider the simplest ordinary differential equation of the first order. It has the form:

It is possible to solve such an equation, simply by injecting its right-hand side.

Examples of such equations:

Equations with separating variables

IN general This type of equations looks like this:

Let us give an example:

Solving such an equation, you need to divide the variables, leading it to the form:

After that, it will remain integrating both parts and get a solution.

Linear differential equations of first order

Such equations look:

Here p (x) and q (x) are some functions of an independent variable, and y \u003d y (x) is the desired function. Let us give an example of such an equation:

Solving such an equation, most often use the variation method of an arbitrary constant or represent the desired function in the form of a product of two other functions y (x) \u003d u (x) v (x).

To solve such equations, a certain preparation is necessary and to take them "from the skill" will be quite difficult.

An example of solving a Du with separating variables

So we reviewed the simplest types of do. Now we will analyze the decision of one of them. Let it be an equation with separating variables.

First, rewrite the derivative in a more familiar form:

Then we split the variables, that is, in one part of the equation, we will collect all the "igraki", and in the other - "Iks":

Now it remains to integrate both parts:

We integrate and get common decision This equation:

Of course, the solution of differential equations is a kind of art. You need to be able to understand how the type of equation relates, and also learn how to see which transformations need to be done with it to lead to one or another thing, not to mention simply on the ability to differentiate and integrate. And to succeed in solving a Du, practice is needed (as in everything). And if you have this moment There is no time to deal with how differential equations or the Cauchy task got up as a bone in the throat or you do not know how to make a presentation correctly, contact our authors. In a short time, we will provide you with a ready and detailed decision, to deal with the details of which you can at any time convenient for you. In the meantime, we suggest watch a video on "How to solve differential equations":

Definition.View equation

, unknown function and its derivatives call differential equation n.-o order.

Definition. View equation

binding an independent variable , unknown function and its derivative called the differential equation of the first order.

The order of the differential equation is called the order of the senior derivative incoming in this equation.

Definition. General decision Differential equation (2) in the area called function , where from - Arbitrary constant, satisfying the following conditions:

1) for each number from The function is a solution of equation (2);

2) if , then there is such a number that the decision satisfies the initial condition .

If the general solution is obtained in an implicit form , it is called a common integral, and – private integral equation (8).

If the differential equation (8) can be resolved relative , it will take the form:

Differential equation (9) called permitted relative to the derivative.

Equation (9) is sometimes written in the form:

where – functions of two variables.

Cauchy theorem. (Theorem of the existence and uniqueness of the solution of the differential equation (9)). If in equation (9) the function and its private derivative of software is defined and continuous in the plane area ( Xoy.) and - an arbitrary point of, then exists, and the only one, the solution of this equation satisfying the initial condition .

The task of finding a solution of equation (9) with a given initial condition called cauchy task.

Definition. Private decision Differential equation (9) call any function , which is obtained from the general solution if an arbitrary constant gives a certain value.

Definition.The differential equation i of the order is called the equation with divided variablesif it can be written in the form

or , (12)

where – set functions.

To solve the equation (11) we divide the variables:

Or divide both parts (12) on :

from

Definition. Equation or (13) is called the equation with separated variables.

Definition. The function is called uniform The function of zero measurement, if it depends only on the relationship, i.e. .

Definition. A homogeneous differential equation called view equation (14)

We introduce a new unknown function, putting , or . Differentiating, we get.

Substitute to equation (14), we transform it to mind . Separating variables and integrating, we will find

From here.

After the integration is performed, you need to return to the function, putting it.

Example. Solve equation.

Expressing the derivative, we get or.

Put. Then. Substituting into the equation, we get. From where.

We split variables.

After integration found

or .

Finally.

Definition. The linear differential equation is the species equation

We introduce two new unknown functions and, putting. Since unknown functions have become two, and the conditions for these functions are only one (their product should satisfy equation (15)), then one more condition on these functions can be imposed arbitrarily than we use it below.

Substitute in (15),

receive

or (16)

As a function, choose any function that satisfies the condition. (17)

We obtain an equation with separating variables to find. Integrate this equation, believing a constant integration equal to zero (the last legally, since we are satisfied with any solution of equation (17)):

We substitute the value found to equation (16):

Integrating, we find a function :. Alternating the found functions and, we obtain the general solution of the equation (15).

Definition.Bernoulli equation is called the view equation

where m. - Any valid number. This equation is solved with the same reception as the linear equation.

Definition. The equation

it is called the equation of a complete differential if its left side is a complete differential of some function. In this case, equation (18) can be rewritten in the form. The total integral of equation (18) will

Theorem. Let functions have continuous private derivatives in some region ( D.) plane ( Xoy.). In order for the expression to be a complete differential of some function, it is necessary and enough to in all points of the region ( D.Equality was performed

Let the equation (18) be given for which the condition (20) is satisfied. The latter means that there is a function such that

To solve equation (18), it is necessary, based on equalities (21), to find the function and write down the common integral of equation (18) in the form (19).

Example. Find a solution equation satisfying condition.

We have: ,.

Find and:

Thus, i.e. There is such a function that

To find it intense x. The first of the equals (22):

Here, an unknown function plays the role of permanent integration. To find indifferentiation (23) by y.:

On the other hand, from (22) we have from these two equalities or.

From here. (24)

Substituting in (24), we obtain, according to (19), the overall integral of this equation.

Comment.Since, according to (19), the function is equated with an arbitrary constant, then when performing integration (24), constant integration can not be written.

Differential equations of first order allowed relative to the derivative

How to solve differential equations of first order

Let we have a differential first order equation allowed relative to the derivative:
.
Dividing this equation on, when we obtain the equation of the form:
,
where.

Further, we look at whether these equations are not to one of the following types. If not, then rewrite the equation in the form of differentials. For this we write and multiply the equation on. We obtain equation in the form of differentials:
.

If this equation is not an equation in complete differentials, we believe that in this equation is an independent variable, and is a function from. We divide the equation to:
.
We further look if this equation does not apply to one of the types listed below considering that and changed places.

If the type is not found for this equation, then we do not see if the simple substitution equation cannot be easier. For example, if the equation looks:
,
That we notice that. Then make a substitution. After that, the equation will take a simpler form:
.

If it does not help, then try to find an integrating multiplier.

Equations with separating variables

;
.
We divide on and integrate. When we get:
.

Equations resulting in equations with dividing variables

Uniform equations

We solve the substitution:
,
where - the function from. Then
;
.
We share variables and integrate.

Equations leading to homogeneous

We enter variables and:
;
.
Permanent and choose so that free members appealed to zero:
;
.
As a result, we obtain a homogeneous equation in variables and.

Generalized homogeneous equations

Make a substitution. We obtain a homogeneous equation in variables and.

Linear differential equations

There are three methods for solving linear equations.

2) Bernoulli method.
We are looking for a solution in the form of a product of two functions and from the variable:
.
;
.
One of these functions we can choose an arbitrary way. Therefore, as choosing any no zero solution of the equation:
.

3) Method of variation of constant (Lagrange).
Here we first solve a homogeneous equation:

Common decision uniform equation It has the form:
,
where is the constant. Next, we replace the function constant depending on the variable:
.
Substitute to the original equation. As a result, we obtain the equation from which we define.

Bernoulli equations

Substitution The Bernoulli equation is given to linear equation.

Also, this equation can be solved by Bernoulli. That is, we are looking for a solution in the form of a product of two functions depending on the variable:
.
Substitute to the original equation:
;
.
As choosing any no zero solution of the equation:
.
Determining, we obtain the equation with separating variables for.

Riccati equations

It is not solved in general. Forceed

Riccati equation is given to mind:
,
where - constant; ; .
Next, for a substitution:

It is given to mind:
,
where.

Properties of Riccati equation and some particular cases of its solutions are presented on the page.
Differential equation Riccati \u003e\u003e\u003e

Jacobi equations

Resolved by substitution:
.

Equations in full differentials

Given that
.
When performing this condition, the expression on the left part of equality is a differential of some function:
.
Then
.
From here we obtain the integral of the differential equation:
.

To find a function, the most convenient way is the method of sequential separation of differential. For this use formulas:
;
;
;
.

Integrating multiplier

If the first order differential equation is not given any of the listed types, then you can try to find an integrating multiplier. The integrating multiplier is such a function when multiplying to which the differential equation becomes the equation in complete differentials. The differential equation of the first order has an infinite number of integrating multipliers. But, common methods There is no integrated multiplier.

Equations that are not resolved relative to the derivative Y "

Equations that make a decision relative to the derivative Y "

First you need to try to resolve the equation relative to the derivative. If possible, the equation can be given to one of the types listed above.

Equations permitting multiplied

If the equation manage to decompose on multipliers:
,
The task is reduced to a sequential solution of simpler equations:
;
;

;
. We believe. Then
or .
Next, integrate the equation:
;
.
As a result, we obtain the expression of the second variable through the parameter.

More general equations:
or
Also resolve in parametric form. To do this, it is necessary to select such a function so that from the source equation it is possible to express or via the parameter.
To express the second variable through the parameter, integrate the equation:
;
.

Equations permitted relative to y

Equations Clero

Such an equation has a general solution

Lagrange equations

Solution we are looking for a parametric form. We assume where the parameter.

Equations leading to the Bernoulli equation

These equations are given to the Bernoulli equation, if you search for their parameter solutions by entering the parameter and making substitution.

References:
V.V. Stepanov, course of differential equations, "LCA", 2015.
N.M. Gunter, R.O. Kuzmin, Collection of tasks on higher mathematics, "Lan", 2003.

The simplest D.U.1 is the equation of the species as is known from the course of integral calculus, the function y. located integration

Definition. The view equation is called a differential equation with separated variables. It can be written in the form

We integrate both parts of the equation, we obtain the so-called common integral (or general solution).

Example.

Decision. We write an equation in the form
We integrate both parts of the equation:

(General integral differential equation).

Definition.The view equation is called the equation with separating variables, If functions can be represented as a piece of functions

i.e. there is an equation

To solve such a differential equation, it is necessary to bring it to the type of differential equation with separated variables, for which we divide the equation for the work
Indeed, dividing all members of the equation
,

- Differential equation with separated variables.

To solve it enough to integrate

When solving a differential equation with separating variables, you can be guided by the following algorithm (rule) separation of variables.

First step. If the differential equation contains a derivative It should be written in the form of differential relations:

The second step. Multiply Equation on
, then grouped the terms comprising differential function and differential independent variable
.

Third step.Expressions obtained at
, to present in the form of a work of two factors, each of which contains only one variable (
). If after that the equation will take a lot, dividing it to the work
, We obtain a differential equation with separated variables.

Fourth step. Integrating the equation, we obtain the overall solution of the original equation (or its common integral).

Consider equations

№ 2.

№ 3.

Differential equation No. 1 is a differential equation with separating variables, by definition. We divide the equation for the work
We get the equation

Integrating, get

or

The last ratio is a common integral of this differential equation.

In the Differential Equation No. 2, replace
multiply by
, get

general solution of the differential equation.

Differential equation number 3 is not a equation with dividing variables, since, writing it in the form

or
,

we see that the expression
in the form of a work of two factors (one -

only from y, the other - only with h.) It is impossible to submit. Note that sometimes you need to perform algebraic transformations to see that this differential equation is with separating variables.

Example number 4.. The equation is given by transforming the equation, making a common multiplier to the left
We divide the left and right parts of the equation on the work
receive

We integrate both parts of the equation:

from
- General integral of this equation. (but)

Note that if constant integration is written as
The total integral of this equation may have another form:

or
- General integral. (b)

Thus, the overall integral of the same differential equation may have various shapes. In any case, it is important to prove that the total integral obtained satisfies this differential equation. To do this, you need to directly delete h.both parts of equality setting a common integral, given that y. There is a function OT. h.. After exception from We obtain the same differential equations (source). If a common integral
, (view ( but))

If a common integral
(view (b))

We get the same equation as in the previous case (a).

We now consider simple and important classes of the first-order equations, which are driven to equations with separating variables.