Second order differential equations method of variation. ODE. Method of variation of an arbitrary constant. Social transformations. State and Church

Consider now the linear inhomogeneous equation
. (2)
Let y 1 ,y 2 ,.., y n be a fundamental system of solutions, and - common decision corresponding homogeneous equation L(y)=0 . Similarly to the case of first-order equations, we will seek a solution to Eq. (2) in the form
. (3)
Let us verify that a solution in this form exists. To do this, we substitute the function into the equation. To substitute this function into the equation, we find its derivatives. The first derivative is
. (4)
When calculating the second derivative, four terms appear on the right side of (4), when calculating the third derivative, eight terms appear, and so on. Therefore, for the convenience of further calculations, the first term in (4) is assumed to be equal to zero. With this in mind, the second derivative is equal to
. (5)
For the same reasons as before, in (5) we also set the first term equal to zero. Finally, the nth derivative is
. (6)
Substituting the obtained values of the derivatives into the original equation, we have
. (7)
The second term in (7) is equal to zero, since the functions y j , j=1,2,..,n, are solutions of the corresponding homogeneous equation L(y)=0. Combining with the previous one, we obtain a system of algebraic equations for finding the functions C" j (x)
(8)
The determinant of this system is the Wronsky determinant of the fundamental system of solutions y 1 ,y 2 ,..,y n of the corresponding homogeneous equation L(y)=0 and therefore is not equal to zero. Therefore, there is a unique solution to system (8). Having found it, we obtain the functions C "j (x), j=1,2,…,n, and, consequently, C j (x), j=1,2,…,n Substituting these values into (3), we obtain the solution of the linear inhomogeneous equation.
The described method is called the method of variation of an arbitrary constant or the Lagrange method.

Example #1. Let's find the general solution of the equation y "" + 4y" + 3y \u003d 9e -3 x. Consider the corresponding homogeneous equation y "" + 4y" + 3y \u003d 0. The roots of its characteristic equation r 2 + 4r + 3 \u003d 0 are equal to -1 and - 3. Therefore, the fundamental system of solutions of a homogeneous equation consists of the functions y 1 = e - x and y 2 = e -3 x. We are looking for a solution to an inhomogeneous equation in the form y \u003d C 1 (x)e - x + C 2 (x)e -3 x. To find the derivatives C " 1 , C" 2 we compose a system of equations (8)
C′ 1 ·e -x +C′ 2 ·e -3x =0
-C′ 1 e -x -3C′ 2 e -3x =9e -3x
solving which, we find , Integrating the obtained functions, we have
Finally we get

Example #2. Solve second-order linear differential equations with constant coefficients by the method of variation of arbitrary constants:

y(0) =1 + 3ln3
y'(0) = 10ln3

Solution:
This differential equation belongs to linear differential equations with constant coefficients.
We will seek the solution of the equation in the form y = e rx . To do this, we compose the characteristic equation of a linear homogeneous differential equation with constant coefficients:
r 2 -6 r + 8 = 0
D = (-6) 2 - 4 1 8 = 4

The roots of the characteristic equation: r 1 = 4, r 2 = 2
Therefore, the fundamental system of solutions is the functions: y 1 =e 4x , y 2 =e 2x
The general solution of the homogeneous equation has the form: y =C 1 e 4x +C 2 e 2x
Search for a particular solution by the method of variation of an arbitrary constant.
To find the derivatives of C "i, we compose a system of equations:
C′ 1 e 4x +C′ 2 e 2x =0
C′ 1 (4e 4x) + C′ 2 (2e 2x) = 4/(2+e -2x)
Express C" 1 from the first equation:
C" 1 \u003d -c 2 e -2x
and substitute in the second. As a result, we get:
C" 1 \u003d 2 / (e 2x + 2e 4x)
C" 2 \u003d -2e 2x / (e 2x + 2e 4x)
We integrate the obtained functions C" i:
C 1 = 2ln(e -2x +2) - e -2x + C * 1
C 2 = ln(2e 2x +1) – 2x+ C * 2

Since y \u003d C 1 e 4x + C 2 e 2x, then we write the resulting expressions in the form:
C 1 = (2ln(e -2x +2) - e -2x + C * 1) e 4x = 2 e 4x ln(e -2x +2) - e 2x + C * 1 e 4x
C 2 = (ln(2e 2x +1) – 2x+ C * 2)e 2x = e 2x ln(2e 2x +1) – 2x e 2x + C * 2 e 2x
Thus, the general solution of the differential equation has the form:
y = 2 e 4x ln(e -2x +2) - e 2x + C * 1 e 4x + e 2x ln(2e 2x +1) – 2x e 2x + C * 2 e 2x
or
y = 2 e 4x ln(e -2x +2) - e 2x + e 2x ln(2e 2x +1) – 2x e 2x + C * 1 e 4x + C * 2 e 2x

We find a particular solution under the condition:
y(0) =1 + 3ln3
y'(0) = 10ln3

Substituting x = 0 into the found equation, we get:
y(0) = 2 ln(3) - 1 + ln(3) + C * 1 + C * 2 = 3 ln(3) - 1 + C * 1 + C * 2 = 1 + 3ln3
We find the first derivative of the obtained general solution:
y’ = 2e 2x (2C 1 e 2x + C 2 -2x +4 e 2x ln(e -2x +2)+ ln(2e 2x +1)-2)
Substituting x = 0, we get:
y'(0) = 2(2C 1 + C 2 +4 ln(3)+ ln(3)-2) = 4C 1 + 2C 2 +10 ln(3) -4 = 10ln3

We get a system of two equations:
3 ln(3) - 1 + C * 1 + C * 2 = 1 + 3ln3
4C 1 + 2C 2 +10ln(3) -4 = 10ln3
or
C * 1 + C * 2 = 2
4C1 + 2C2 = 4
or
C * 1 + C * 2 = 2
2C1 + C2 = 2
From: C 1 = 0, C * 2 = 2
A particular solution will be written as:
y = 2e 4x ln(e -2x +2) - e 2x + e 2x ln(2e 2x +1) – 2x e 2x + 2 e 2x

A method for solving linear inhomogeneous differential equations of higher orders with constant coefficients by the method of variation of the Lagrange constants is considered. The Lagrange method is also applicable to solving any linear inhomogeneous equations if the fundamental system of solutions of the homogeneous equation is known.

Content

Lagrange method (variation of constants)

Consider a linear inhomogeneous differential equation with constant coefficients of an arbitrary nth order:
(1) .
The method of constant variation, which we considered for the first order equation, is also applicable to equations of higher orders.

The solution is carried out in two stages. At the first stage, we discard the right side and solve the homogeneous equation. As a result, we obtain a solution containing n arbitrary constants. In the second step, we vary the constants. That is, we consider that these constants are functions of the independent variable x and find the form of these functions.

Although we are considering equations with constant coefficients here, but the Lagrange method is also applicable to solving any linear inhomogeneous equations. For this, however, the fundamental system of solutions of the homogeneous equation must be known.

Step 1. Solution of the homogeneous equation

As in the case of first-order equations, we first look for the general solution of the homogeneous equation, equating the right inhomogeneous part to zero:
(2) .
The general solution of such an equation has the form:
(3) .
Here are arbitrary constants; - n linearly independent solutions of the homogeneous equation (2), which form the fundamental system of solutions of this equation.

Step 2. Variation of Constants - Replacing Constants with Functions

In the second step, we will deal with the variation of the constants. In other words, we will replace the constants with functions of the independent variable x :
.
That is, we are looking for a solution to the original equation (1) in the following form:
(4) .

If we substitute (4) into (1), we get one differential equation for n functions. In this case, we can connect these functions with additional equations. Then you get n equations, from which you can determine n functions. Additional equations can be written in various ways. But we will do it in such a way that the solution has the simplest form. To do this, when differentiating, you need to equate to zero terms containing derivatives of functions. Let's demonstrate this.

To substitute the proposed solution (4) into the original equation (1), we need to find the derivatives of the first n orders of the function written in the form (4). Differentiate (4) by applying the rules for differentiating the sum and the product:
.
Let's group the members. First, we write out the terms with derivatives of , and then the terms with derivatives of :

.
We impose the first condition on the functions:
(5.1) .
Then the expression for the first derivative with respect to will have a simpler form:
(6.1) .

In the same way, we find the second derivative:

.
We impose the second condition on the functions:
(5.2) .
Then
(6.2) .
And so on. Under additional conditions, we equate the terms containing the derivatives of the functions to zero.

Thus, if we choose the following additional equations for the functions :
(5.k) ,
then the first derivatives with respect to will have the simplest form:
(6.k) .
Here .

We find the nth derivative:
(6.n)
.

We substitute into the original equation (1):
(1) ;

.
We take into account that all functions satisfy equation (2):
.
Then the sum of the terms containing give zero. As a result, we get:
(7) .

As a result, we have a system linear equations for derivatives:
(5.1) ;
(5.2) ;
(5.3) ;
. . . . . . .
(5.n-1) ;
(7') .

Solving this system, we find expressions for derivatives as functions of x . Integrating, we get:
.
Here, are constants that no longer depend on x. Substituting into (4), we obtain the general solution of the original equation.

Note that we never used the fact that the coefficients a i are constant to determine the values of the derivatives. That's why the Lagrange method is applicable to solve any linear inhomogeneous equations, if the fundamental system of solutions of the homogeneous equation (2) is known.

Examples

Solve equations by the method of variation of constants (Lagrange).

Solution of examples > > >

See also: Solution of first order equations by constant variation method (Lagrange)
Solving higher-order equations by the Bernoulli method
Solving Linear Inhomogeneous Higher-Order Differential Equations with Constant Coefficients by Linear Substitution

Method of Variation of Arbitrary Constants

Method of variation of arbitrary constants for constructing a solution to a linear inhomogeneous differential equation

a n (t)z (n) (t) + a n − 1 (t)z (n − 1) (t) + ... + a 1 (t)z"(t) + a 0 (t)z(t) = f(t)

consists in changing arbitrary constants c k in the general decision

z(t) = c 1 z 1 (t) + c 2 z 2 (t) + ... + c n z n (t)

corresponding homogeneous equation

a n (t)z (n) (t) + a n − 1 (t)z (n − 1) (t) + ... + a 1 (t)z"(t) + a 0 (t)z(t) = 0

to helper functions c k (t) , whose derivatives satisfy the linear algebraic system

The determinant of system (1) is the Wronskian of functions z 1 ,z 2 ,...,z n , which ensures its unique solvability with respect to .

If are antiderivatives for taken at fixed values of the constants of integration, then the function

is a solution to the original linear inhomogeneous differential equation. Integration of an inhomogeneous equation in the presence of a general solution of the corresponding homogeneous equation is thus reduced to quadratures.

Method of variation of arbitrary constants for constructing solutions to a system of linear differential equations in vector normal form

consists in constructing a particular solution (1) in the form

where Z(t) is the basis of solutions of the corresponding homogeneous equation, written as a matrix, and the vector function , which replaced the vector of arbitrary constants, is defined by the relation . The desired particular solution (with zero initial values at t = t 0 has the form

For a system with constant coefficients, the last expression is simplified:

Matrix Z(t)Z− 1 (τ) called Cauchy matrix operator L = A(t) .

Lecture 44. Linear inhomogeneous equations of the second order. Method of variation of arbitrary constants. Linear inhomogeneous equations of the second order with constant coefficients. (special right side).

Social transformations. State and Church.

Social politics Bolsheviks was largely dictated by their class approach. By a decree of November 10, 1917, the estate system was abolished, pre-revolutionary ranks, titles and awards were abolished. The election of judges has been established; the secularization of civil states was carried out. Established free education and medical service(Decree of October 31, 1918). Women were equalized in rights with men (decrees of December 16 and 18, 1917). The decree on marriage introduced the institution of civil marriage.

By a decree of the Council of People's Commissars of January 20, 1918, the church was separated from the state and from the education system. Much of the church property was confiscated. Patriarch of Moscow and All Russia Tikhon (elected on November 5, 1917) anathematized on January 19, 1918 Soviet power and called for a fight against the Bolsheviks.

Consider a linear inhomogeneous second-order equation

The structure of the general solution of such an equation is determined by the following theorem:

Theorem 1. The general solution of the inhomogeneous equation (1) is represented as the sum of some particular solution of this equation and the general solution of the corresponding homogeneous equation

Proof. We need to prove that the sum

is the general solution of equation (1). Let us first prove that function (3) is a solution of equation (1).

Substituting the sum into equation (1) instead of at, will have

Since there is a solution to equation (2), the expression in the first brackets is identically equal to zero. Since there is a solution to equation (1), the expression in the second brackets is equal to f(x). Therefore, equality (4) is an identity. Thus, the first part of the theorem is proved.

Let us prove the second assertion: expression (3) is general solution of equation (1). We must prove that the arbitrary constants included in this expression can be chosen so that the initial conditions are satisfied:

whatever the numbers x 0 , y 0 and (if only x 0 was taken from the area where the functions a 1 , a 2 and f(x) continuous).

Noticing that it is possible to represent in the form . Then, based on conditions (5), we have

Let's solve this system and find From 1 and From 2. Let's rewrite the system as:

Note that the determinant of this system is the Wronsky determinant for the functions 1 and at 2 at the point x=x 0. Since these functions are linearly independent by assumption, the Wronsky determinant is not equal to zero; hence system (6) has a definite solution From 1 and From 2, i.e. there are such values From 1 and From 2, for which formula (3) determines the solution of equation (1) that satisfies the given initial conditions. Q.E.D.

Let's move on to common method finding particular solutions to an inhomogeneous equation.

Let us write the general solution of the homogeneous equation (2)

We will look for a particular solution of the inhomogeneous equation (1) in the form (7), considering From 1 and From 2 as some as yet unknown features from X.

Let us differentiate equality (7):

We select the desired functions From 1 and From 2 so that the equality

If this additional condition is taken into account, then the first derivative takes the form

Now differentiating this expression, we find:

Substituting into equation (1), we obtain

The expressions in the first two brackets vanish because y 1 and y2 are solutions of a homogeneous equation. Therefore, the last equality takes the form

Thus, function (7) will be a solution to the inhomogeneous equation (1) if the functions From 1 and From 2 satisfy equations (8) and (9). Let us compose a system of equations from equations (8) and (9).

Since the determinant of this system is the Vronsky determinant for linearly independent solutions y 1 and y2 equation (2), then it is not equal to zero. Therefore, solving the system, we find how certain functions from X:

Solving this system, we find , whence, as a result of integration, we obtain . Next, we substitute the found functions into the formula , we obtain the general solution of the inhomogeneous equation , where are arbitrary constants.

Theoretical minimum
In the theory of differential equations, there is a method that claims to have a sufficiently high degree of universality for this theory.
We are talking about the method of variation of an arbitrary constant, applicable to the solution of various classes of differential equations and their
systems. This is exactly the case when the theory - if you take the proof of the statements out of brackets - is minimal, but allows you to achieve
significant results, so the main focus will be on examples.
The general idea of the method is quite simple to formulate. Let given equation(system of equations) is difficult to solve or not clear at all,
how to solve it. However, it can be seen that when some terms are excluded from the equation, it is solved. Then they solve just such a simplified
equation (system), get a solution containing a certain number of arbitrary constants - depending on the order of the equation (the number
equations in the system). Then it is assumed that the constants in the found solution are not really constants, the found solution
is substituted into the original equation (system), a differential equation (or system of equations) is obtained to determine the "constants".
There is a certain specificity in applying the method of variation of an arbitrary constant to different problems, but these are already details that will be
shown with examples.
Let us separately consider the solution of linear inhomogeneous equations of higher orders, i.e. equations of the form
.
The general solution of a linear inhomogeneous equation is the sum of the general solution of the corresponding homogeneous equation and the particular solution
given equation. Let us assume that the general solution of the homogeneous equation has already been found, namely, the fundamental system of solutions (FSR) has been constructed
. Then the general solution of the homogeneous equation is .
It is necessary to find any particular solution of the inhomogeneous equation. For this, constants are considered to be dependent on the variable.
Next, you need to solve the system of equations
.
The theory guarantees that this system of algebraic equations with respect to derivatives of functions has a unique solution.
When finding the functions themselves, the integration constants do not appear: after all, any one solution is sought.
In the case of solving systems of linear inhomogeneous equations of the first order of the form

the algorithm remains almost unchanged. First you need to find the FSR of the corresponding homogeneous system of equations, compose the fundamental matrix
system , the columns of which are the elements of the FSR. Next, the equation
.
Solving the system, we determine the functions , thus finding a particular solution to the original system
(the fundamental matrix is multiplied by the found feature column).
We add it to the general solution of the corresponding system of homogeneous equations, which is built on the basis of the FSR already found.
The general solution of the original system is obtained.
Examples.
Example 1 Linear inhomogeneous equations of the first order.
Let us consider the corresponding homogeneous equation (we denote the required function by ):
.
This equation is easily solved by separation of variables:

.
Now we represent the solution of the original equation in the form , where the function is yet to be found.
We substitute this type of solution into the original equation:
.
As you can see, the second and third terms on the left side cancel each other out - this is feature method of variation of an arbitrary constant.

Here already - indeed, an arbitrary constant. In this way,
.
Example 2 Bernoulli equation.
We act similarly to the first example - we solve the equation

method of separation of variables. It will turn out , so we are looking for the solution of the original equation in the form
.
Substitute this function into the original equation:
.
And again there are cuts:
.
Here you need to remember to make sure that when dividing by, the solution is not lost. And the case corresponds to the solution of the original
equations. Let's remember him. So,
.
Let's write .
This is the solution. When writing the answer, you should also indicate the solution found earlier, since it does not correspond to any final value
constants .
Example 3 Linear inhomogeneous equations of higher orders.
We note right away that this equation can be solved more simply, but it is convenient to show the method on it. Although some advantages
the method of variation of an arbitrary constant also has it in this example.
So, you need to start with the FSR of the corresponding homogeneous equation. Recall that in order to find the FSR, the characteristic
the equation
.
Thus, the general solution of the homogeneous equation
.
The constants included here are to be varied. Compiling a system