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) .

The method of variation of an arbitrary constant, or the Lagrange method, is another way to solve first-order linear differential equations and the Bernoulli equation.

Linear differential equations of the first order are equations of the form y’+p(x)y=q(x). If the right side is zero: y’+p(x)y=0, then this is a linear homogeneous 1st order equation. Accordingly, the equation with a non-zero right side, y’+p(x)y=q(x), — heterogeneous linear equation of the 1st order.

Arbitrary constant variation method (Lagrange method) consists of the following:

1) We are looking for a general solution to the homogeneous equation y’+p(x)y=0: y=y*.

2) In the general solution, C is considered not a constant, but a function of x: C=C(x). We find the derivative of the general solution (y*)' and substitute the resulting expression for y* and (y*)' into the initial condition. From the resulting equation, we find the function С(x).

3) In the general solution of the homogeneous equation, instead of C, we substitute the found expression C (x).

Consider examples on the method of variation of an arbitrary constant. Let's take the same tasks as in , compare the course of the solution and make sure that the answers received are the same.

1) y'=3x-y/x

Let's rewrite the equation in standard form (in contrast to the Bernoulli method, where we needed the notation only to see that the equation is linear).

y'+y/x=3x (I). Now we are going according to plan.

1) We solve the homogeneous equation y’+y/x=0. This is a separable variable equation. Represent y’=dy/dx, substitute: dy/dx+y/x=0, dy/dx=-y/x. We multiply both parts of the equation by dx and divide by xy≠0: dy/y=-dx/x. We integrate:

2) In the obtained general solution of the homogeneous equation, we will consider С not a constant, but a function of x: С=С(x). From here

The resulting expressions are substituted into condition (I):

We integrate both parts of the equation:

here C is already some new constant.

3) In the general solution of the homogeneous equation y=C/x, where we considered С=С(x), that is, y=C(x)/x, instead of С(x) we substitute the found expression x³+C: y=(x³ +C)/x or y=x²+C/x. We got the same answer as when solving by the Bernoulli method.

Answer: y=x²+C/x.

2) y'+y=cosx.

Here the equation is already written in standard form, no need to convert.

1) We solve a homogeneous linear equation y’+y=0: dy/dx=-y; dy/y=-dx. We integrate:

To get a more convenient notation, we will take the exponent to the power of C as a new C:

This transformation was performed to make it more convenient to find the derivative.

2) In the obtained general solution of a linear homogeneous equation, we consider С not a constant, but a function of x: С=С(x). Under this condition

The resulting expressions y and y' are substituted into the condition:

Multiply both sides of the equation by

We integrate both parts of the equation using the integration-by-parts formula, we get:

Here C is no longer a function, but an ordinary constant.

3) Into the general solution of the homogeneous equation

we substitute the found function С(x):

We got the same answer as when solving by the Bernoulli method.

The method of variation of an arbitrary constant is also applicable to solving .

y’x+y=-xy².

We bring the equation to the standard form: y’+y/x=-y² (II).

1) We solve the homogeneous equation y’+y/x=0. dy/dx=-y/x. Multiply both sides of the equation by dx and divide by y: dy/y=-dx/x. Now let's integrate:

We substitute the obtained expressions into condition (II):

Simplifying:

We got an equation with separable variables for C and x:

Here C is already an ordinary constant. In the process of integration, instead of C(x), we simply wrote C, so as not to overload the notation. And at the end we returned to C(x) so as not to confuse C(x) with the new C.

3) We substitute the found function С(x) into the general solution of the homogeneous equation y=C(x)/x:

We got the same answer as when solving by the Bernoulli method.

Examples for self-test:

1. Let's rewrite the equation in standard form: y'-2y=x.

1) We solve the homogeneous equation y'-2y=0. y’=dy/dx, hence dy/dx=2y, multiply both sides of the equation by dx, divide by y and integrate:

From here we find y:

We substitute the expressions for y and y’ into the condition (for brevity, we will feed C instead of C (x) and C’ instead of C "(x)):

To find the integral on the right side, we use the integration-by-parts formula:

Now we substitute u, du and v into the formula:

Here C = const.

3) Now we substitute into the solution of the homogeneous

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.

The social policy of the 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 care (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 Tikhon of Moscow and All Russia (elected November 5, 1917) on January 19, 1918, anathematized 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 us turn to the general method for finding particular solutions of 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 will find both certain functions of 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.

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

See also:

Lagrange method (variation of constants)

Consider a linear inhomogeneous differential equation with constant coefficients of an arbitrary nth order:
(1) .
The constant variation method, 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) .
Etc. 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 got a system of 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. So 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