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 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 sides of the equation:

here C is already some new constant.

3) In the general solution of the homogeneous equation y \u003d C / x, where we considered C \u003d C (x), that is, y \u003d C (x) / x, instead of C (x) we substitute the found expression x³ + C: y \u003d (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

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 various classes 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 algebraic equations with respect to derivatives of functions, there is only one 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 characteristic method of variation of an arbitrary constant.

Here already - indeed, an arbitrary constant. Thus,
.

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
.
We 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

The method of variation of arbitrary constants is used to solve inhomogeneous differential equations. This lesson is intended for those students who are already more or less well versed in the topic. If you are just starting to get acquainted with the remote control, i.e. If you are a teapot, I recommend starting with the first lesson: First order differential equations. Solution examples. And if you are already finishing, please discard the possible preconceived notion that the method is difficult. Because he is simple.

In what cases is the method of variation of arbitrary constants used?

1) The method of variation of an arbitrary constant can be used to solve linear inhomogeneous DE of the 1st order. Since the equation is of the first order, then the constant (constant) is also one.

2) The method of variation of arbitrary constants is used to solve some linear inhomogeneous equations of the second order. Here, two constants (constants) vary.

It is logical to assume that the lesson will consist of two paragraphs .... So I wrote this proposal, and for about 10 minutes I painfully thought about what other clever crap to add for a smooth transition to practical examples. But for some reason, there are no thoughts after the holidays, although it seems that I did not abuse anything. So let's jump right into the first paragraph.

Arbitrary Constant Variation Method
for a linear inhomogeneous first-order equation

Before considering the method of variation of an arbitrary constant, it is desirable to be familiar with the article Linear differential equations of the first order. In that lesson, we practiced first way to solve inhomogeneous DE of the 1st order. This first solution, I remind you, is called replacement method or Bernoulli method(not to be confused with Bernoulli equation!!!)

We will now consider second way to solve– method of variation of an arbitrary constant. I will give only three examples, and I will take them from the above lesson. Why so few? Because in fact the solution in the second way will be very similar to the solution in the first way. In addition, according to my observations, the method of variation of arbitrary constants is used less often than the replacement method.

Example 1

(Diffur from Example No. 2 of the lesson Linear inhomogeneous DE of the 1st order)

Solution: This equation is linear inhomogeneous and has a familiar form:

The first step is to solve a simpler equation:
That is, we stupidly reset the right side - instead we write zero.
The equation I'll call auxiliary equation.

In this example, you need to solve the following auxiliary equation:

Before us separable equation, the solution of which (I hope) is no longer difficult for you:

Thus:
is the general solution of the auxiliary equation .

On the second step replace a constant of some yet unknown function that depends on "x":

Hence the name of the method - we vary the constant . Alternatively, the constant can be some function that we have to find now.

IN original inhomogeneous equation Let's replace:

Substitute and into the equation :

control moment - the two terms on the left side cancel. If this does not happen, you should look for the error above.

As a result of the replacement, an equation with separable variables is obtained. Separate variables and integrate.

What a blessing, the exponents are shrinking too:

We add a “normal” constant to the found function:

At the final stage, we recall our replacement:

Function just found!

So the general solution is:

Answer: common decision:

If you print out the two solutions, you will easily notice that in both cases we found the same integrals. The only difference is in the solution algorithm.

Now something more complicated, I will also comment on the second example:

Example 2

Find the general solution of the differential equation
(Diffur from Example No. 8 of lesson Linear inhomogeneous DE of the 1st order)

Solution: We bring the equation to the form :

Set the right side to zero and solve the auxiliary equation:

General solution of the auxiliary equation:

In the inhomogeneous equation, we will make the substitution:

According to the product differentiation rule:

Substitute and into the original inhomogeneous equation :

The two terms on the left side cancel out, which means we are on the right track:

We integrate by parts. Tasty letter from the integration-by-parts formula we have already involved in the solution, so we use, for example, the letters "a" and "be":

Now let's look at the replacement:

Answer: common decision:

And one example for independent solution:

Example 3

Find a particular solution of the differential equation corresponding to the given initial condition.

,
(Diffur from Lesson 4 Example Linear inhomogeneous DE of the 1st order)
Solution:
This DE is linear inhomogeneous. We use the method of variation of arbitrary constants. Let's solve the auxiliary equation:

We separate the variables and integrate:

Common decision:
In the inhomogeneous equation, we will make the substitution:

Let's do the substitution:

So the general solution is:

Find a particular solution corresponding to the given initial condition:

Answer: private solution:

The solution at the end of the lesson can serve as an approximate model for finishing the assignment.

Method of Variation of Arbitrary Constants
for a linear inhomogeneous second order equation
with constant coefficients

One often heard the opinion that the method of variation of arbitrary constants for a second-order equation is not an easy thing. But I guess the following: most likely, the method seems difficult to many, since it is not so common. But in reality, there are no particular difficulties - the course of the decision is clear, transparent, and understandable. And beautiful.

To master the method, it is desirable to be able to solve inhomogeneous equations of the second order by selecting a particular solution according to the form of the right side. This method is discussed in detail in the article. Inhomogeneous DE of the 2nd order. We recall that a second-order linear inhomogeneous equation with constant coefficients has the form:

The selection method, which was considered in the above lesson, works only in a limited number of cases, when polynomials, exponents, sines, cosines are on the right side. But what to do when on the right, for example, a fraction, logarithm, tangent? In such a situation, the method of variation of constants comes to the rescue.

Example 4

Find the general solution of a second-order differential equation

Solution: There is a fraction on the right side of this equation, so we can immediately say that the method of selecting a particular solution does not work. We use the method of variation of arbitrary constants.

Nothing portends a thunderstorm, the beginning of the solution is quite ordinary:

Let's find common decision corresponding homogeneous equations:

We compose and solve the characteristic equation:


– conjugate complex roots are obtained, so the general solution is:

Pay attention to the record of the general solution - if there are brackets, then open them.

Now we do almost the same trick as for the first order equation: we vary the constants , replacing them with unknown functions . That is, general solution of the inhomogeneous We will look for equations in the form:

Where - yet unknown functions.

It looks like a garbage dump, but now we'll sort everything.

Derivatives of functions act as unknowns. Our goal is to find derivatives, and the found derivatives must satisfy both the first and second equations of the system.

Where do "games" come from? The stork brings them. We look at the previously obtained general solution and write:

Let's find derivatives:

Dealt with the left side. What's on the right?

is the right side of the original equation, in this case:

The coefficient is the coefficient at the second derivative:

In practice, almost always, and our example is no exception.

Everything cleared up, now you can create a system:

The system is usually solved according to Cramer's formulas using the standard algorithm. The only difference is that instead of numbers we have functions.

Find the main determinant of the system:

If you forgot how the “two by two” determinant is revealed, refer to the lesson How to calculate the determinant? The link leads to the board of shame =)

So: , so the system has a unique solution.

We find the derivative:

But that's not all, so far we've only found the derivative.
The function itself is restored by integration:

Let's look at the second function:

Here we add a "normal" constant

At the final stage of the solution, we recall in what form we were looking for the general solution of the inhomogeneous equation? In such:

The features you need have just been found!

It remains to perform the substitution and write down the answer:

Answer: common decision:

In principle, the answer could open the brackets.

A full check of the answer is performed according to the standard scheme, which was considered in the lesson. Inhomogeneous DE of the 2nd order. But the verification will not be easy, since we have to find rather heavy derivatives and carry out a cumbersome substitution. This is a nasty feature when you're solving diffs like this.

Example 5

Solve the differential equation by the method of variation of arbitrary constants

This is a do-it-yourself example. In fact, the right side is also a fraction. We remember trigonometric formula, by the way, it will need to be applied in the course of the solution.

The method of variation of arbitrary constants is the most generic method. They can solve any equation that can be solved the method of selecting a particular solution according to the form of the right side. The question arises, why not use the method of variation of arbitrary constants there as well? The answer is obvious: the selection of a particular solution, which was considered in the lesson Inhomogeneous equations of the second order, significantly speeds up the solution and shortens the notation - no messing around with determinants and integrals.

Consider two examples with Cauchy problem.

Example 6

Find a particular solution of the differential equation corresponding to given initial conditions

,

Solution: Again a fraction and an exponent in interesting place.
We use the method of variation of arbitrary constants.

Let's find common decision corresponding homogeneous equations:



– different real roots are obtained, so the general solution is:

The general solution of the inhomogeneous we are looking for equations in the form: , where - yet unknown functions.

Let's create a system:

In this case:
,
Finding derivatives:
,

Thus:

We solve the system using Cramer's formulas:
, so the system has a unique solution.

We restore the function by integration:

Used here method of bringing a function under a differential sign.

We restore the second function by integration:

Such an integral is solved variable substitution method:

From the replacement itself, we express:

Thus:

This integral can be found extraction method full square , but in examples with diffurs, I prefer to expand the fraction method of uncertain coefficients:

Both functions found:

As a result, the general solution of the inhomogeneous equation is:

Find a particular solution that satisfies the initial conditions .

Technically, the search for a solution is carried out in a standard way, which was discussed in the article. Inhomogeneous Second Order Differential Equations.

Hold on, now we will find the derivative of the found general solution:

Here is such a disgrace. It is not necessary to simplify it, it is easier to immediately compose a system of equations. According to the initial conditions :

Substitute the found values ​​of the constants into a general solution:

In the answer, the logarithms can be packed a little.

Answer: private solution:

As you can see, difficulties can arise in integrals and derivatives, but not in the algorithm of the method of variation of arbitrary constants. It was not I who intimidated you, this is all a collection of Kuznetsov!

To relax, a final, simpler, self-solving example:

Example 7

Solve the Cauchy problem

,

The example is simple, but creative, when you make a system, look at it carefully before deciding ;-),




As a result, the general solution is:

Find a particular solution corresponding to the initial conditions .



We substitute the found values ​​of the constants into the general solution:

Answer: private solution:

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 considered by us for first order equations, 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 sum differentiation rules And works :
.
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