Indicate the structure of the general solution of the differential equation. Structure of the general solution of a linear homogeneous differential equation

D U higher orders

As we have already said, differential equations can contain derivatives of various orders.

Such differential equations have solutions that contain as many arbitrary constants of integration → what is the order of the differential equation, i.e. for a differential equation of the 2nd order, there will be two arbitrary constants C1 and C2, for the 3rd →C1, C2, and C3, etc.

Thus, the general solution (general integral) of such a differential equation will be the function

.

To obtain a particular solution of such differential equations, it is necessary to set as many initial conditions as the order of the differential equation, or how many arbitrary constants are obtained in the general solution.

D U in total differentials. Integrating factor

A differential equation of the form is called a differential equation in complete differentials if its left-hand side is the total differential of some smooth function , i.e. if , . A necessary and sufficient condition for the existence of such a function is:

To solve a differential equation in total differentials, you need to find a function. Then the general solution of the differential equation can be written in the form for an arbitrary constant C.

An integrating factor for the differential equation

such a function is called, after multiplication by which the differential equation turns into an equation in total differentials. If the functions M and N in the equation have continuous partial derivatives and do not vanish at the same time, then an integrating factor exists. However, there is no general method for finding it.

Structure of the general solution of LNDE

Consider a linear nonhomogeneous differential equation

+ (x) + ... + (x)y" + (x)y = f(x).

− whatever the initial point (x0, y0, ) , x0∈ , there are values ​​C1 =C10 , ..., Cn = Cn0 such that the function y = Φ(x, C10 , ..., Cn0) satisfies the initial conditions y(x0) = y0, y "(x0) ,..., (x0) = .

The following assertion is true (a theorem on the structure of the general solution of a linear inhomogeneous equation).

If all coefficients of the equation of a linear homogeneous differential equation are continuous on the segment , and the functions y1(x), y2(x),..., yn(x) form a system of solutions of the corresponding homogeneous equation, then the general solution of the inhomogeneous equation has the form

y(x,C1,..., Cn) = C1 y1(x) + C2 y2(x) + ... + Cn yn(x) + y*(x),

where C1,...,Cn are arbitrary constants, y*(x) is a particular solution of an inhomogeneous equation.

LNDE of the 2nd order

Linear inhomogeneous differential equations of the second order.

An equation of the form y "+ py" + qy \u003d f (x), where p and q are real numbers, f (x) is a continuous function, is called a second-order linear non-homogeneous equation with constant coefficients.

The general solution of the equation is the sum of the particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. Finding the general solution of the homogeneous equation has been studied. To find a particular solution, we use the method of indefinite coefficients, which does not contain an integration process.

Consider different types of right-hand sides of the equation y" + py" + qy = f(x).

1) The right side has the form F(x) = Pn(x), where Pn(x) is a polynomial of degree n. Then a particular solution y can be sought in the form where Qn (x) is a polynomial of the same degree as Pn (x), and r is the number of zero roots of the characteristic equation.

Example. Find the general solution of the equation y "- 2y" + y \u003d x + 1.

Decision: The general solution of the corresponding homogeneous equation has the form Y = ex (C1 + C2x) . Since none of the roots of the characteristic equation k2 – 2k + 1 = 0 is equal to zero (k1 = k2 = 1), we are looking for a particular solution in the form where A and B are unknown coefficients. Differentiating twice and substituting, " and " in this equation, we find -2A + Ax + B = x + 1.

Equating the coefficients at the same powers of x in both parts of the equation: A = 1, –2A + B = 1, - we find A = 1, B = 3. So, the particular solution of this equation has the form = x + 3, and its general solution is y \u003d ex (C1 + C2x) + x + Z.

2) The right side has the form f(x) = eax Pn(x), where Рn (х) is a polynomial of degree n. Then a particular solution should be sought in the form where Qn(x) is a polynomial of the same degree as Pn(x), and r is the number of roots of the characteristic equation equal to a. If a = 0, then f(x) = Pn(x), i.e. case 1 takes place.

LODU with constant coefficients.

Consider the differential equation

where are real constants.

To find the general solution of Eq. (8), we proceed as follows. We compose the characteristic equation for equation (8): (9)

Let be the roots of equation (9), and among them there may be multiples. The following cases are possible:

a) - real and different. The general solution of the homogeneous equation will be ;

b) the roots of the characteristic equation are real, but there are multiples among them, i.e. , then the general solution will be

c) if the roots of the characteristic equation are complex (k=a±bi), then the general solution has the form .

The structure of the total. LODE solutions of the 2nd order

Consider the linear homogeneous differential equation

+ (x) + ... + (x)y" + (x)y = 0.

A general solution of this equation on an interval is a function y = Φ(x, C1,..., Cn) depending on n arbitrary constants C1,..., Cn and satisfying the following conditions:

− for any admissible values ​​of the constants C1,..., Cn, the function y = Φ(x, C1,..., Cn) is a solution to the equation on ;

− whatever the initial point (x0, y0, ) , x0∈ , there are values ​​C1 =C10 , ..., Cn = Cn0 such that the function y = Φ(x, C10 , ..., Cn0) satisfies the initial conditions y(x0) = y0, y "(x0) = y1,0 ,..., (x0) = .

Knowledge of the fundamental system of solutions to the equation makes it possible to construct a general solution to this equation. Recall the definition of the general solution of the differential equation P-th order

Function
, defined in some range of variables
, at each point of which the existence and uniqueness of the solution of the Cauchy problem takes place, and which has continuous partial derivatives with respect to X up to order P inclusive, is called the general solution of equation (15) in the specified area, if:

system of equations

is solvable in the specified region with respect to arbitrary constants
, so

(16)

2. function
is a solution to equation (15) for all values ​​of arbitrary constants
, expressed by formulas (16), when the point
belongs to the area under consideration.

Theorem 1. (on the structure of the general solution of a linear homogeneous differential equation). If functions
,
, …,
form a fundamental system of solutions to the homogeneous linear equation P-th order
in the interval
, i.e. in the interval of continuity of the coefficients, then the function
is the general solution of this equation in the domain D:
,
,
.

Proof. At each point of the indicated region, the existence and uniqueness of the solution of the Cauchy problem takes place. Let us now show that the function
satisfies the definition of a general solution to the equation P-th order.

system of equations

solvable in the area D with respect to arbitrary constants
since the determinant of this system is the Wronsky determinant for the fundamental system of solutions (12) and, therefore, is nonzero.

2. Function
by the property of solutions of a homogeneous linear equation is a solution of the equation
for all values ​​of arbitrary constants
.

Therefore, the function
is a general solution to the equation
in area D. The theorem has been proven.

Example.

.

The solutions of this equation are obviously the functions
,
. These solutions form a fundamental system of solutions, since

.

Therefore, the general solution to the original equation is the function .

The structure of the general solution of an inhomogeneous linear equation of the nth order.

Consider the inhomogeneous linear equation P-th order

Let us show that, as in the case of a linear inhomogeneous equation of the first order, the integration of equation (1) reduces to the integration of a homogeneous equation if one particular solution of the inhomogeneous equation (1) is known.

Let be
- a particular solution of equation (1), i.e.

,
. (2)

Let's put
, where z is a new unknown function from X. Then equation (1) takes the form

or
,

whence, by virtue of identity (2), we obtain

. (3)

This is a homogeneous linear equation, the left side of which is the same as the considered inhomogeneous equation (1). Those. we have obtained a homogeneous equation corresponding to this inhomogeneous equation (1).

,
, …,
,

is the fundamental system of solutions of the homogeneous equation (3). Then all solutions of this equation are contained in the formula for its general solution, i.e.

.

Substitute this value z into the formula
, we get

.

The resulting function is a general solution of equation (1) in the region D.

Thus, we have shown that the general solution of the linear inhomogeneous equation (1) is equal to the sum of some particular solution of this equation and the general solution of the corresponding homogeneous linear equation.

Example. Find a general solution to the equation

.

Decision. We have, a particular solution of this inhomogeneous linear equation has the form

.

General solution of the corresponding homogeneous equation
, as we showed earlier, has the form

Therefore, the general solution to the original equation is:
.

In many cases, the task of finding a particular solution to an inhomogeneous equation is facilitated by using the following property:

Theorem. If in equation (1) the right side has the form

and it is known that
, a - particular solution of the equation
, then the sum of these particular solutions +will be a particular solution of equation (1).

Proof. Indeed, since by condition is a particular solution to the equation
, a - particular solution of the equation
, then

,
.

those. +is a particular solution of equation (1).

For a linear inhomogeneous differential equation n- order

y(n) + a 1(x)y(n- 1) + ... + an- 1 (x) y" + an(x)y = f(x),

where y = y(x) is an unknown function, a 1(x),a 2(x), ..., an- 1(x), an(x), f(x) - known, continuous, fair:
1) if y 1(x) and y 2(x) are two solutions of the inhomogeneous equation, then the function
y(x) = y 1(x) - y 2(x) is the solution of the corresponding homogeneous equation;
2) if y 1(x) solution of the inhomogeneous equation, and y 2(x) is the solution of the corresponding homogeneous equation, then the function
y(x) = y 1(x) + y 2(x) is the solution of the inhomogeneous equation;
3) if y 1(x), y 2(x), ..., yn(x) - n linearly independent solutions of a homogeneous equation, and ych(x) is an arbitrary solution of an inhomogeneous equation,
then for any initial values
x 0, y 0, y 0,1, ..., y 0,n- 1
Expression
y(x)=c 1 y 1(x) + c 2 y 2(x) + ... + cn yn(x) +ych(x)
called common solution linear inhomogeneous differential equation n-th order.

To find particular solutions to inhomogeneous differential equations with constant coefficients with right-hand sides of the form:
pk(x)exp(a x)cos( bx) + Q m(x)exp(a x)sin( bx),
where pk(x), Q m(x) - degree polynomials k and m accordingly, there is a simple algorithm for constructing a particular solution, called selection method.

The selection method, or the method of uncertain coefficients, is as follows.
The desired solution of the equation is written as:
(Pr(x)exp(a x)cos( bx) + QR(x)exp(a x)sin( bx))xs,
where Pr(x), QR(x) - degree polynomials r=max( k, m) with unknown coefficients
pr , pr- 1, ..., p 1, p 0, qr, qr- 1, ..., q 1, q 0.
Thus, to find a general solution to a linear inhomogeneous differential equation with constant coefficients, one should
find the general solution of the corresponding homogeneous equation (write the characteristic equation, find all the roots of the characteristic equation l 1, l 2, ... , ln, write down the fundamental system of solutions y 1(x), y 2(x), ..., yn(x));
find any particular solution of the inhomogeneous equation ych(x);
write an expression for the general solution
y(x)=c 1 y 1(x) + c 2 y 2(x) + ... + cn yn(x) + ych(x);

Linear inhomogeneous differential equations of the second order with constant coefficients with the right side of a special form. Method of indefinite coefficients.

Differential equation of the form (1)

where , f is a known function, is called a linear differential equation of the nth order with constant coefficients. If , then equation (1) is called homogeneous, otherwise it is called inhomogeneous.

For linear inhomogeneous equations with constant coefficients and with a right-hand side of a special form, namely, consisting of sums and products of functions , a particular solution can be sought by the method of indefinite coefficients. The form of the particular solution depends on the roots of the characteristic equation. Below is a table of types of particular solutions of a linear inhomogeneous equation with a special right-hand side.

complex plane. Modulus and argument of a complex number. The main value of the argument. geometric sense

Complex numbers are written as: a + bi. Here a and b are real numbers, and i is an imaginary unit, i.e. i 2 \u003d -1. The number a is called the abscissa, and b is called the ordinate of the complex number a + bi. Two complex numbers a + bi and a - bi are called conjugate complex numbers.

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here point A means the number -3, point B is the number 2, and O is zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a + bi will be represented by the point P with the abscissa a and the ordinate b (see Fig.). This coordinate system is called the complex plane.

The modulus of a complex number is the length of the vector OP representing the complex number on the coordinate (complex) plane. The modulus of a complex number a+ bi is denoted by | a + bi | or the letter r and is equal to:

Conjugate complex numbers have the same modulus. __

The argument of a complex number is the angle between the OX axis and the vector OP representing that complex number. Hence, tan = b / a .

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 at h and the general solution of the corresponding homogeneous equation

Proof. We need to prove that the sum (3)

There is a general solution to equation (1).

Let us first prove that function (3) is a solution of equation (1). Substituting instead of at sum into equation (1) we will have:

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

Let us now prove that expression (3) is the general solution of equation (1), i.e. let us prove that the arbitrary constants included in it can be chosen so that the initial conditions (5) are satisfied

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

Noting that it can be represented in the form , where at 1, at 2 linearly independent solutions of equation (2), and From 1 and From 2 are arbitrary constants, we can rewrite equality (3) as . Then, based on condition (5), we will have the system

.

From this system of equations, you need to determine From 1 and From 2. Let us rewrite the system in the form

(6)

System qualifier – there is a Vronsky determinant for solutions 1 and at 2 at point . Since these functions are linearly independent by condition, the Wronsky determinant is not equal to zero, therefore, system (6) has a unique 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.

Thus, if the general solution of the homogeneous equation (2) is known, then the main task in integrating the inhomogeneous equation (1) is to find some of its particular solutions at h.

Linear inhomogeneous differential equations of the second order with constant coefficients with the right side of a special form. Method of indefinite coefficients.

Sometimes it is possible to find a simpler solution without resorting to integration. This happens in special cases when the function f(x) has a special look.

Let we have the equation , (1)

where p and q real numbers, and f(x) has a special look. Consider several such possibilities for equation (1).

Let the right side of equation (1) be the product of an exponential function and a polynomial, i.e. has the form , (2)

where is a polynomial of the nth degree. Then the following cases are possible:

a) number - is not a root characteristic equation .

In this case, a particular solution must be sought in the form (3)

those. as a polynomial too n-th degree, where А 0 , А 1 ,…,А n coefficients are to be determined.

In order to determine them, we find the derivatives and .

Substituting at h, and into equation (1) and reducing both parts by a factor, we will have:

Here, is a polynomial of the nth degree, is a polynomial of the (n-1)th degree, and is a polynomial of the (n-2)th degree.

Thus, to the left and to the right of the equal sign are polynomials n-th degree. Equating the coefficients at the same powers X(the number of unknown coefficients is ), we obtain a system of equations for determining the coefficients А 0 , А 1 , …, А n .

if the right side of equation (1) has the form: