Linear differential systems equations.
The system of differential equations is called linear, if it is linear with respect to unknown functions and their derivatives. system n-linear equations 1st order is written as:
The system coefficients are const.
It is convenient to write this system in matrix form: ,
where is a column vector of unknown functions depending on one argument.
Column vector of derivatives of these functions.
Column vector of free members.
Coefficient matrix.
Theorem 1: If all matrix coefficients A are continuous on some interval and , then in some neighborhood of each m. 
TS&E conditions are met. Consequently, through each such point there passes a single integral curve.
Indeed, in this case, the right-hand sides of the system are continuous with respect to the set of arguments and their partial derivatives with respect to (equal to the coefficients of matrix A) are limited, due to continuity on a closed interval.
Methods for solving SLDs
1. A system of differential equations can be reduced to one equation by eliminating the unknowns.
Example: Solve the system of equations: 
(1)
Solution: exclude z from these equations. From the first equation we have . Substituting into the second equation, 
after simplification we get: 
.
This system of equations (1) reduced to a single second-order equation. After finding from this equation y, should be found z, using equality.
2. When solving a system of equations by eliminating unknowns, one usually obtains an equation more high order, therefore, in many cases it is more convenient to solve the system by finding integrated combinations.
Continued 27b
Example: Solve the system 
Solution:
Let's decide this system Euler's method. Let us write down the determinant for finding the characteristic
equation: , (since the system is homogeneous, in order for it to have a non-trivial solution, this determinant must be equal to zero). We obtain a characteristic equation and find its roots: 
The general solution is: 
;
- eigenvector.
We write down the solution for: 
;
- eigenvector.
We write down the solution for: 
;
We get the general solution: 
.
Let's check:
let's find : and substitute it into the first equation of this system, i.e. .
We get:
- true equality.
Linear diff. nth order equations. Theorem on the general solution of an inhomogeneous linear equation of the nth order.
A linear differential equation of the nth order is an equation of the form: (1)
If this equation has a coefficient, then dividing by it, we arrive at the equation: (2) .
Usually equations of the type (2). Suppose that in ur-i (2) all odds, as well as f(x) continuous on some interval (a,b). Then, according to TS&E, the equation (2) has a unique solution that satisfies the initial conditions: , , …, for . Here - any point from the interval (a,b), and all - any given numbers. Equation (2) satisfies TC&E , therefore does not have special solutions.
Def.: special the points are those at which =0.
Properties of a linear equation:
- A linear equation remains linear no matter how the independent variable is changed.
- A linear equation remains so for any linear change of the desired function.
Def: if in the equation (2) put f(x)=0, then we get an equation of the form: (3) , which is called homogeneous equation relatively not homogeneous equation (2).
Let us introduce the linear differential operator: (4). Using this operator, you can rewrite in short form the equation (2) And (3): L(y)=f(x), L(y)=0. Operator (4) has the following simple properties:
From these two properties a corollary can be deduced: .
Function y=y(x) is a solution to the inhomogeneous equation (2), If L(y(x))=f(x), Then f(x) called the solution to the equation. So the solution to the equation (3) called the function y(x), If L(y(x))=0 on the considered intervals.
Consider inhomogeneous linear equation: , L(y)=f(x).
Suppose that we have found a particular solution in some way, then .
Let's introduce a new unknown function z according to the formula: , where is a particular solution.
Let's substitute it into the equation: , open the brackets and get: .
The resulting equation can be rewritten as: 
Since is a particular solution to the original equation, then , then .
Thus, we have obtained a homogeneous equation with respect to z. The general solution to this homogeneous equation is a linear combination: , where the functions - constitute the fundamental system of solutions to the homogeneous equation. Substituting z into the replacement formula, we get: 
(*) for function y– unknown function of the original equation. All solutions to the original equation will be contained in (*).
Thus, the general solution of the inhomogeneous line. equation is represented as the sum of a general solution of a homogeneous linear equation and some particular solution of an inhomogeneous equation.
(continued on the other side)
30. Theorem of existence and uniqueness of the solution to differential. equations
Theorem: If the right side of the equation is continuous in the rectangle 
and is limited, and also satisfies the Lipschitz condition: , N=const, then there is a unique solution that satisfies the initial conditions and is defined on the segment 
, Where .
Proof:
Consider the complete metric space WITH, whose points are all possible continuous functions y(x) defined on the interval , the graphs of which lie inside the rectangle, and the distance is determined by the equality: 
. This space is often used in mathematical analysis and is called space of uniform convergence, since the convergence in the metric of this space is uniform.
Let's replace the differential. equation with given initial conditions to an equivalent integral equation: 
and consider the operator A(y), equal to the right side of this equation: 
. This operator matches each continuous function
Using Lipschitz's inequality, we can write that the distance . Now let’s choose one for which the following inequality would hold: .
It should be chosen so that , then . Thus we showed that .
According to the principle of contraction mappings, there is a single point or, what is the same, a single function - a solution to a differential equation that satisfies the given initial conditions.
Theorem on the general solution (on the structure of the general solution) of a normal system of inhomogeneous linear odes.
Let us consider an inhomogeneous linear system of ordinary differential equations of the nth order
Here 
A 
The following is true general solution structure theorem this heterogeneous linear system ODU.
If matrix A(x) and vector function b
(x) are continuous on [ a,
b], and let Φ
(x) is the fundamental matrix of solutions of a homogeneous linear system, then the general solution of the inhomogeneous system Y"
= A(x) Y
+ b(x) has the form: 
Where C- an arbitrary constant column vector, x 0 - an arbitrary fixed point from the segment.
From the above formula it is easy to obtain a formula for solving the Cauchy problem for a linear inhomogeneous ODE system - the Cauchy formula.
Solving the Cauchy problem, Y(x 0) = Y 0 is a vector function
Method of variation of arbitrary constants for finding partial solutions of a normal system of inhomogeneous linear odes.
Definition of a system of inhomogeneous linear ODEs. ODU system type:
called linear heterogeneous . Let
System (*) in vector-matrix form: .- the system is homogeneous, otherwise it is inhomogeneous.
The method itself.
Let there be a linear inhomogeneous system 
, then is a linear homogeneous system corresponding to a linear inhomogeneous one. Let be the fundamental matrix of the decision system, 
, where C is an arbitrary constant vector, is the general solution of the system. Let us look for a solution to system (1) in the form 
, where C(x) is an unknown (yet) vector function. We want the vector function (3) to be a solution to system (1). Then the identity must be true:
(an arbitrary constant vector, which is obtained as a result of integration, can be considered equal to 0). Here the points x 0 , are any.
We see, therefore, that if in (3) we take as C(t) 
, then the vector function 
will be a solution to system (1).
The general solution of the linear inhomogeneous system (1) can be written in the form 
. Let it be necessary to find a solution to system (1) that satisfies the initial condition 
. Substitution (4) of the initial data (5) gives 
. Therefore, the solution to the Cauchy problem (1)-(5) can be written as: 
. In the special case when the last formula takes the form: 
.
Fundamental system of solutions to a normal system of homogeneous linear equations with constant coefficients in the case of simple real roots of the characteristic equation.
Normal linear homogeneous systemnabout s constant coefficients
-
or 
,The coefficients of linear combinations of the sought functions are constant. This system is in matrix form 
–matrix form, where A is a constant matrix. Matrix method: From characteristic equation 
we will find different roots and for each root (taking into account its multiplicity) we will determine the corresponding particular solution. The general solution is: 
. In this case 1) if -
is a real root of multiple 1, then 
, where is the eigenvector of matrix A corresponding to the eigenvalue, that is. 2)
–
multiplicity root, then the system solution corresponding to this root is sought in the form of a vector 
(**), whose coefficients 
are determined from a system of linear equations obtained by equating the coefficients at the same powersx as a result of substituting the vector (**) into the original system.
Fundamental system of NLOS solutions is a collection of arbitrary n linearly independent solutions
A fundamental system of solutions to a normal system of homogeneous linear ODEs with constant coefficients in the case when all the roots of the characteristic equation are simple, but there are complex roots.
The question has been removed.
General view of the system
, i = 1, 2, ..., m; j = 1, 2, ..., n, - system coefficients; - free members; - variables;
If all = 0, the system is called homogeneous.
General solution of a system of linear equations
Definition 1. Homogeneous system m linear algebraic equations For n unknowns is called a system of equations
type (1) or in matrix form (2)
where A is a given matrix of coefficients of size mxn,
Column n of unknowns is the zero column of height m.
A homogeneous system is always consistent (the extended matrix coincides with A) and has obvious solutions: x 1 = x 2 = ... = x n = 0.
This solution is called zero or trivial. Any other solution, if there is one, is called non-trivial.
Theorem 1. If the rank of matrix A is equal to the number of unknowns, then system (1) has a unique (trivial) solution.
Indeed, according to Cramer’s theorem, r=n and the solution is unique.
Theorem 2. In order for a homogeneous system to have a non-zero solution, it is necessary and sufficient that the rank of the system matrix be less than the number of unknowns ( follows from the theorem on the number of solutions).
Þ if there are non-zero solutions, then the solution is not unique, then the determinant of the system is equal to zero, then r Ü if r Theorem 3. A homogeneous system of n equations with n unknowns has a nonzero solution if and only if detA = 0. Þ if there are non-zero solutions, then there are infinitely many solutions, then according to the theorem on the number of solutions r Ü if detA = 0, then r Theorem 4. In order for a homogeneous system to have a non-zero solution, it is necessary that the number of equations of the system be less than the number of unknowns. Since the rank of a matrix of coefficients cannot be greater than the number of its rows (as well as the number of columns), then r Definition 2. The system variables located on the basis columns of the original coefficient matrix are called basic variables, and the remaining variables of the system are called free. Definition 4. Private decision inhomogeneous system AX = B is called the column vector X obtained by zero values free variables. Theorem 6. General solution of an inhomogeneous system linear equations AX = B has the form , where is a particular solution to the system of equations AX = B, and is the FSR of the homogeneous system AX = 0. A non-homogeneous system of linear equations is a system of the form: Its extended matrix. Theorem (on the general solution of inhomogeneous systems). · if , where is the number of variables of system (2), then solution (2) exists and it is unique; · if , then the general solution of system (2) has the form , where is the general solution of system (1), called general homogeneous solution, is a particular solution of system (2), called private inhomogeneous solution. A homogeneous system of linear equations is a system of the form: The zero solution of system (1) is called trivial solution. Homogeneous systems are always compatible, because there is always a trivial solution. If there is any non-zero solution to the system, then it is called non-trivial. Solutions of a homogeneous system have the property of linearity: Theorem (on the linear solution of homogeneous systems). Theorem (about the structure of the general solution). · if , where is the number of system variables, then only a trivial solution exists; · if , then there are linearly independent solutions to the system under consideration: , and its general solution has the form: , where are some constants. 2. Permutations and substitutions. Determinant of nth order. Properties of determinants. Definition of the determinant - th order. Let a square matrix of the first order be given: Definition. The product of the elements of matrix A, taken one from each row and each column, is called a member of the determinant of matrix A.3 If any two rows or two columns are interchanged in the determinant, then the determinant changes its sign to the opposite. 4If a matrix contains a zero row (column), then the determinant of this matrix is equal to zero.5 If two rows (columns) of a matrix are equal to each other, then the determinant of this matrix is equal to zero.6 If two rows (columns) of a matrix are proportional to each other, then the determinant of this matrix is equal to zero.7 The determinant of a triangular matrix is equal to the product of the elements on the main diagonal.8 If all elements k the th row (column) of the determinant are presented as sums a k j + b k j, then the determinant can be represented as the sum of the corresponding determinants.9 The determinant will not change if the corresponding elements of another row (or the corresponding column) are added to the elements of any of its rows (or the corresponding column), multiplied by the same number.10. Let A And B are square matrices of the same order. Then the determinant of the product of matrices is equal to the product of determinants: Cycle – such a sequence of cells in the transport table (i 1 ,j 1), (i 1 ,j 2), (i 2 ,j 2),…(i k ,j 1), in which two and only two adjacent cells are located in one row or column, with the first and last cells also being in the same row or column. (?)Permutation along the cycle - (shift along the cycle by value t)- an increase in volumes in all odd cells of the cycle marked with a “+” sign by t and a decrease in transportation volumes in all even cells marked with a “-” sign by t. The optimal transportation plan corresponds to the minimum of the linear objective function f(X)= min under restrictions on consumption and supply An equation of the form F(n; x n; x n +1 ;…; x n + k) = 0, where k is a fixed number and n is an arbitrary natural number, x n ; x n +1 ;…; x n + k are terms of some unknown number sequence, called a difference equation of order k. Solving a difference equation means finding all sequences (x n) satisfying the equation. The general solution of a kth order equation is its solution x n = φ(n, C 1 , C 2 , …, C k ), depending on k independent arbitrary constants C 1 , C 2 , …, C k . The number of k constants is equal to the order of the difference equation, and independence means that none of the constants can be expressed in terms of the others. Consider a linear difference equation of order k with constant coefficients: a k x n+k + a k-1 x n+k-1 + … + a 1 x n+1 + a 0 x n = f n , where a i R (a k ≠ 0, a 0 ≠ 0) and (f n ) – given numbers and sequence. ^
Theorem on the general solution of an inhomogeneous equation.
The general solution x n of a linear inhomogeneous difference equation is the sum of the particular solution x n * of this equation and the general solution n of the corresponding homogeneous equation. ^
Theorem on the general solution of a homogeneous equation.
Let x n 1 ,…, x n k be a system consisting of k linearly independent solutions of a linear homogeneous difference equation. Then the general solution of this equation is given by the formula: x n = C 1 x n 1 + … + C k x n k. Knowledge of roots characteristic equation allows you to construct a general solution to a homogeneous difference equation. Let's consider this using the example of a second-order equation: The resulting solutions can be easily transferred to the case of higher-order equations. Depending on the values of the discriminant D=b 2 -4ac of the characteristic equation, the following cases are possible: The set of solutions to a linear homogeneous difference equation of the kth order forms a k-dimensional linear space, and any set of k linearly independent solutions (called the fundamental set) is its basis. A sign of linear independence of solutions of a homogeneous equation is that the Casoratti determinant is not equal to zero: The equation is called the characteristic equation of a homogeneous linear equation. ^
In what form should one look for its particular solution? Explain the answer.
X n +2 -4x n +1 +3x n =n 2 2 n +n 3 3 n In what form should one look for its particular solution? The answer must be explained. X n +2 -4x n +1 +3x n =n 2 2 n +n 3 3 n X n +2 -4x n +1 +3x n =0 X n =C 1 3 n +C 2 1 n X 1 n =(a 1 n 2 +b 1 n+C 1)2 n X 2 n =(d 2 n 3 +a 2 n 2 +b 2 n+C 2)n2 n X n = C 1 3 n + C 2 1 n + X 1 n + X 2 n x n +2 -4x n +1 +3x n =n 2 +2 n +3 n 1) x n +2 -4x n +1 +3x=0 λ 1 =3, λ 2 =1 x n o =C 1 (3) n +C 2 (1) n = C 1 (3) n +C 2 2) f(n)=2 n , g(n)=3 n , z(n)=n 2 Since the base of the exponential power f(n)=2 n, equal to 2, does not coincide with any of the roots of the characteristic equation, we look for the corresponding particular solution in the form Y n =C(2) n. Since the base of the exponential function g(n)=3 n, equal to 3, coincides with one of the roots of the characteristic equation, we look for the corresponding particular solution in the form X n =Bn(3) n. Since z(n)=n 2 is a polynomial, we will look for a particular solution in the form of a polynomial: Z n =A 1 n 2 +A 2 n+A 3 . x n +2 +2x n +1 +4x n =cos +3 n +n 2 1) x n+2 +2x n+1 +4x n =0 λ 1 =-1+i, λ 2 =-1-i Since the base of the exponential power f(n)=3 n, equal to 3, does not coincide with any of the roots of the characteristic equation, we look for the corresponding particular solution in the form Y n =B(3) n. Since g(n)=n 2 is a polynomial, we will look for a particular solution in the form of a polynomial: X n =A 1 n 2 +A 2 n+A 3. Z n = Ccos x n +2 +2x n +1 +4x n = cos +3 n +n 2 λ 1 =-1+i, λ 2 =-1-i X n 0 =(2) n (C 1 cos +C 2 sin ) 2) f(n)=3 n , g(n)=n 2 , z(n)=cos Since the base of the exponential power f(n)=3 n, equal to 3, does not coincide with any of the roots of the characteristic equation, we look for the corresponding particular solution in the form Y n =B(3) n. Since g(n)=n 2 is a polynomial, we will look for a particular solution in the form of a polynomial: X n =A 1 n 2 +A 2 n+A 3. Z n = Cncos The Samuelson-Hicks business cycle model assumes direct proportionality of investment volumes to the increase in national income (acceleration principle), i.e. where coefficient V>0 is the acceleration factor, I t - the amount of investment in period t, X t -1 ,X t -2 - the value of national income in periods (t-1) and (t-2), respectively. It is also assumed that demand at this stage where a, b are the coefficients of the linear expression of demand at this stage: Stationary sequence 40. Formulate the problem of determining the current value of a coupon bond. What is the Cauchy problem for a difference equation? Find an equilibrium solution to the Cauchy problem of determining the current value of a coupon bond. Check that the value found matches the amount that must be paid at the moment in order to receive the coupon amount in each coupon period for an infinitely long time at a given interest rate for one coupon period. Let F
– the par value of a coupon bond (i.e. the amount of money paid by the issuer at the time of redemption coinciding with the end of the last coupon period), K
– coupon value (i.e. the amount of money paid at the end of each coupon period), X Those. p Where C 1 and C 2 are unknown. All y are known numbers, calculated at x = x 0. For the system to have a solution for any right-hand side, it is necessary and sufficient that the main determinant be different from 0. If , then the Wronski determinant is not equal to 0, for any values of x 0. Proof. Let the determinant be equal to 0, but let us choose the initial non-zero conditions y=0, y’=0. Then we get the following system: This system has an infinite number of solutions when the determinant is 0. C 11 and C 12 are solutions to the system. This contradicts the first case, which means that the Wronski determinant is not equal to 0 for any x 0 if . It is always possible to select a particular solution from the general solution for . Ticket No. 33 A theorem on the structure of the general solution of a linear homogeneous differential equation of the 2nd order with proof. Theorem on the general solution of a differential equation: solutions to this equation, then the function Proof: Ticket No. 34 A theorem on the structure of the general solution of a linear inhomogeneous differential equation of the 2nd order with proof. Let an equation with the right side be given: . Equation without right side A theorem on the structure of the general solution of an equation with the right-hand side. T.1 The general solution of an equation with the right-hand side can be composed as the sum of the general solution of the equation without the right-hand side and some particular solution of this equation. Proof. Let us denote by the general solution and some particular solution of this equation. Let's take the function Substituting the expressions for y, y’, y’’ into the left side of the equation, we find: The expression in the first square bracket is equal to 0. And the expression in the second bracket is equal to the function f(x). Therefore, the function Ticket No. 35 Linear homogeneous differential equations of the 2nd order with constant coefficients, F.S.R. and general solution in the case of different real roots, characteristic equations with proof. Let us take a homogeneous linear equation of the second order with constant coefficients: where a are numbers. Let's try to satisfy the equation with a function of the form . From here we have: From this we can see what the solution to this equation will be if r is the root of the quadratic equation. This equation is called characteristic. To create a characteristic equation, you need to replace y by one, and each derivative by r to a power of the order of the derivative. 1) The roots of the characteristic equation are real and different. In this case, both roots can be taken as indicators of the r function. Here you can immediately get two equations. It is clear that their ratio is not equal to a constant value. The general solution in the case of real and different roots is given by the formula: Ticket No. 36 Linear homogeneous differential equations of the 2nd order with constant coefficients, F.S.R. and general solution in the case of multiple roots, characteristic equations with proof. The roots of a real equation are real and equal.
Let (i.e. system (2) be consistent), then:
Let be the solutions of the homogeneous system (1), and let be arbitrary constants. Then is also a solution to the system under consideration.
Let then:
1
| | | | | | | | | | |
Free cell assessment– (see potential method)
The optimal plan should determine the minimum total cost of transportation, without exceeding the production volume of each of the suppliers and fully covering the needs of each of the consumers.
^
Condition for the optimality of the reference plan.
No. 32. Formulate the definition of a difference equation of order k and its general solution. State the definition of a linear difference equation of order k with constant coefficients. Formulate theorems on the general solution of homogeneous and inhomogeneous linear difference equations (without proof).
No. 33. Describe an algorithm for solving a homogeneous linear difference equation with constant coefficients. Formulate definitions of the following concepts: fundamental set of solutions of a linear difference equation, characteristic equation, Casoratti determinant.
C 1 , C 2 are arbitrary constants.
№ 34. Given a linear difference equation with constant coefficients X n +2 – 4x n +1 + 3x n = n 2 2 n + n 3 3 n.
No. 35. Given a linear difference equation with constant coefficients x n +2 -4x n +1 +3x n =n 2 +2 n +3 n. In what form should one look for its particular solution?
No. 36. Given a linear difference equation with constant coefficients x n +2 +2x n +1 +4x n =cos+3 n +n 2 . In what form should one look for its particular solution?
No. 37. Given a linear difference equation with constant coefficients x n +2 +2x n +1 +4x n = cos+3 n +n 2 . In what form should one look for its particular solution?
#38: Describe the Samuelson-Hicks model. What economic assumptions underlie it? In what case is the solution to the Hicks equation a stationary sequence?
depends on the amount of national income at the previous stage 
linearly 
. The condition for equality of supply and demand has the form 
. Then we come to the Hicks equation
is a solution to the Hicks equation only for 
; factor 
is called the Keynes multiplier (a one-dimensional analogue of the total cost matrix).
№ ^
39. Describe the spider market model. What economic assumptions underlie it? Find the equilibrium state of the web market model.
- current value of the bond at the end of the nth coupon period, 
coincides with the amount that must be paid at the moment in order to receive the coupon amount in each coupon period for an infinitely long time at a given interest rate for one coupon period.
Vronsky's determinant. If the determinant is 0, then the system has a solution only if there is a proportion of the initial conditions. Therefore, it follows from this that the choice of initial conditions is subject to the law, so that any initial conditions cannot be taken, and this is a violation of the conditions of the Cauchy problem.
also a solution. Based on this theorem, we can conclude about the structure of the general solution of a homogeneous equation: if 1 and 2 have solutions to the differential equation such that their ratios are not equal to a constant, then the linear combination of these functions is the general solution to the differential equation. A trivial solution (or a null one) cannot serve as a solution to this equation.
if we put 0 instead of a function, we call it characteristic.
. We have
, 
.
there is a solution to this equation.,
.
