Linear systems of differential equations. Basic systems 


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Linear systems of differential equations. Basic systems



Existence and uniqueness theorem. System of the form

(1)

where aij , fi are given functions, xi is unknown functions called a linear system of differential equations.

Eq (1) also can be represented as =A(t) (2)

Systems of linear equations. General theory.

System (2) is called homogeneous, if f ≡¯0, otherwise inhomogeneous.

We consider the linear homogeneous system

(3) where A(t) is a continuous matrix on I of dimension n × n.

We introduce the operator L = − on the set of differentiable columns. We prove its linearity.

For this we consider a differentiable vector

y(t)= , where α1,...,α n are constants.

It is easy to see which proves linearity of operator L.

Theorem 1. Linear combination of solutions of the homogeneous linear system with arbitrary constant coefficients is also a solution of (3).

Remark. From the existence and uniqueness follows, that the unique solution to the problem

, (t0)= a<t0<b is = (the trivial solution).

Theorem 2. If = + is a complex solution of the system (3), then 1 = , 2 = is the real solutions of the system (3).

Definition 1. Vector system of the solutions 1, 2,..., m of (3) is linearly dependent on the interval (a, b), if there are constants α1,...,α m not all zero at the same time, that , a<t<b. Otherwise, the system solutions of (3) is linearly independent on (a, b).

Theorem 3. If for any t 0 ∈(a, b) the system of initial vectors 1(t 0), 2(t 0),..., m (t 0) are linearly dependent, then the corresponding solutions i, i =1, m are also linearly dependent on (a,b).

Let 1, 2,..., n is the solution of (3). Determinant of their components

W[ 1, 2,..., ] = called the Wronskian.

Theorem 4. In order to ¯ y 1, ¯ y 2 ,…, ¯ yn would be linearly independent solutions necessarily and sufficient that W ≠ 0, t ∈(a,b).

Definition 2. System of n linearly independent solutions ¯ y 1, ¯ y 2 ,…, ¯ yn of system (3) is called a fundamental system of solutions or basis.

Theorem 5. The system (3) has a fundamental system of solutions. If ¯ y 1, ¯ y 2 ,…, ¯ yn is the basis, that the general solution has the form (t)= , where c 1 ,c 2 ,…,,cn are arbitrary constants.

 

44. Some methods of the system integration (leading to one equation etc)

We know some methods of the system integration: Euler method of integration of homogeneous linear differential equations with constant coefficients, Method of variation of constants, Method of undetermined coefficients etc. We will discuss in detail these 3 more basic methods.We consider a linear system of differential equations:

(1)

where fi (t)-continuous functions in an interval;the coefficients aij (i, j = 1 ,2,…,n)- constants. The easiest way to integrate a system by reducing it to one equation of higher order, and this would also be a linear equation with constant coefficients. We write the system in matrix form:

(2)


The general solution of (2) has the structure: X=Xg.s+Xp.s, where Xg.sthe general solution of the homogeneous system

(If the vector f (t) is identically equal to zero: f(t)=0, then the system is said to be homogeneous)

We consider Euler method of integration of homogeneous linear differentialequations with constant coefficients.

According to this method, we find the solution of (4) as:

X = Be λt, (5)

Where B= is unknown column vector, λis unknown number.

Substituting (5) into (4), we obtain the matrix equation:AB=λB, (B ) (6)

λ (λ ≠ 0) is the eigenvalue of the matrix A; vector B is eigenvector corresponding to λ, is found from det (A −λ E) = 0, (7))

where E is the identity matrix n × n.

For a given eigenvalue λcomponents corresponding eigenvector B is found of the system of linear homogeneous equations:

Under solving of the system the following cases is possible:

1. All the roots of (7) are real and different.

2. The roots of (7) are real, but some of them are multiple.

3. The characteristic equation (7) has a complex variety of roots

According to the theory, in order to make a valid decision, it is enough to take the real and imaginary parts of one of the solutions: X 1= ReY 1, X 2= ImY 1.

Constructed in this way, the system solutions, the so-called fundamental system of solutions would consist of real functions.

Among the roots of the characteristic equation there are multiple complex roots. In this case the solution is to be found by analogy with the case of 2. Then separating the real and imaginary parts, we obtain so-called fundamental system of

(linearly independent) solutions of real functions.

Let us, finally, to find a particular solution Xp.s of (1) has the right part of a special type, if the vector F is:

F = (P (t) cosbt + Q (t) sinbt) eat, where P (t) ,Q (t) - vector polynomials of the form m H0 + H1 t + H2 t +…+ Hmtm

(not necessarily the same order) with vector coefficients H 0 ,H 1 ,H 2, … ,Hm. For linear systems with right-special form a particular solution can be found in the same form as the right-hand side, but with unknown coefficients. These coefficients are determined by substituting the source solution to the equation and equating like terms in the left and right sides of the matrix equation. The above method is known as the method of undetermined coefficients.



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