Fundamental system of solutions.

Definition. System of n linearly independent solutions ₁(t), ₂(t),… _n(t)

of system = A(t) (1) is a fundamental system of solutions or basis.

Theorem. The system (1) has a fundamental system of solutions. If

₁(t), ₂(t),… _n(t) is basis, then the general solution has the form

(t)= _{i i}(t) where c 1 ,c 2, … ,cn are arbitrary constants.

Concept of the fundamental matrix. Ostrogradsky-Liouville formula.

We consider the system

y 1(t) ,y 2 (t),.. ,yn (t) (2)

Y(t) =

and the Wronskian. If (2) is linearly independent, then detY(t) =W(t)

Y (t) is called an integral or a fundamental matrix for the system (1).

If Y (t 0) = E, the matrix is called integral normalized at the point t = t 0.

Theorem. If Y (t) is integral matrix of vector equation (1), we have detY(t)=detY(t₀)*exp (2)

the Ostrogradsky-Liouville formula. Linear transformations

= B(t) (3)

Theorem. The general solution of the inhomogeneous vector equation = A(t) + (t) (4)is equal to the total solution (5)of the homogeneous equation (1) and a particular solution of the inhomogeneous vector equation (4)

Y_h=Y_p+Y_inh (5)

 

17) Fundamental matrix. Liouville formula A square matrix Φ(t) whose columns are formed by linearly independent solutions x ₁(t), x ₂(t),..., x_n (t) is called the fundamental matrix of the system of equations. It has the following form:

where x_ij (t) are the coordinates of the linearly independent vector solutions x ₁(t), x ₂(t),..., x_n (t).

Φ(t) is nonsingular, i.e. there always exists the inverse matrix Φ ⁻¹(t). Since the fundamental matrix has n linearly independent solutions, after its substitution into the homogeneous system we obtain the identity

Ф’(t) A(t)Ф(t)

Ф’(t) A(t)Ф(t) Ф’(t)

The general solution of the homogeneous system is expressed in terms of the fundamental matrix in the form

(t)= Ф(t)C where C is an n -dimensional vector consisting of arbitrary numbers. The fundamental matrix Φ(t) for such a system of equations is given by

Ф(t) Liouville formula. Consider the n -dimensional first-order homogeneous linear differential equation

on an interval I of the real line, where A (x) for x ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning that each Φ(x) is a square matrix of dimension n with real or complex entries and the derivative satisfies

Ф’(t) A(t)Ф(t) x ∈ I

Let trA(ξ)= ξ ∈ I

A(ξ) = (a_i, j(ξ))_{i, j ∈ {1,...,n}}, the sum of its diagonal entries. If the trace of A is a continuous function, then the determinant of Φ satisfies

detФ(x)

for all x and x ₀ in I.

 

 

Integration of linear inhomogeneous system with quasi-polynomial right side.

(x)=A (x)+ (x), (L[ ]= (x)), x?(a,b)

Theorem: Condition: The vector-function (x) has the form (x) +…+ (x) and vector-function (x) +…+ (x) is the solution of followin inhomogeneous systems

(x) =A(x) + (x), … (x) =A(x) + (x),

Statement: vector- function (x) +…+ (x) is the solution of inhomogeneous systems

(x) =A(x) + (x)=A(x) (x) …+ (x)

Proof: adding the equality

(x) =A(x) + (x), … (x) =A(x) + (x),

We get that (x) +…+ (x) is the solution of inhomogeneous systems (x) =A(x) + (x)

Def:Vector-function (x) called quasi-polinom if it has the form

(x)= (x)* , where (x)- quasi-polinom

Statement: If (x)= (x)* + (x)* (1) then there (x) and (x) polinoms of r=k degree such that Vector-function (x)= (x)* + (x)* is particular solution of inhomogeneous linear system.

Proof: right size of (1) we can submit (x)= (x)* + (x)*

Particular sol. of system (x)= (x) * and (x)= (x) *

(x), (x) quasi-polinoms

Solution of sourse function is (x)+ (x) =

(x) * + (x) * =)= (x)* + (x)*

19) Studying of different cases (resonance and in resonance cases) Theorem 1 (Non-resonance). Assume that 'ϕ(λ) has no roots on the imaginaryaxis, g 2 BC and p 2 C. Then equation has at least one bounded solution if and only if p? C0 + BC: To state the result in the case of resonance we follow and define the notion of upper and lower average. Given p?C0 + BC,

(p):=

(p):=

It is easy to verify that

−1 < AL (p) <= AU (p) < +1:

Moreover, if p is periodic the identity AL (p) = AU (p) = p holds and the concept of average is recovered.

Theorem 2 (Resonance). Assume that λ = 0 is a simple root of ϕ(λ) and there are no other roots on the imaginary axis. In addition, g? BC and p? C. Then a sufficient condition for the existence of a bounded solution

p? C 0 + BC; (−∞) < (p) <= (p) < g (+∞):

As mentioned in the introduction, this theorem extends results.

+ c + g (y) = p (t) (c > 0) ;

and it was proved that if g? BC there exists a bounded solution if and only if p= p*+p** with p *? C 0 g (−∞) < inf p** <= sup p **<g(+∞)