16) 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 c1 ,c2 ,…,cn are arbitrary constants.

Concept of the fundamental matrix. Ostrogradsky-Liouville formula.

We consider the system

y1(t),y2 (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(t0 ) = E, the matrix is called integral normalized at the point t = t0 .

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 x₁(t), x₂(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 x₁(t), x₂(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 x₀ in I.