42. Systems of linear differential equations with constant coefficients

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=X_g.s+X_p.s, where X_g.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 −λE) = 0, (7))

whereE 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 takethe real and imaginary parts of one of the solutions:X₁= ReY₁, X₂= ImY₁.

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

Among the roots of the characteristic equation there are multiple complex roots. Inthis 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.sof (1) has the right part of a special type,if the vector F is:

F = (P(t)cosbt+Q(t)sinbt)e^at ,where P(t),Q(t) - vector polynomials of the form mH₀+ H₁ t + H₂ t +…+ H_mt^m

(not necessarily the same order) with vector coefficients H₀,H₁,H₂,…,H_m. For linearsystems with right-special form a particular solution can be found in the same form as theright-hand side, but with unknown coefficients. These coefficients are determined bysubstituting the source solution to the equation and equating like terms in the left and rightsides of the matrix equation. The above method is known as the method of undeterminedcoefficients.

43. Linear systems of differential equations. Basic systems

Existence and uniqueness theorem. System of the form

(1)

where a_ij , fi are given functions, x_i 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 t0 ∈(a,b) the system of initial vectors 1(t0 ), 2(t0 ),..., m(t0 ) 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 ¯y1, ¯ y2 ,…, ¯yn would be linearly independent solutions necessarily and sufficient that W ≠ 0, t∈(a,b) .

Definition 2. System of n linearly independent solutions ¯y1, ¯ y2 ,…, ¯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 ¯y1, ¯ y2 ,…, ¯yn is the basis, that the general solution has the form (t)= , where c1 ,c2 ,…,,cn are arbitrary constants.