Fundamental system of solutions

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

of system is called a fundamental system of solutions or basis.

Theorem. The system has a fundamental system of solutions. If

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

where are arbitrary constants.

Concept of the fundamental matrix. Ostrogradsky-Liouville formula.

We consider the system

of n arbitrary vector solutions of the vector equation and form a matrix of the order n x n

Y(t) =

and the Wronskian. If the system of vectors is linearly independent, then detY (t)= W (t)does not vanish for any value t of the interval of continuity of the matrix A (t). In this case the matrix Y (t) is called an integral or a fundamental matrix for the system . If Y ()= E, where E is the unit

40 Integration of linear inhomogeneous equation with quasipolinomial right side.

The particular solution y * (x) can be solved by trial, if the right side of the equation - quasipolynomial - function of the form

f (x) = exp(α x)(M_m (x)cos(β x) + N_n (x)sin(β x)). Here M_m (x) - a polynomial of degree m, N_n (x) - a polynomial of degree n, α and β - are real numbers.

Calculation method of selection of a particular solution of the inhomogeneous linear equations with quasipolynomial in the right side is the following. Carefully look at the right side of the equation and write down the number of α ± βi.

Then the characteristic equation of the homogeneous equation and find its roots. There are two cases: the roots of the characteristic polynomial is no root, equal to the number of α ± βi (nonresonantce case), and among the roots of the characteristic polynomial is r roots equal to the number of α ± βi (resonanCE case).

Consider the non-resonant case (among the roots of the characteristic polynomial is no root, equal to the number of α ± βi). Then the particular solution will search in the form

y *(x) = exp(α x)(P_k (x)cos(β x) + Q_k (x)sin(β x)),

We consider the resonance case (among the roots of the characteristic polynomial is r roots equal to the number of α ± βi). Then the particular solution will be sought in the form

y *(x) = exp(α x)(P_k (x)cos(β x) + Q_k (x)sin(β x)) x^r,

Wronskian determinant. Liouville formula.

 

Wronskian determinant

Given functions , then the Wronskian determinant is the determinant of the square matix

 

 

where f ^(k) indicates the k th derivative of f (not exponentiation).

The Wronskian of a set of functions is another function, which is zero over any interval where is linearly dependent. Just as a set of vectors is said to be linearly dependent when there exists a non-trivial linear relation between them, a set of functions is also said to be dependent over an interval when there exists a non-trivial linear relation between them, i.e.,

for some , not all zero, at any

Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solution of a second- order differential equation are independent, we may use the Wronskian.

Consider the functions x ², x, and 1. Take the Wronskian:

Note that W is always non-zero, so these functions are independent everywhere. Consider, however, x ² and x:

 

 

Note that W is always non-zero, so these functions are independent everywhere. Consider, however, x ² and x:

 

 

Here W is always zero, so these functions are always dependent. This is intuitively obvious, of course, since

2 x ²+3=2(x ²)+3(1)

Given n linearly independent functions , we can use the Wronskian to construct a linear differential equation whose solution space is exactly the span of these functions. Namely, if g satisfies the equation;

then for some choice of

As a simple illustration of this, let us consider polynomials of at most second order. Such a polynomial is a linear combination of and . We have

Hence, the equation is which indeed has exactly polynomials of degree at most two as solutions.

 

Liouville formula.

Definition. Let are real-valued functions on , which are times differentiable on . Then their Wronskian is defined by

 

Theorem (The Liouville formula) Let be a sequence of solutions of is continuous. Then the Wronskian of this sequence satisfies the identity

for all .

Recall that the trace trace A of the matrix A is the sum of all the diagonal entries of the matrix.

Proof. Let the entries of the matrix ( then

We use the following formula for differentiation of the determinant, which follows from the full expansion of the determinant and the product rule:

Indeed, if are real-valued differentiable functions then the product rule implies by induction

Hence, when differentiating the full expansion of the determinant, each term of the determinant gives rise to n terms where one of the multiples is replaced by its derivative. Combining properly all such terms, we obtain the derivative of the determinant is the sum of n determinants where one of the rows is replaced by its derivative, that is, (1).

The fact that each vector satisfies the equation can be written in the coordinate form as follows