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(αx)(M_m(x)cos(βx) + N_n(x)sin(βx)). 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(αx)(P_k(x)cos(βx) + Q_k(x)sin(βx)),

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(αx)(P_k(x)cos(βx) + Q_k(x)sin(βx))x^r,

41) Cauchy function. Fundamental solutions.

Def1:   Linear inhomogeneous system of differential equations is called the system of the form:

(t)=A(t)x(t)+f(t) t€I=(t₁,t₂)

A(t)= f(t)= x(t)=

Suppose, that all the elements are continuous on the I. Def2:  Fundamental system of solutions of inhomogeneous system is called n arbitrary linear independent solutions of this equation. If is fundamental system of solutions, then the matrix Ф(t)= is called the fundamental matrix of the system.

The fundamental matrix satisfies to the equation =A Ф . General solution of homogeneous system can be written as x(t)= Ф(t)C, where – some permanent vector. The general form of solutions of homogeneous system with constant coefficients x(t)= ,(здесь P_k, _k) where - eigenvalues of matrix A, with corresponding multiplicity m₁,…m_l. P_k(t) – vector quasi-polynomials with degree, that is not more than m_k-1.

In the case of linear equation, we can show , that x(t)=t inhomog. (t)=x homog. (t)+x­^-(t)

Partial solution of inhomogeneous system x­^-(t) we will find in the form x­^-(t) = Ф(t)C(t) (method of variation of constants). Substituting in inhomogeneous system, we get: C(=AФ) + Ф =AФC+f = Ф =f => = Ф ^-1f

Hence, C(t)=

We don` t care about constant, because we are finding arbitrary solution.By substituting C(t)in the expression for :

= Ф(t) , Then x(t)= Ф(t)C+Ф(t)

Suppose, x(t₀)=x⁰. Substitute t₀ to the expression for x(t) : x(t₀)= Ф(t₀)C, hence C= Ф^-1(t₀)x⁰

Suppose, K(t,s)= Ф(t)Ф^-1(s) – matrix of 2 variables. Then the general solution has the form: x(t)=K(t,t₀)x⁰+ . Judging by the introduced matrix K, this is the Cauchy formula.If the fundamental matrix is normalized in t_0, then x(t)= Ф(t) x⁰+Ф(t) If matrix A is permanent, then Ф(t) = and That is why

x(t)= x⁰+Ф(t)