29. Linear nonhomogeneous systems of differential equations with constant coefficients.

(1)

,x f(x)=P_k (x)- e^αx

n=1 y’+p₁y=f(x)

z-solution of homogeneous DE.

y=C(x)z y’=C’(x)z+C(x)z’

C’(x)z+ C(x)z’+ P₁C(x)z=f(x)

C’(x)z+C(x) [z’+p;z]=f(x)

C’(x)z=f(x)

C’= C=

n=2 y’’+p₂y’+p₂y=f(x) z₁z₂-solutions of HDE.

P₁*( y=C₁z₁ +C₂z₂ y=C₁(x)z₁+C₂(x)z₂ )

P₂*(y’=C₁z₁+C₂z₂ y=C₁(x)z₁+C₂(x)z₂₎

y’= Requirement

y’’=

C₁(z’’+P₁ +P₁z₁)+C₂(z’’+P₁ +P₂ )=f(x)

z₁…z_n- fundamental system of solutions

y=C₁z₁+…+C_nz_n

R₁

P_j =0

R_n-1

W=

30. Method of undetermined coefficients.

Ly= (1)

f(x)=P_k(x)e^αx

Ly=f₁+f₂

Ly₁=f₁, Ly₂=f₂

y₁+y₂y=y

1. - isn’t root of CE

2. α- root of CE. S-multiple

3. 

F= P_k(x)e^αx

Method of variation of constants.

X=C₁X₁V₁+…+C_nX_nV_n C=c(t)

31. Theorems of a continuity and differentiability of the solution as functions of parameters and initial data. Concept of stability of the solution in Lyapunov’s sense.

32. The theorem of continuous dependence of the solution of normal system from parameters.

Definition(continuous) y=f(x)

|x- |< and |f(x)-f( )|<

Ɐ >0

= (t,x) (1)

33) DE of second order. Reduction to simple forms Integration with series.

34) Oscillatory character of solution of the homogeneous linear equations of second order.

Using replacement

ODE (1) could be reduced to simple form

Example:

=

Case 1:

Case 2:

In Case 1 solution of DE has oscillatory character.

Definition. The solution of DE (3) is called oscillating on (a,b) if it has at least two zeros in the interior of (a,b)

q(x)=const

q(x) q(x)

(1)=> +

Approximate solution y=

1. We put (4)→(1)

2. Collecting coefficients of get new equation for