Matritsali differensial tenglamalarni integrallash. Koshi integral formulasi, eksponensial matrisa

“MATRITSALI DIFFERENSIAL TENGLAMALARNI INTEGRALLASH.KOSHI INTEGRAL FORMULASI, EKSPONENSIAL MATRISA”


REJA:

1.Differensial tenglamalar sistemasini vektorli ko’rinishi.

2.Xarakteristik tenglamaning ko’rinishi.
3.Matirisali tenglamaning yechimlarining xossalari.

“ Matritsali differensial tenglamalarni integrallash.Koshi integral formulasi Eksponensial matrisa” mavzusi bo‘yicha tarqatma material

O’zgarmas koeffisiyentli chiziqli differensial tenglamalar sistemasi

^dyi  ⁿ a_ij y _j (1) (i 1, n) dx j1
berilgan bo’lsin.
Ma’lumki (1) sistemani vektorli
dy
 Ay (2) dx
ko’rinishda xam yozish mumkin. Bunda

a11  A  a21  an1 

a12 a22  an2

   

a1n   a2n  ,    ann 

 y1     y2  y        yn  

birustunlimatrisayoki o’ (2) vektorlitenglamauchunKoshimasalasi

y(x0 )  y ^0,

 y10   y 0     y20  colon(y10 , y20 ,...,yn0 )

 ^ 
 y_n0 _
(2)tenglamani yechimini
yBe^x (3)
ko’rinishda izlaymiz. Bunda B, n1 tartibli matrisa
₁ 
 
B  2 
n 
(3) ni (2) ga keltirib qo’ysak

Be^x  ABe^x

yoki
(AE)B 0 (4)
tenglama ega bo’lamiz. Bunda E birlik matrisa

 1  E 0   0 

0 1  0

   

0  0,   1

0
 
^{trivial bo’lmagan} B ^⁰^ matrisa (4) tenglamani qanoatlantirishi uchun
 0

(AE) (5)
matirisaning maxsus bo’lishi zarur va yetarlidir. Ya’ni uning determinanti det(A E)  A E  0 (6).
(6) ga (2) sistemaga mos bo’lgan xarkteristik tenglama deyiladi.
soniga A matrisaning xos qiymati, V vektor esa λ ga mos bo’lgan xos vektor deyiladi.
(6) xarkteristik tenglamaning xar bir λ_k ildizi uchun (4) tenglamadan nolga teng bo’lmagan ₁(k ) _
B  2(k ) 
 
n(k ) 
Matrisani aniqlaymiz.
(2) vektorli tenglamaning ixtiyoriy ta chiziqli bog’liq bo’lmagan

Y₁(x),Y₂(x),...,Y_n (x)

vektorli yechimlarga (2) tenglamaning fundamental yechimlar sistemasi deyiladi. Bunda quyidagi xollar bo’lishi mumkin.
1xol
Xrakteristik tenglamaning ildizlari haqiqiyva bir-biriga teng emas. U xolda (2) tenglama n-ta yechimlarga ega bo’lib ularni

Y_k  B^(k)e^kx (k 1, n) (7)
ko’rinishda yozish mumkin. Isbot etish mumkinkim bular (2) tenglamaning fundamental yechim sistemasini tashkil etadi.U xolda (2) tenglamaning umumiy yechimi n n
y c_kB ^(k)e^kx c_k y_k (8)
k1 k1
dan iborat bo’ladi.

(AE)B 0 6  1  0 3 2  (A1E)B(1)

AE  0 1^2815 0 1 3 33 1112((11))  0 

2 5

Misol-1 y  Ay A  ^_36 2^1^_
3^₁(1) ^₂(1)  0 3^₁(1) ^₂(1) ^₁(1) 1 ^₂(1)  3

3 1 (A 1E)B(2)  0 3  ^1(2) ^2(2) ^1(2) 1 B(2)  11 y2  e5t 11

3 1312((22))   0 ^2(2) 1

B(1)  ^_1^_ y₁  B(1)e3t  e3t ^_1^_

1 1

y  c1y1  c2 y2  c1e3t 3  c2e5t 1


2 xolxarakteristiktenglama_k  p  qi kompleksildizgaegabo’lsinBuxolda (2) tenglamaningyechimibuildizgamosbo’lganyechimi
^y_k  B(k)e(pqi)x  B(k)e pxeiqx  B(k)e px (cosqx isin qx)
B^(k) kompleks son bo’lgani uchun uni
11(k)  21(k) 
Bk  ^B₁(k)  ^iB₂(k) ^₁₂(k)  i^₂₂(k) 
   
1(nk)  2(kn) 

_{ko’rinishda yozish mumkin} (A B)C  AC  BC ga asosan
^11(k) _ _^21(k) __

~yk  epx ....12(k)   i..22(k) (cosqx  isin qx) 
1(nk)  2(kn) 
11(k) cosqx 21(k) sin qx  i21(k) cosqx 11(k) sin qx
epx12(k) cosqx 22(k) sin qx  i22(k) cosqx 12(k) sin qx

................................................................ 
(k) cosqx 2(kn) sin qx  i2(kn) cosqx 1(nk) sin qx
 1n
(1) (2)
11(k) cosqx 21(k) sin qx 11(k) sin qx 21(k) cosqx
 y_1k  e ^px  ........................... ; y_2k  e ^px  ........................... 
1(nk) cosqx 2(kn) sin qx 1(nk) sin qx 2(kn) cosqx

yc1y1k c2 y2k c1epx(1) c2epx(2)
Misol

dy  Ay dx A EB  0 3   2  0

A  34 12  A E  0 y  y12  y ^2 2 5  0 1,2 1 2i

4 1
A 1EB  0 _3 14 2i _1_12_ 2i____₁₂ _  0
_
2  2i₁  2₂  0 1 i₁ ₂ ₁ 1 ₂ 1 i
~y  B(1)e12ix  e x B(1)e2ix  e x 11 icos2x  isin 2x 
 ^{ e} _x ^cos2x  sincos22xxiisinsin22xx cos2x_

_

yc1excos2cosx2sinx 2xc2exsin 2sinx2cosx 2x

y₁ c₁e^x cos2xc₂e^x sin 2x y₂ c₁e^x(cos2x sin 2x) c₂e^x(sin2x cos2x)

3 xol.
Agar xarakteristik tenglama r-karrali λ_s ildizga ega bo’lsa, u xolda, bu ildizga mos bo’lgan (2) tenglamaning yechimi
y(x) B1(s) B2(s)xB3(s)x2 ...Br(s)xr1esx
dan iborat buladi.
Misol-2
dy 1  2
 Ay A  2  3^_
dx _
A EB  0 A E  0
1  2(1)2  0 1,2  1 2  3 
y  BA¹₁ ^  ^BA²₂ ^x^e^x  0
buni berilgan tenglamaga qo’yamiz
 A

• 
   B₁₁    BA₂₂ xe_x   BA₂₂ e_x   1223 BA₁₁    BA₂₂ xe_x
  


bundan
 A₁  A₂x  A₂  A₁ 2B₁  A₁x 2B₂^x
_ B₁  B₂x  B₂ 2A_1` 3B₁  2A₂x 3B₂x
A₁ , A₂ ixtiyoriy

• 
  А₁  А₂  А₁  2B₁ 2B₁  2А₁  А₂
  
• 
  А₂  А₂  2B₂ 2B₂  2A₂ B₁  B₂  2А₁ 3B₁
  

A²
B₂  2А₂ 3B₂ B₁  A₁ 
2
B2  A2
 A
y   A1 1 A22    AA22 xex   AA11    A22 A2 xA2 xex 

 c1e_x 11  c2e_x 2x2x1
A2 c2
A1  c1 
2

Endi (2) tenglamaning ta chiziqli bog’liq bo’lmagan yechimlaridan n x n y(x) matrisani tuzamiz.
 y₁(x)   y₁(x) y₁₂(x).....y_1n (x) 
   
Y(x)  y₂ (x) y₂₁(x) y₂₂(x).....y_2n (x)
 yn (x)  yn1(x) yn2 (x).....ynn (x)
u xolda
dY
 A(x)Y (9) dx
ga matrisali tenglama deyiladi.

detY(x) W(x)ga

Vronskiy determinanti deyiladi. Agar U(x) matrisa, (9) matrisiali tenglamani qanoatlantirsa, unga (9) tenglamaning integrali yoki fundamental matrisasi deyiladi. (matrisali yechim) Bundan ko’rinadikim chiziqli differensiali tenglamalar sistemasini
dyi  n Pij (x)y j ni dy A(x)y vektorli ravishda yoki dY A(x)Y dx _j1 dx dx
matrisali ravishda yozish mumkin. Bu tenglamalr orasidagi boglanish shundan iboratki nxn Y(x) matrisali yechimning ustunlari (2) tenglamaning uzaro chiziqli bog’liq bo’lmagan vektorli yechimlarni tashkil etadi.
Agar A(x) matrisa funksiya, o’zgarmas matrisa bo’lsa.
a11 a12.....a1n 
 

А a₂₁ a₂₂.....a_2n  an1 an2.....ann

o’zgarmas koeffisiyentli matrisali
dY
 AY dx
tenglamaning yechimini

Y  Be^x

ko’rinishda izlaymiz bunda B n 1 tartibli matrisa
₁ 
 
B₂ 
n 
Agar o’zgarmas matrisa uchun
Ah h
tenglik bajarilsa, u xolda  son A matrisaning xos soni (xos qiymati), h vektorga esa  ga mos bo’lgan xos vektor deyiladi.
TEOREMA.Y(x) matrisa (9) tenglamaning fundamental matrisasi bo’lishi x(,) oraliqdagi qiymatlar uchun detY(x) W(x)  0
shartining bajarilishi zarur va yetarlidir.
TEOREMA 2.Agar ^y₁ (x) matrisa (9) tenglamaning biror intervalda aniqlangan matrisali yechimi bo’lsa u xolda ^y₁(x)c xam bu tenglamaning yechimi buladi.
Ya’ni d(Y1c)  A(t) (Y,c) dx
S, nx1 tartibli ixtiyoriy o’zgarmas matrisa xakikatan xam
^dY1  A(x)Y₁ (10) dx
tenglamaning ikki tomonini ungdan C matrisaga kupaytiramiz.
^dY1 C  A(x)Y₁C
dx
\C o’zgarmas matrisa bo’lgani uchun d(Y1C)  A(x)(Y1C) dx ya’ni Y₁C (9) tenglamani yechimi buladi.