1. Интегралы ФНП Диф_ур (853736), страница 21
Текст из файла (страница 21)
# (24.24) L, (' /#, W(t)c { ". .(( Lz = 0, " '0101ttZZLy L(W(t)c) + L B@W(t) W;1(s)b(s)dsCA L B@W(t) W;1(s)b(s)dsCA :t0t08 '01tZL B@W(t) W;1(s)b(s)dsCA =t00(24:25)1d B@W(t) Zt W;1 (s)b(s)dsCA ;dtt0ZtZ_ t) W;1(s)b(s)ds + W(t)W;1(t)b(t);;A(t)W(t) W;1 (s)b(s)ds = W(tt0t0Zt ;1Zt ;1_;A(t)W(t) W (s)b(s)ds = b(t) + W(t) W (s)b(s)ds;t0t0Zt ;1;A(t)W(t) W (s)b(s)ds:t0310_ t) A(t)W(t), $ (24.25) 3A, W(ZtZ;1Ly = b(t) + A(t)W(t) W (s)b(s)ds ; A(t)W(t) W;1 (s)b(s)ds b(t)tt0t0# #$, #-/#0 (24.24) ( (24.23) " # c.? ' Ly = b(t) y(t0) = y { $' $ B (24.23), /#0 (24.24) ' $ y(t0) = y, W(t0)c = y, # ( c = c = W;1(t0)y:&', #-/#0 (24.24), c = W;1 (t0)y , #$( $ B.
F $, (24.24) { ". //0'( (24.23). , #$." # 24.2. $ B;A(t)y=b(t)y(t(24:26)Ly dy0) = ydt( $# Ca b] 0( A(t) ' b(t)) #-/#0Z;1y = W(t)W (t0)y + W(t) W;1 (s)b(s)dstt0(24:27) W(t) { /' 0 ( .( ( Lz = 0, t0 2 Ca b].F # $ $# '( #$' 24.6. J, / (24.27) $ $) '( $ (24.26) ( //0'( (24.23). 24.5. @( ". $ B8< y_ = z(24:28): z_ = ;y + 1: /' 0 ( .( ( y_ = z , z_ = ;y. # (. $. 24.2), (//0 t ) # # #:yU = ;y , yU + y = 0:311". (. 24.3) y = C1 cos t + C2 sin t:A, z = y_ , ".
( y_ = z , z_ = ;y:0 1 01 0101yCcost+CsintcostsintC12@ A=@A @A@ 1 Az;C1 sin t + C2 cos t = ; sin t cos t C2 :&. $' 001costsintW(t) = @ ; sin t cos t A /'( 0( ( ( y_ = z , z_ = ;y (' , '$ 24.5!). /B (24.27) ". (24.28):0 1 01 201 Zt 01;1 0 1 3@ y A = @ cos t sin t A 64@ C1 A + @ cos s sin s A @ 0 A ds75 =z; sin t cos tC2; sin s cos s1001 20 1 Zt 0= @ cos t sin t A 64@ C1 A + @ cos s; sin t cos tC20sin s10 1 3; sin s A @ 0 A ds75 =cos s101 20 1 Zt 01 3costsintC;sinsA 64@ 1 A + @A ds75 ==@; sin t cos tC2cos s01301 20 1 0costsintCcost;1= @ ; sin t cos t A 4@ C1 A + @ sin t A5 :2,', # $ # ". (( ///0'( , 3 ' $' Y0 ( ( (24.17) ' "$." # 24.3. Y0 ( ( (24.17) ( $# Ca b] 0( A(t) Cn1Ca b] $ n ( n { $' (#) (24.17)). 8 # "$ Y0 3$' " n ( $% ( (24.17) ( n"0 "( /'( 0 ( W(t)).312J, ( ( Y ( , b(t) 6 0.
$ Y0 ( ( (24.17) # y(t), .( ( (24.23), .. Y = y(t) + Y0, # ## y = y + y (. (24.24)). . $1.L B (24.27) $ ' ". (24.23), $ /' 0 ( W(t) .( ( (24.17). B# #%(, #% ' /'( 0 /' (, 0 W(t) '$ .. # (24.23) 0 A(t) A , 3 '.@ ' $', " 0 A # " $. $' /'( 0 % #( %## . @ . .' A = (aij )nij=1 { 0 # n.#$ 24.6. 8# x = fx1 ::: xng $ A, . " $ 0 ) x 6= 06 ") Ax = 0x , (A ; 0E )x = 0:(24:29)J' E = diag(1 ::: 1) { 0 $ n n:#'# "# (24.29) ' x 6= 0 det(A ; 0E ) = 0: , ": det(A ; 0E ) = 0 (24.29) ' x 6= 0:#$ 24.7.
Adet(A ; 0E ) = 0(24:30)$ 0 A ( %## //0'( (24.17)), 313# = j { ( 0. 3 !(A) = f1 ::: mg $ 0 A. #3 " $ = j 3 ' .( "( # x = xj ($3, ().#$ 24.8. &# !(A) = f1 ::: mg 0 A $ , $ n $% "% $((.. m = n i 6= j i 6= j i j = 1 n). 8 ,# !(A) 0 A .@ #$' . 3.60) 4 $ " $ 1 2 0 A ( $% "% # 1 x2.70) ? # !(A) = f1 ::: ng 0 A (, #' % "% # x1 ::: n ( 0 "$"$ Rn ( C n n{% ##% #).? " = (1 ::: n) { 0 $ "% # xj 0A, 0101CC...T ;1 AT = b b = diag(1 ::: n) = BB@(24:31)A0n( , 0 A '( 0b, 0 T $ ). ' ( //0'% (Ly dy(24:32)dt ; Ay = 0 ( 0( A.
&$' # 0 A ( .( (. :D. " ! -y = xe t fx1e t ::: xne tg ) (24.32), , x = f1 ::: xng A, = 0.". #-/#0 y = xe t (24.32).I '0000x 0e t ; Axe t 0 , (A ; 0E )x = 0:31400 , y = xe t (24.32), = 0 { " $ 0 A, x { "( #( 0, .( " $ = 0., ": = 0 x 6= 0 { . " $ "( # 0 A, 0(A ; 0E )x = 0 , (A ; 0E )xe t 0 ,, 0xe t ; Axe t 0 , dtd (xe t) ; A(xe t) 0: 3 $, #-/#0 y = xe t { (24.32). , #$.8'$' ( F(, ' /' 0 ( (24.32) % #(%## (24.30). 24.7. !(A) = f1 ::: ng A , 000000W(t) = (x1e t ::: xnent)1(24:33)! xj = fxj1 ::: xjng { A, = j (j = 1 n), ) (24.32) ( R = (;1 +1)).". F( #3( "0 xj ej t 0 (24.33) (24.32). #3 ', 0 (24.33) 3. 4 ' 3' # t = 0. =W(0) = (x1 ::: xn):#'# "0 ( 0 " # 0 A, . $ " $ = j j = 1 n, ( $, $ det W(0) 6= 0:&', (24.33) /'( 0( ( (24.32).
, #$.315" # 24.4. 8 # !(A) = f1 ::: ng -0 A ". ( (24.32) y = yo:o: = W(t)c = c1x1e t + ::: + cnxnent(24:34) xj { "( # 0 A, .( " $ = j , (j = 1 n)6 c1 ::: cn { $' .J, /' 0 W(t) (24.32) 3 ' #% #( %## (24.30). # 3 #' /0 "% % #% 0 A.
,# /0 % %#% $%, /'( 0 ( # # !(A)$' $ . 24.6. ' ((x_ = x + et , x_ ! = 1 0 ! x ! + et ! :(24:35)y_3 2 y0y_ = 3x + 2y20 ( !10A= 3 2 :? " $ % $ %## "1;01det(A ; E ) = 0 , 3 2 ; = 0 , (1 ; )(2 ; ) = 0 , 1 =2 = 2:,# "$, # !(A) = f16 2g 0 A (. 8 " # ( 0. 0 = 1 = 1 (24.29) 0 0 ! x11 ! = 0 ! , ( 0 x11 + 0 x12 = 0(24:36)3 1 x1203 x11 + 1 x12 = 0:#'# ( "% x11 x12, (24.36) # 3x11 =;x12, ". (24.36) 1! x1 = c1 ! x12;3c13161 c1 { $' .
@ ( "( #, $' 3 3' c1 = 1: "( # !x1 = ;13 .( " $ = 1 = 1: 4 " #, . " $ = 2 = 2 ' ;1 0 ! x21 ! = 0 ! , ( ;1 x21 + 0 x22 = 03 0 x2203 x21 + 0 x22 = 0: ". 2! !x1 = 0 c2x223 $' c2 = 1, "( # !2x = 01 .( " $ = 2 = 2: 24.7 /' 0 ( ( //0'( , .( (24.33), ! ! ! t!10e01 t2 tt2tW(t) = (x e x e ) = ;3 e 1 e = ;3et e2t :4 ".
( (24.35) '$ /( (24.24):1223x = y = W(t) 64c + Zt W;1(s) es ! ds75 :(24:37)o::0y08 0 W;1(t) :! ;t! t!;1 1 2te0e0e0;1W (t) = ;3et e2t = e3t 3et et = 3e;2t e;2t : !317,' (24.37) !xy! 2 ! Zt ;s= y:: = ;3et e02t 64 cc1 + 3ee;2s20et20 Zt1ds t! 666 ! BBBe0c1= ;3et e2t 666 c + BBB Zt 04 2@;s03e ds13CC77 tCC77eCC77 = ;3etA5e! s! 30e ds75 =;2s 00 ! " c1 ! + t !# :e2t c2;3e;t + 3+ -/! ( +!+) 0'** ,(" + !E " //0' ( #% () '( /$#( 0 ( ' . ", /# . /$# , . $ . $ (, '% () ' . 23 #$', ' '% % $ "." $ ( //0'% (.
8 ( 0 '$', " # ' ' 3 /$# . =, ", $. '% ( $ 3# % ., . # ' $' (. ,# "$, $# 3( 3( %3 (, #% # ', //0' , "((. B %.'"( . , //0' dy = f (t y)dt318(24:38) y = fy1 ::: yng, f (t y) = ff1 ::: fng, t t0. I ', (24.38) . ( B) # /# y = '(t) ( . % t t0:#$ 24.9.
y = '(t) (24.38) $ 2 t t0, " " > 0 . > 0 ($., ". , ") #, :1) y = y(t) (24.38) ' y(t0), . jy(t0) ; '(t0)j < , . %t t062) % #% ( #$jy(t0) ; '(t0)j < ) jy(t) ; '(t)j < " (8t t0):(24:39)0 n 11=2XJ' jyj jfy1 ::: yngj = @ yj2 A :j =1,# "$, (' '( #( y = '(t) $, "$# # ( ' # y = y(t) '( t = t0 "$# # y = '(t) "#3# Ct06 +1) (. . 24.1).J, "$' '% #% # 3# Ct0 T ], # . $ "$ % '% #, $ ! ) (24.38). = ' ' ( # # (24.38) (, //0( " D ( ' f (y t)). # $ "((' y = '(t) t t0 (.. "$' ( "# 3# Ct0 1)).#$ 24.10.
y = '(t) (24.38) $ t ! +1, :) y = '(t) ( E t t06") . 4 > 0 #, " y = y(t) (24.38), . jy(t0) ; '(t0)j < 4 ' lim jy(t) ; '(t)j = 0:(24:40)t!+1319,# "$, # (' y = '(t)$ '# "$' # % ( y = y(t) (24.38), # . $ "$ % # '(t) '( t = t0, "3 % c '(t) t ! +1 (. . 24.3).;. 24.2. 24.3#$ 24.11. y = '(t) (24.38) $ 2 t t0 .
" > 0#, " > 0 ( y = y (t) (24.38) t1 = t1() > t0 #, jy (t0) ; '(t0)j < jy (t1) ; '(t1)j ": ' ( //0'% (dy = A(t)y + b(t):(24:41)dt#$ (. 24.8), (% (" ( (# (), " (. (% . .#$ 24.12. I ', ( (24.41)( E t t0 (# ( t ! +1), ( ) y = '(t) ( ( t t0 ( # ( t ! +1). 24.8. * (24.41) A(t) b(t) ( Ct06 +1): D (24.41) t t0 ( t ! +1) ! !, ! ) z 0 cdz = A(t)z(24:42)dt320 t t0 ( t ! +1).". ' y = '(t) { $' /# (24.41), y = y(t) { " ( .? y = '(t) ( t t0, " " > 0 . = (") > 0 #, #$jy(t0) ; '(t0)j < ) jy(t) ; '(t)j < " (8t t0):@ $' z (t) = y(t) ; '(t) % ( ( (24.41) ( (24.42), .
#$ jz (t0)j < ) jz (t)j < " (8t t0):(24:43)V, $ % ( y = y(t) ( (24.41) z (t) = y(t) ; '(t) ( (24.42). 8#$ (24.43) $, ' z 0 ( (24.42) ( t t0: =#,$ ( $' (/# ) y = '(t)( # (' ' ( (24.42).3 ', ' z (t) 0 ( (24.42) ( t t0. , #$ (24.43) " z = z (t) ( (24.42).
? ' y = '(t) { /# ( (24.41) y = y(t) { $' ( , . jy(t0) ; '(t0)j < , (#'# $'z (t) = y(t) ; '(t) ( (24.42)) ojy(t) ; '(t)j jz (t)j < " (8t t0):F $, y = '(t) ( (24.41)( t t0. ,# ## y = '(t) /#' $' "$, ( (24.41)( t t0: ; ( #( (.321, #$.,# "$, $ #( ( ( () ( (( (24.41) $' # (' ( (') z 0 .( ( (24.42). 4 ( 0( A(t) = A = const 3 ' '. 24.9. " ! ) z (t) 0 (24.42) t ! +1 , !(A) = f1 ::: mg A ( Re < 0: 24.10. * (24.42) A(t) = A !(A) = fj g i 6= j i 6= j i j = 1 n(..
). /! !) z (t) 0 , A ( Re 0: 24.11. * (24.42) A(t) = A !(A) = f1 ::: mg ( (Rej >0). /! ) z (t) 0 t t0: 24.1. 8 24.10 ( ' z (t) 0 (24.42) "', " # !(A) 0 A " (. @ 3 "', " #" "$' . 3 "' '# # #.J, 3 " (, / (% ( # //0' :(24:44)y(n) + an;1y(n;1) + ::: + a1y0 + a0y = f (t):4(', #-/#0 fy1(t) ::: yn(t)g #y1 = y y2 = y0 y3 = y00 ::: yn;1 = y(n;2) yn = y(n;1) 322 y = y(t) { (24.44). , y10 = y0 = y2 y20 = y00 = y3 ::: yn0 ;1 = y(n;1) = ynyn0 = y(n) = ;an;1y(n;1) ; an;2y(n;2) ; ::: ; a1y0 ; a0y + f (t) ;an;1yn ; an;2yn;1 ; ::: ; a1y2 ; a0y1 + f (t) #-/#0 fy1(t) ::: yn(t)g 1 0 y 1 010 y0 1 00010:::0011BB y0 CC BB0.1.
: .: : 0.0. CCCC BBBB y2 CCCC BBBB 0. CCCCBB .2 CC BB 0...... CC BB .. CC + BB . CC :BB . CC = BB .BB 0 CC BB00 ::: 01 CCA BB@ yn;1 CCA BB@ 0 CCA@ yn;1 A @ 0;a0 ;a1 ;a2 : : : ;an;2 ;an;1f (t)ynyn0(24:45)V, y = y(t) { (24.44), #-/#0fy y0 ::: y(n;2) y(n;1)g (24.45). ": fy1(t) ::: yn(t)g { (24.45), y = y1 (t) { p (24.44) (. (24.45)).=#, (24.44) # ( #$ '# ) (24.45). 23 #$', (' ( (24.44) $ "( (' ( (24.45), ".
8 " $ 0 A (24.45),, n + an;1n;1 + ::: + a1 + a0 = 0(24:46)# %## # //0' (24.44). ,# "$, # (24.44) # 0 A (24.45), " ( (#( ( (), / (24.45), #% ( (24.44). 24.7. =' (' ((x_ = ;3x + 2y + ety_ = ;5y + sin t:20 ( !;32A = 0 ;5 :(24:47)323? # f1 2g $ %## ;3;2 = 0 , (3 + )(5 + ) = 0 ,det(A ; E ) = 0 , 0;5 ; ";3, 1 =2 = ;5:,# ## # # 1 = ;3 2 = ;5 3 #Re < 0 (24.47) # ( t ! +1(. 24.8 24.9). 24.8. =' (' //0' y000 + 4y0 = 0:&# "$ # %## 266 1 = 032 + 4 = 0 , ( + 4) = 0 , 4 2 = ;2i3 = +2i:8 # # 3 ( . #'# , #3 % //0' ( E t t0 ($' t0 2 R { $' /# ). F # $ 24.10 $ 24.1., "&- ( (24.38).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
