1. Интегралы ФНП Диф_ур (853736), страница 17
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n ; 1 3.8 . 3 "$ 256 "#% (:8>1 y1 (x) + ::: + n yn (x) 0>>< 1y10 (x) + ::: + nyn0 (x) 0,>:::>>: y(n;1) (x) + ::: + y(n;1) (x) 01nn1010y1 (x) : : : yn (x) C BBBBy10 (x) : : : yn0 (x) CCC BBBB, BB .... CCC BB...B@ (n;1)A@y1 (x) : : : yn(n;1) (x)1 0 1CC BB 0 CCBB 0 CC2 CC=.. CC BB .. CC A @ A10# ( (( $ /#0( y1(x) ::: yn(x)) #3 x 2 Ca6 b] ( 1 2 ::: n): @ ' ( , .( 8# /#0( y1(x) ::: yn(x), ". ' #3 x 2 Ca6 b] ..W (x) W Cy1 ::: yn] 0 (8x 2 Ca6 b]): , #$.J, " 3 $'( /#0( y1(x) ::: yn(x) 2 C n;1Ca6 b] .
23.3. #$', /#0(( 0 0 x 12(x;1)0x1y1(x) = 0 1 < x 2y2 (x) = (x ; 1)2 1 < x 2( $ $# C06 2] W Cy1(x) y2 (x)] 0 (8x 2 C06 2]):, ##% % 1 2 3 0 (8x 2 Ca6 b]): x 2 C06 1] 1 y1 (x) +2 y2 (x)21 (x ; 1) + 2 0 0: F 3 1 = 0 $' 2: @ 3# (16 2] 1 0 + 2(x ; 1)2 0# , 2 = 0: =#, 3 1y1(x) + 2y2 (x) 0 3# C06 2] ' 1 = 2 = 0: J,/#0 y1(x) y2(x) ( $ $# C06 2]: & (, y1(x) y2(x) (x ; 1)2 0 =W Cy1 y2]jx201] = y0 (x) y0 (x) 012x201] 2(x ; 1) 0 0 (x ; 1)2 W Cy1 y2]jx2(12] = 0 2(x ; 1) 0257n..
' 8# W Cy1 y2] 3 ". ' $# C06 2].&0, , $, y1(x) y2(x) Ly = 0 $# Ca6 b] #//0. F " #$ 3., 23.4. (( $/#0(." # 23.1. ? # /#0( y1(x) : : : yn(x) 2n;C 1Ca6 b] % " ( # x = x0 2 Ca6 b] #$ /#0 ( $ $# Ca6 b]:4(', " y1 (x) ::: yn(x) " ( $ $# Ca6 b], W (x) W Cy1 ::: yn] 3 ". " ' $#, $, , " " # x = x0 "' 3.0 "( ( //0' Ly y(n) + an;1(x)y(n;1) + ::: + a1(x)y0 + a0(x)y = 0:(23:5)4#3 .( 3( $'. 23.5. * y1(x) ::: yn(x) ) ! (23.5) Ca6 b] an;1(x) ::: a0(x): /! y1 (x) ::: yn(x) Ca6 b] , ! W (x) W Cy1 ::: yn] Ca6 b]:".
4' # $ 23.1. 4#3 "%'. ' y1(x) ::: yn(x) (23.5)( $ $# Ca6 b]: #3, # ". ' ( # $# Ca6 b]: 3, .. . # x0 2 Ca6 b] #, #258W (x) ". ' ( #: y1(x0) : : : yn(x0 ) y0 (x ) : : : y0 (x ) W (x0 ) = 1 .. 0 . . . n .. 0 = 0: y(n;1) (x0) : : : y(n;1) (x0) 1n, "0 ( $, .. . 1 ::: n , #, 0101y(x)y(x)1 0n0BCBBCCBC..BCBCA = 0 ,+:::+..1Bn@C@ (n;1)Ayn(n;1) (x0)y1 (x0 )8>< 1y1(x0) + ::: + nyn (x0) = 0(23:6), > (n;1) : : :: y(n;1)(x0 ) + ::: + nyn (x0) = 0:1 1& .' #$% /#0 y = '(x) = 1y1 (x) + ::: + nyn(x).
#'# yj (x) (23.5), $ ( //0' L , L'(x) = 1Ly1(x) + ::: + nLyn(x) 1 0 + ::: + n 0 0:F $, /#0 y = '(x) (23.5). =$ (23.6) , /#0 ' , ..'(x0) = 0 '0 (x0) = 0 ::: '(n;1)(x0) = 0:@ # 3 ' ' y 0 . 8 (. 23.1) /#0 y = '(x) y 0 $# Ca6 b] $ '(x) 0 $1 y1 (x) + ::: + n yn (x) 0 (8x 2 Ca6 b]):#'# $' 1 ::: n , 3 $, /#0 y1(x) ::: yn(x) ( $ 259$# Ca6 b]. 2 , # #$, 3 W (x0 ) = 0 . &', # ". ' ( # $# Ca6 b]: , #$.=$ 23.4 #$' 23.5 # . ( # W (x) W Cy1 ::: yn] (y1(x) ::: yn(x) ( //0' (23.5) $# Ca6 b] #//0 an;1(x) ::: a0(x):50) ? # W (x) ". ' #( #x = x0 $# Ca6 b] 3 $#Ca6 b] (.e. W (x) 0 (8x 2 Ca6 b]):60) ? # W (x) % " ( # x = x0$# Ca6 b], $# Ca6 b]:&( 50 60 # #3 $ /; xRx an;1(t)dtW (x) = W (x0 )e 0(8x 2 Ca6 b])(23:7)$( /( # {E. J' an;1(x) { #//0 $( y(n;1) (23.5), x = x0 { $' /# # $# Ca6 b]:"$ ' $ Y0 3 % ( y = y(x) (23.5).
B# # 3 Y0? 8%, ( . 4(', y1(x) y2(x) $'% 3 Y0 3 Lyj (x) 0 j = 1 2 $ $'% C1 C2 ( ( L) 3LCC1y1(x) + C2y2(x)] = C1Ly1(x) + C2Ly2(x) 0:F 3 #$, " ( #"0 3 Y0 3 Y0 (.. (23.5)). &', Y0 { ( .=$ (( " $, ( X#, 3 ' "$, .. # f1 ::: fm 2 X # " (:) f1 ::: fm ( $6") ## " " f 2 X , . 1 ::: n#, f = 1f1 + ::: + m fm :260 1 ::: m $ # f "$ f1 ::: fm (#$, # "$ ).8 Y0 #3 3 ' "$. 8 //0'% ( ( #3 "( (( () "$ ( $' /'( ( (.
2 # $ " .&. "$ Y0 .( (. 23.6 ( C?C BGH 3$I ##HC$I). (23.5) ai(x) - Ca6 b], ! n Ca6 b] ) y1(x) ::: yn(x) (n { (23.5)). * ! ) (23.5) )y1(x) ::: yn(x), .. ) (23.5) (23:8)y = y(x6 C1 ::: Cn) = C1y1(x) + ::: + Cnyn(x)! C1 ::: Cn { .". #3 , (23.5) . n ( $% $# Ca6 b] (.
8$'$' 0 B det B 6= 0 "0 fb1j ::: bnjg j = 1 n: ,# 0 B 3 # n. B3( "0 ( 0 " '$' # '( # $ B (23.5). n $ B:( Ly = 0y(x0 ) = b11 y0 (x0) = b21 : : : y(n;1) = bn16 : : :( Ly = 0: : : y(x ) = b y0 (x ) = b : : : y(n;1) (x ) = b :01n02n0nnB3 $ % $ ( #//0 ai(x)) . "$ $ y1(x) ::: yn(x) 261% $ . 8# # x = x0 % (: y1(x0)W Cy1 ::: yn]jx=x0 = : : : y(n;1) (x )01: : : yn(x0) b11 : : : b1n : : : : : : = ..
. . . .. : : : yn(n;1) (x0) bn1 : : : bnn det B 0 , . , y1(x) ::: yn(x) ( $ $# Ca6 b]: &. #% ( #$ (% 3 3' " 3, " $' 0 B det B 6= 0).#3 ', (23.8) { ". (23.5). "% $% % C1 ::: Cn /#0 (23.8) (23.5), # ## Y0 ( (23.5) ( . ' ' y = y(x) { $'( $ BLy = 0 y(x0) = y0 y0 (x0) = y10 ::: y(n;1)(x0 ) = yn0;1(23:9) x0 2 Ca6 b]: #3, . $ %C1 = C10 ::: Cn = Cn0 #, /#0 y = C10y1(x) + ::: + Cn0yn(x) y = y(x) $ B (23.9).
(23.8)' (23.9), C1y1(x0 ) + : : : + Cnyn(x0) = y0C1y10 (x0 ) + : : : + Cnyn0 (x0) = y10,(n;1)C1y1 (x0) + : : : + Cnyn(n;1) (x0) = yn0;11001 B C1 C 01yy(x):::y(x)010n0BCBB01(x0 ) : : : yn0 (x0 ) CCC BBB C.2 CCC BBB y10 CCCyB, BBB : : : : : : : : : CCC BB . CC = BBB .. CCC :(23:10)BC@ (n;1)ABC @ 0 Ay1 (x0) : : : yn(n;1) (x0) @ Cn Ayn;1,# ## y1(x) ::: yn(x) ( $ $# Ca6 b] % # W (x) W Cy1 ::: yn] $'(# $# Ca6 b].
' (23.10) # W (x0 ), $ . @ ( (23.10) C1 = C10 ::: Cn = Cn0: 262 /#0 y = C10y1(x) + ::: + Cn0yn(x) ' (23.5), ' (23.9) ( " C10 ::: Cn0). &', /#0 (23.8) ". (23.5). , #$.=$ ( , " $ n ( $% ( y1(x) ::: yn(x) (23.5) # n "$ "$ Y0. &', Y0 ( (23.5) $' n.#$ 23.3. E " $ n ($% $# Ca6 b] ( y1(x) ::: yn(x) (23.5)(n{ #) $ ) ( "$ (). ' //0' Ly y(n) + an;1(x)y(n;1) + ::: + a1(x)y0 + a0(x)y = h(x)(23:11) #3 .
3. 23.7 ( C?C BGH 3$I ##HC$I). (23.11) ai(x) - h(x) Ca6 b], ) (23.11) ( ) y = y(x6 C1 ::: Cn) = C1y1(x) + ::: + Cnyn(x) + y(x) ,, y:: = :: + y::(23:12)! y1(x) ::: yn(x) { ) ! ! Ly = 0 y = y(x) { )! (23.11), C1 ::: Cn { .". L # /#0 (23.12), "'Ly = C1Ly1(x) + ::: + CnLyn(x) + Ly(x) 0 + ::: + 0 + h(x) h(x):F $, /#0 (23.12) (23.11) $'% $% % C1 ::: Cn. ' '(x0 y0 y10 ::: yn0;1) { $' # Rn+1 (x0 2 Ca6 b]). #3, y = y(x) $ BLy = h(x) y(x0 ) = y0 y0 (x0) = y10 ::: y(n;1)(x0 ) = yn0;1(23:13)2633 ' $ (23.12) " % $(C = C10 ::: Cn = Cn0 %. (23.12) (23.13)," 'C1y1(x0 ) + : : : + Cnyn(x0) = y0 ; y(x0 )C1y10 (x0 ) + : : : + Cnyn0 (x0) = y10 ; y0 (x0 ),: : :C1y1(n;1) (x0) + : : : + Cnyn(n;1) (x0) = yn0;1 ; y(n;1)(x0 )1 01010y;y(x)Cy(x):::y(x)00110n0C BCBB 0CBBB y1(x0) : : : yn0 (x0 ) CCC BBB C2 CCC BBB y10 ; y0 (x0) CCC, BB : : : : : : : : : CC BB ..
CC = BB..CC : (23:14)A @ 0A@ (n;1)A@(n;1)y1 (x0) : : : yn(n;1) (x0) Cnyn;1 ; y (x0 )' ( #W (x) W Cy1 ::: yn] # x = x0 #'# /' ( y1(x) ::: yn(x) ( $ $# Ca6 b], #$( ' (23.14) . &', (23.14) C1 = C10 :::Cn = Cn0 $ /#0 y = C10y1(x) + ::: + Cn0yn(x) $ B (23.13). , #$, /#0 (23.12) ". (23.11).
,#$.$ %&& 23.7 # ". //0' (23.11) # 0:1) /'( ( y1(x) ::: yn(x) . 62) y = y(x) (23.11).& . ( 0. # ( #//0 (. .($) 3 $'. ? 3 ( /' ( y1(x) ::: yn(x) Ly = 0 $' 0 " .264 23.8. * y1(x) ::: yn(x) { ) ! (23.5) Ca6 b] ai(x): h(x) !! (23.11) Ca6 b] ! ) ( y = y(x) = C1(x)y1 (x) + ::: + Cn(x)yn (x)(23:15)! C1(x) ::: Cn(x) ( ! ) ! (23.5)) 010 0 1 01C0y(x):::y(x)1nBB 0CB 1 C BCBB y1(x) : : : yn0 (x) CCC BBB C20 CCC BBB 0 CCC(23:16)BB : : :CC BB ..
CC = BB .. CC :::::::@ (n;1)A@A @Ay1 (x) : : : yn(n;1) (x) Cn0h(x)". #$' #:(23:17)Ly = y00 + a1(x)y0 + a0(x)y = h(x):8 (23.16) 010 0 1 01( 00y(x)y(x)C0C1211 y1 + C2 y2 = 0@ 0A@A@Ay1(x) y20 (x) C20 = h(x) , C10 y10 + C20 y20 = h(x): (23:18), /#0y = y (x) = C1(x)y1(x) + C2(x)y2(x)(23:19) C1(x) C2(x) (23.18), (23.17).8 $ y0 y00 /#0 (23.19) (23.18):y0 = C10 y1 + C20 y2 + C1y10 + C2y20 = C1y10 + C2y20 y00 = C10 y10 + C20 y20 + C1y100 + C2y200 = h(x) + C1y100 + C2y200 : , y00 + a1(x)y0 + a0(x)y 265 (C1y100 + C2y200 + h(x)) + a1(x)(C1 y10 + C2y20 ) + a0(x)(C1 y1 + C2y2):M $' #//0 ' #3( /#0( C1(x) C2(x) y00 + a1(x)y0 + a0(x)y h(x) + C1(y100 + a1y10 + a0y1) + C2(y200 + a1y20 + a0y2):#'# y1 (x) y2(x) { .
Ly = 0 Lyj (x) 0 $ y00 + a1y0 + a0y h(x): ,# "$, /#0 y = C1(x)y1 (x) + C2(x)y2 (x) (23.17). , #$. 23.4. ', /#0 y1 = cos x y2 = sin x "$ /' y00 + y = 0 (". y00 + y = 1= cos x:#'# y10 (x) = ; sin x y100 = ; cos x /#0 y1 = cos x y00 + y = 0: , # 3 "3, /#0 y2(x) = sin x #3 y00 + y = 0: 8 #W Cy1 y2] = yy101y2 = cos x sin x = cos2 x + sin2 x = 1:y20 ; sin x cos x 8, ". ' 3# (;16 +1) $/#0 y1 = cos x y2 = sin x "$ /' ( y00 + y = 0:@( ' y = y(x) 00y + y0 = 1= cos x / y = C1(x) cos x + C2(x) sin x: /#0C1(x) C2(x) 3 ' 1;1 010 01 0Ccosxsinx0A @A@ 10 A = @C2; sin x cos x1= cos x ,0 0 1 0101Ccosx;sinx0, @ C10 A = @ sin x cos x A @ 1= cos x A ,2sin x C 0 = 1 , C = ; Z sin xdx = Z d cos x = ln j cos xj + C C10 = ; cos11x 2cosxcosxZ0C2 = 1dx = x + C 2:266#'# y00 + y = 1= cos 1() 2() 3 $' C1(x) = ln j cos xjC2(x) = x: % y = C1(x) cos x + C2(x) sin x, y = y(x) = (ln j cos xj) cos x + x sin x $ ".
$ /y = C1 cos x + C2 sin x + (ln j cos xj) cos x + x sin x == (C1 + ln j cos xj) cos x + (C2 + x) sin x:+ . % @, $ p z = a + ib a b { , i = ;1 { 0 (i2 = ;1). a = Re z $ , b = Im z { ## z . Hz = a ; ib $ z = a + ib 0p 2 (' jz j = a + b2 $ z . 23% ##% "$ "#( CI . B3 ## z = a + ib # M (a6 b) #;! ( #. ' 0X $ 0 {# ;OM , ' 0Y { . & #'X 0 $ 6 3 "$ "#( CI . A ' = (;!0M b6 ;0!x) $ ! ## z . V, $. < $ ' = (;!0M b6 ;0!x),3.( %0 ' < 2 ( % ; < ' ).
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