1. Интегралы ФНП Диф_ур (853736), страница 18
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M $ "$ #: ' = arg z .267;. 23.1=$ . 23.1 , a = jz j cos ' b = jz j sin ': J, ## 3 $' . a + ib = jz j(cos ' + i sin '): F/ $ ! z , ' / z = a + ib { "#( /( ## z . ? '$' /( F(e+i = e(cos + i sin ) 3 ' . / ## z = jz jei' $ z .4 ##% $ , $' (' % , .. a1 = a2a1 + ib1 = a2 + ib2 , b1 = b2:4( ## :1) (a1 + ib1) + (a2 + ib2) = (a1 + a2) + i(b1 + b2)(a1 + ib1) ; (a2 + ib2) = (a1 ; a2) + i(b1 ; b2)62) (a1 + ib1) (a2 + ib2) a1a2 + ia1b2 + ib1a2 + i2b1b2 == (a1a2 ; b1b2) + i(a1b2 + a2b1)6+ ib1) = (a1 + ib1)(a2 ; ib2) :3) ((aa1 +ib )a2 + b22222A3 ##% ( % ) $' #( /:Cjz1j(cos '1 + i sin '1)] Cjz2j(cos '2 + i sin '2)] == jz1j jz2j C(cos('1 + '2) + i sin('1 + '2)] :268 # $ /o(jz j(cos ' + i sin '))n = jz jn(cos(n') + i sin(n'))$ =.
& .' 3 ' 0 $ # n{( $ ##% .#$ 23.4. "3 z = z (x) : D ! CI , . #3 (' x 2 D R ## z = u(x) + iv(x) (u(x) v(x) 2 R), $ x "' D:@ # /#0 3 $' " /# (. @,Cu(x) + iv(x)]2 = u2(x) + 2u(x)v(x)i ; v2(x):B# # / F(eu(x)+iv(x) = eu(x)Ccos v(x) + i sin v(x)]: # ' / # ## :iz ; e;iziz + e;izeesin z = 2i cos z = 2 :(23:20)$ ##( /#0 (' :d Cu(x) + iv(x)] = u0 (x) + iv0(x)z 0(x) dxnd(n)z (x) dxn Cu(x) + iv(x)] = u(n) (x) + iv(n)(x) { ZbaZbZbZbaaaz (x)dx Cu(x) + iv(x)]dx = u(x)dx + i v(x)dx:@,(e(+i)x )(n) = ( + i )ne(+i)x ( + i = const):269' ' //0' y(n) = f (x y y0 ::: y(n;1))(23:21) # '# /#0 (' ( ## /#0 ('( ( ).
B# # 3, ## (' (. 20.2): /#0 y = u(x) + iv(x)$ (23.21) $# Ca6 b] , " (23.21), ". 3 $# Ca6 b]: J B (23.21) ('# $' ' # 3 "' ##().@ " ' ( //0' (23:22)Ly y(n) + an;1(x)y(n;1) + ::: + a1(x)y0 + a0(x)y = h(x) (' #//0 ai(x) ##( ' h(x) = h1(x)+ ih2 (x) (h1 (x) h2(x) { (' /#0).V, L ( C nCa6 b]##$% /#0(. 4#3 . 3.70) ? y = u(x) + iv(x) ## (23.22) (' #//0 ai(x) (i = 0 n ; 1) (' u(x) v(x) (Lu = h1(x) Lv = h2(x):8 , ( L Ly(x) LCu(x) + iv(x)] Lu(x) + iLv(x) h1(x) + ih2(x): $' ' (' , 3Lu(x) h1(x) Lv(x) h2(x)... $' h(x) 0 .( $'.80) 4(' ## y = u(x) + iv(x) //0' Ly = 0270 (' #//0 ai(x) #3 (.. Lu(x) 0 Lv(x) 0).E( $' ( $' $# Ca6 b] ##$% /#0( y1(x) ::: yn(x) #3, ## ('% /#0( (. 23.1).,'# $' #//0 1 ::: n (( #"0 "'## . 8 23.1{23.8 ##% (.8 '( " /# (( $ #% ##$% ('% /#0(, %3.90) & /#0(y1 = e x ::: yn = enx1 j (## (') $(..
i 6= j i 6= j i j = 1 n) ( $ $'$# Ca6 b]:". J # 1x e e1xx1xnW Ce ::: e ] = 1 : n;1 1x 1 e 1: : : enx : : : nenx = e x: : :enx 1 ::: n;1 1: : : nn;1enx 1: : : 1 : : : n : : : : : nn;1 ($' $ #3 "0 ".( 3' { . #). &.( $' ' ( $ 9) $ (i ; j ) %Yi j 2 f1 2 ::: ng #%, j < i, .. 4n =(i ; j ): F /#1j<in3 #$' #0, " '. '# 8 1 1 1 43 = 1 2 3 2 2 2 123' #. " # ( # 43 #, 3 (;1 ), # '( # { ,2713 (;1), " ' 1 11;;43 = 0 2 ; 1 3 ; 1 = (2 ; 1 ) (3 ; 1 ) = 0 2 ; 2 ; 2 213 312 13 123 1 1 = (2 ; 1)(3 ; 1) = (2 ; 1)(3 ; 1)(3 ; 2):23=#, ' 8c# W (x) W Ce x ::: enx] 1Y(i ; j ):j<in#'# i 6= j i 6= j (i j = 1 n) # ' "., W () 6= 0 " $# Ca6 b]: &( 90 #$.100) ' ('% /#0(W (x) = e( +:::+n)x11e x ::: emx1em x cos m+1x ::: esx cos sx(23:23)em x sin m+1x ::: esx sin s x:? 1 ::: m m+1 = m+1 + im+1 ::: s = s + is $, #$ /#0( ( $ "$# Ca6 b]:".
/#0(e x e x e x cos 3x e x sin 3x:(23:230)+1+11233& % ( #"0 :C1e x + C2e x + C3e x cos 3x + C4e x sin 3x 0:123(23:24)3='$ /cos 3x = 21 (ei x + e;i x) sin 3x = 21i (ei x ; e;i x) . 3 333C1e x + C2e x + C5e x + C6e x 027212333(23:240) "$: C5 = (C3 ; iC4)=2 C6 = (C3 + iC4)=2: 1 2 3 , 1 2 3 3 (', 1 2 { (' ).
( 90 /#0( e x e x e x e x ( $ " $# Ca6 b]. J, 3 (23:240) '# , # C1 = C2 = C5 = C6 =0:=$ 8>C3 ; iC4 >(< C5 =C3 + iC4 = 02,>C + iC4C3 + iC4 = 0>: C6 = 32, C3 = C4 = 0: =#, 3 (23.24) C1 = C2 = C3 = C4 = 0 a $ /#0( (23:230)( $ " $# Ca6 b]: &( 100 #$.; 3 3 "' #$ . 3.110) ? 1 :: m , /#0(e x xe x ::: xr ;1e x11121331::::::::::::::::::::::::::::::::::::(23:25)em x xem x ::: xrm ;1em x r1 ::: rm { ' (rj 1 j = 1 m), ( $ $' $# Ca6 b]: E( $( " #3, "$ (' /#0((23.25).$ 1 "( # Ly y(n) + an;1y(n;1) + ::: + a1y0 + a0y = 0273(23:26) #//0 an;1 ::: a1 a0: "# p() n + an;1n;1 + ::: + a1 + a0 = 0()$ (23.26) $ y(k) k (k = 0 n).#$ 23.5.
2 p() n +an;1n;1 +:::+a1+a0 $ ! (23.26), () { , . (23.26).= 3L(ex) exp()(23:27) { , # ## (ex)(k) k ex k = 0 n: :D. " ! y = e x (0 {) ) (23.26), , = 0 ! !p() (, ( , ! ()).4(', p(0) = 0 $ (23.27) 3L(e x) 0 #$ ., # y = e x (23.26). ": y = e x { (23.26), L(e x) 0 $ (23.27) , p(0) = 0 .. = 0 {#' %## p():=$ F( $ 3 # .( $'. 23.9. 1 ::: n ! () (.. i 6= j i 6= j i j = 1 n), y1 = e x :::yn = enx(23:280) ) (23.26). 9 ) ( ?a,b]) (23.26) y = C1e x + ::: + Cnenx(23:28)! C1 ::: Cn { .4#$' $ $ F( 3 90.0000011274". (23.28) (23.26) 3 "' ##, % " $ #( j %## p()##(.
4 ( (23.26) (' #//0 ai $' ". ('( /.F ', '$' 3 80 ## e(+i)x = ex(cos x + sin x) (' : y = ex cos x y~ = ex sin x: & 80 (' /#0 y y~ #3 (23.26) (' #//0. # #3( ##( #( (23:280), .( $'. 23.10. * 1 ::: n ! () , aj (23.26) . *, , 1 ::: m { , = j :=m+1 im+1 :::=s is(2(s ; m) = n ; m):/! ) (23.26) ( y1 = e x ::: ym = em x1(23:29)ym+1 = em x cos m+1x ::: ys = esx cos sxy~m+1 = em x sin m+1x ::: y~s = esx sin sx ) (23.26) +1+1y=mXj =1Cj ej x +sXk=m+1CCk ekx cos kx + Dk ekx sin k x] ! Ck Dk { .4#$' $ , /#0 (23.29) (23.26) (3 80) "$ ( $ " $# Ca b] (3 90).J, (' % #//0 aj (23.26) %## () # = + i ##-3( #' = ; i:275 23.5. @( ". y000 ; y00 + y0 ; y = 0:& %## ():p() 3 ; 2 + ; 1 = 0:$ ' 3, " '24 1 = 1p() 2( ; 1) + ( ; 1) = (2 + 1)( ; 1) = 0 , = i:23=#, # %## $.
& 23.10 . /' ( " ' y1 = ex y2 = cos x y~2 = sin x $ ". % $ /y = yo:o: = C1ex + C2 cos x + C3 sin x: "( @ , #' = 0 %## p() $ r 1 p(0) = p0 (0) = ::: = p(r;1)(0) = 0 p(r)(0) 6= 0:(23:30)$ $', p() n $% #(1 ::: n (n { ' p()), #'r = 1: # # $ . p().J p() / ,(0(r;1)0) ( ; )r;1+p() = p(0) + p (1!0) ( ; 0) + ::: + p(r ; (1)!0276(r)(n)+ p r(!0) ( ; 0)r + ::: + p n(!0) ( ; 0)n(( 3 ), (23.30), = 0 { #' # r, p() p() = ( ; 0)r g()(23:300) g() { n;r #(, g(0) 6= 0: , ": p() (23:300), g(0) 6= 0 = 0 { #' # r g(0): /'( ( #%#( %## () #'# '% 3(.dn + a dn;1 + ::: + a { //0'( 120) ? L dxn;1 n;10ndx #//0 a /idk (exp()) (8 2 CI 8x 2 k 0):dk4(', (23.27) L(xk ex) (23:31)L(ex) exp():4//0 3 , dk =dk L % # "# //0( x /#0 ex, " '01dk Cexp()] dk L(ex) L @ dk exA L(xk ex):dkdkdk,# "$, 3 (23.31).130) ' = 0 { #' # r %## p() (23.26) #//0 ai: , r /#0(y1 = e x y2 = xe x ::: yr = xr;1e x(23:32)000( $ " $# Ca6 b] (23.26).277".
' k { " ' , . k < r. & 120 32 k3kddL(xk ex)= dk (ex ())= 4 dk (( ; 0)r W (x ))5 = W (x ) exg() (. (23:300)). =000d (( ; )r W (x )) r( ; )r;1W (x ) + ( ; )r @W 000d@ ( ; 0)r;1W1(x )6d2 (( ; )r W (x )) (r ; 1)( ; )r;2W (x ) + ( ; )r;1 @W1 0010d2@r;2 ( ; 0) W2(x )6::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::dk (( ; )r W (x )) ( ; )r;k W (x ) r > k:00kdk 3 = 0, " '2 k3dL(xk e x) 4 dk (( ; 0)r W (x ))5 0:=F $, /#0 (23.32) (23.26). F /#0 ( $ " $# Ca6 b] (.3 110). &( 130 #$.? = 0 = i { ##( #' # r () (' #//0 aj , (23.32) (' , 2r ( $% ('% (y1 = ex cos x y2 = xex cos x ::: yr = xr;1ex cos x(23:33)y~1 = ex sin x y~2 = xex sin x ::: y~r = xr;1 ex sin x:=$ /# .% 3( # .( /'( ( (23.26) (' #//0aj .27800 ! 23.1.1) (23.26) %## (), $ (23.26) $ dk =dxk k (k = 0 n).2) @( # = j %## () % #.3) B3 (' # = 0 # r r ( $% ( e x xe x ::: xr;1e x:4) B3( ##-3% #( = i # l 2l ( $% (ex cos x xex cos x ::: xl;1ex cos x000ex sin x xex sin x ::: xl;1ex sin x:5) "Z ( $ .
/' ( (23.26), . $ n /#0( (n { # (23.26)).". (23.26) y = C1y1(x) + ::: + Cnyn(x) y1(x) ::: yn(x) { 23.1 /' (, C1 ::: Cn { $' . 23.6. @( ". yV ; yIV + 18y000 ; 18y00 + 81y0 ; 81y = 0:& %## (), % # % #:5 ;4+183 ;182 +81;81 = 0 , 4(;1)+182 (;1)+81(;1) = 0( ; 1)(4 + 182 + 81) = 0 , ( ; 1)(2 + 9)2 = 0 ,21)22( ; 1)( ; 3i) ( + 3i) = 0 , 4 1 ==13i (#(# 2):23& 23.1 ( $ , . #3 # :1 = 1 : y1 = e x = ex23 = 3i : y2 = cos 3x y3 = x cos 3x y~2 = sin 3x y~3 = x sin 3x:2791&', ".
% y = C1ex + C2 cos 3x + C3x cos 3x + C4 sin 3x + C5x sin 3x == C1ex + (C2 + xC3) cos 3x + (C4 + xC5) sin 3x: "( #. $ " 4 (23.22) $#Ca6 b] #//0 ai(x) ' h(x) " $3 y = y::(x) $( . ## ( , ". /y:: = y:: + y:: yo:o: { ". . Ly = 0: 4 . " (23.22), #( , # #//0 .140) ' (23.22) #//0 ai(x) ' h(x) $# Ca6 b] ' z = z (x s) { . Lz = 0 .' z (s s) = zx0 (s s) = ::: = zx(n;2)(s s) = 0 zx(n;1)(s s) = 1(23:34) " /# $ s 2 Ca6 b].
, (23.22) ' y(x0) = y0 (x0) = ::: = y(n;1) (x0) = 0 3 "' $ Zxy(x) = z (x s)h(s)ds (x0 x s 2 Ca6 b]):(23:35)x0". @( $ y0 (x) ::: y(n)(x) /#0 (23.35),'$' /(01d @Zx g(x s)dsA = g(x x) + Zx g0 (x s)ds:xdx xx00280& '% ( (23.34), " '0y (x) = z (x x)h(x) + Z zx0 (x s)h(s)ds = Z zx0 (x s)h(s)ds6xxx0x0ZZ00000y (x) = zx(x x)h(x) + zx (x s)h(s)ds = zx00(x s)h(s)ds6x0x0:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZxZx(n;1)(n;2)(n;1)y (x) = zx (x x)h(x) + zx (x s)h(s)ds = zx(n;1)(x s)h(s)ds6xxx0x0ZZ(n);y (x) = zx (x x)h(x) + zx (x s)h(s)ds = h(x) + zx(n)(x s)h(s)ds:(n)x(n 1)x0x0&',x23Zx (n)Zx (n;1)45Ly(x) h(x) + zx (x s)h(s)ds + an;1(x) zx (x s)h(s)ds+x0x0Zx 0Zx+::: + a1(x) zx(x s)h(s)ds + a0(x) zx(x s)h(s)ds x0x023Zx 4h(x) + (Lz (x s))h(s)ds5 h(x)x0# ## Lz (x s) 0: ,# "$, /#0 (23.35) Ly = h(x).
=$ % $% /#0 (23.35) , ' . A3 140 #$. 23.7. @( ". Lz y00 + !2y = h(x)(23:36) ! 6= 0 { , h(x) { $' $#Ca6 b] /#0. ". . Lz = z 00 + !2 z = 0: ,# ## %## 2 + !2 = 0 $% ##-3% # = i! ". z = C1 cos !x + C2 sin !x:281 ' z (s) = 0 z 0(s) = 1 (s 2 Ca6 b] { ):I '(C1 cos !s + C2 sin !s = 0;C1! sin !s + C2! cos !s = 16 ,010 1 0 1cos!ssin!s, @ ;! sin !s ! cos !s A @ CC1 A = @ 01 A ,20 1 01;1 0 1Ccos!ssin!s, @ C12 A = @ ;! sin !s ! cos !s A @ 01 A =010 11!cos!s;sin!s= ! @ ! sin !s cos !s A @ 01 A :=#, C1 = ; sin!!s C2 = cos!!s , $z = z (x s) = !1 C; sin !s cos !x + cos !s sin !x] = !1 sin !(x ; s):&', (23.36) Zx1y = y:: = ! sin !(x ; s) h(s)ds (x0 2 Ca6 b])x0 ". $ /y = yo:: = y:: + y:: = C1 cos !x + C2 sin !x + y:: =Zx1= C1 cos !x + C2 sin !x + ! sin !(x ; s) h(s)ds:x( ' # (23.22) .' # $ .
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