1. Интегралы ФНП Диф_ур (853736), страница 19
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$ 3, ( (23.22) #//0 ai 0'( ( ' 0h(x) = exCMn(x) cos x + Nm (x) sin x]282(23:37) { , Mn(x) Nm (x) { n m. J, /#0 f (x) = P1(x)e x + ::: + Pl (x)el x P1(x) ::: Pl(x) { , 1 ::: l { ( ". ##), $ ( #$ ) .? $' (23.37) cos x sin x $ # (./ (23.20)), (23.37) 3 ' #$ ## #//0. /#0 (23.37) " #3 $' !.
" ', (', = = + i " $' (23.37). F 3 ' (23.22).1 ! 23.2 (!# !#%!).' " ( Ly y(n) + an;1y(n;1) + ::: + a1y0 + a0y = h(x)(23:38) #//0 ai 2 R ' h(x), .( #$ (23.37). 4 ' .:1) ' #' $ = + i ( h(x) (23.38)62) #' $ = # %## p() n + an;1n;1 + ::: + a1 + a0 = 0() #' y = y:: = exCPk (x) cos x + Qk (x) sin x](23:39) Pk (x) Qk (x) { ( #//0) k = max(m6 n)63) #' $ = # #r %p## (), #' (23:40)y = y:: = xr exCPk (x) cos x + Qk (x) sin x]283 Pk (x) Qk (x) { ( #//0) k = max(m6 n)64) % #//0 '/#0 (23.39) ( (23.40)) (23.38), #' " # ex $ #//0 "% % #% xs cos x xs sin x $ ' ( "#% ( '% #//0.J, #' $ = # %## (), , (23.38) .2 " $' " .
,# " 3 ( % "% "% "#//0' (., , CI]). #3, ## " ##. , 23.2 '$ .( 0 $0.$F$ C6$F$$. ? ' h(x) (23.11) (( #"0( % $# Ca6 b] /#0(h1(x) ::: hm(x), ..h(x) 1 h1(x) + ::: + m hm (x) ( #"0y 1y1(x) + ::: + mym (x)( y1(x) ::: ym(x) ( Ly1 = h1(x) ::: Lym = hm (x) (23.11) ($' 1 ::: m "' ##).4(', ( L $ 3Ly1 h1(x) ::: Lym hm (x)# 3LC 1y1(x) + ::: + m ym (x)] 1Ly1(x) + ::: + m Lym(x) 1h1(x) + ::: + mhm (x): #'# .284 23.8.
@( ". Ly y00 + y = x2 + 1:(23:41),# ## ( (23.41) ex, = 0 #'# ( #, = 0. J, #'$ ( + i = 0: ^## 2 + 1 = 0 $% % # 12 = i #' $ + i = 0 o %, (23.41) #' y = y:: = Ax2 + Bx + C(23:42)(.. # 3 , ' (23.41)). 4 % #//0 A B C '(23.42) (23.41) $ #//0 o#% % x. 8 $ (23.42):y0 = 2Ax + B y00 = 2A:J, Ly:: 2A + Ax2 + Bx + C = x2 + 1: #//0 :x2 8>< A = 1x1 > B = 0, A = 1 B = 0 C = 1 ; 2A = ;1::x0 2A + C = 1 ( #//0 (23.42), ( #' (23.41): y = y:: = x2 ; 1:#'# # 12 = i %## () $, ". . yo:o: = C1 cos x + C2 sin x $ ".
% (23.41) $ / y = yo:: = C1 cos x+C2 sin x+x2;1: 23.9. 8' ". Ly y00 + y0 = x + 2:(23:43)@( ". . 00y + y0 = 0 , # ## $' # %## 2 + = 0 $: 1 = 0 2 = ;1: 285 yo:o = C1e0x + C2e;1x = C1 + C2 e;x: J( (23.43). #'# ( # ex #, #' $ = + i = 0: # %## # r = 1: & .3 23.2 (23.43) #' y = y:: = x(Ax + B ) Ax2 + Bx:(23:44)8 $ y0 = 2Ax + B y00 = 2A /#0 (23.44) (23.43), " '2A + 2Ax + B = x + 2: $' #//0 #% % x, x1 ( 2A = 1x0 2A + B = 2 , A = 1=2 B = 2 ; 2A = 1: A B (23.44), ( (23.43) y:: = 12 x2 + x $ ". $ /y = C1 + C2e;x + 12 x2 + x: 23.10. ' Ly y00 ; 3y0 + 2y = (x2 + x)e3x:(23:45)&#' $ ( = + i = 3, ### = 3, # .
^## 2 ; 3 + 2 = 0 $% ('% #:1 = 1 2 = 2. ,# ## #' $ = 3 #p %## , (23.45) #' (. .2 23.2):y = y:: = e3x(Ax2 + Bx + C ):(23:46)4 ( ' % #//0 A B C , ( $ /#0 (23.46):y0 = e3x(3Ax2 +3Bx+3C )+ e3x (2Ax+B ) = e3x(3Ax2 +3Bx+2Ax+3C +B )286y00 = e3x(9Ax2 + 9Bx + 6Ax + 9C + 3B ) + e3x(6Ax + 3B + 2A) == e3x(9Ax2 + 9Bx + 12Ax + 9C + 6B + 2A):&',Ly:: e3x(9Ax2 + 9Bx + 12Ax + 9C + 6B + 2A);;3e3x(3Ax2 + 3Bx + 2Ax + 3C + B ) + 2e3x(Ax2 + Bx + C ) (x2 + x)e3x:&#. $' #, 32Ax2 + 2Bx + 6Ax + 2C + 3B + 2A x2 + x:p #//0 :x2 8>< 2A = 1x1 > 2B + 6A = 1, A = 1=2 B = ;1 C = 1:0 :x 2C + 3B + 2A = 0 A, B C (23.46), y = y:: = e3x(x2=2 ; x +1): ".
. y # 1 = 1 2 = 2 %## . "$: yo:o = C1ex + C2e2x $". o (23.45) $ 1!y = yo:: = C1ex + C2e2x + e3x 2 x2 ; x + 1 : 23.11. ' ". Ly = y00 + y = 2 sin x ; 6 cos x + 2ex(23:47) ( ' # , . ' y(0) = y0 (0) = 0:J' ' #$ (23.39), $ '$' 23.2 '$. $"' (23.47) Ly1 y100 + y1 = 2ex(23:48)Ly2 y200 + y2 = 2 sin x ; 6 cos x287(23:49)# # 3 ' ". ? " ( y1 (x) y2(x) % (, , 0 $0, % (23.47) " ' y:: = y1(x)+ y2 (x): J( # (23.48).&#' $ = + i ( = 1# ## (23.48) #. ^## 2 + 1 = 0 $% # 12 = i: &#' $ , (23.48) #' (.
.2 23.2) y1 = Aex: (23.48), Aex + Aex 2ex , A = 1:J, y1(x) = ex:&#' $ ( (23.49) = i: # # r = 1 %## , (23.49) #' (..3 23.2):y2(x) = x(B sin x + C cos x):8 $y20 (x) = (B ; Cx) sin x + (C + Bx) cos xy200 (x) = ;2C sin x + 2B cos x ; Cx cos x ; Bx sin x (23.49), 3Ly2(x) ;2C sin x + 2B cos x ; Cx cos x ; Bx sin x + xB sin x + xC cos x 2 sin x ; 6 cos x;2C sin x + 2B cos x 2 sin x ; 6 cos x: $' #//0 sin x cos x, sin x ( ;2C = 2 , C = ;1 B = ;3:cos x 2B = ;6288J, (23.49) y2(x) = x(;3 sin x ; cos x):& 0 $0 (23.47) ( % ( y1(x) y2(x) ..y::(x) = y1(x) + y2(x) = ex ; x(3 sin x + cos x):o # 12 = i %## () ".
. Ly = 0 :yo:o (x) = C1 cos x + C2 sin x ". % (23.47):(23:50)y = y:: = C1 cos x + C2 sin x + ex ; x(3 sin x + cos x):@( ' ' # , . ' y(0) = y0 (0) = 0: 4//0 (23.50), (, y0 (x) = ;C1 sin x+C2 cos x+ex ;(3 sin x+cos x);x(3 sin x+cos x)0 : (23:51) (23.50) (23.51) ' y(0) = y0(0) = 0(C1 + 1 = 0 , C = ;1 C = 0:12C2 + 1 ; 1 = 0C', # ' # y = y(x) = ; cos x + ex ; x(3 sin x + cosx): #'( :) y = C1 cos x + C2 sin x + ex ; x(3 sin x + cos x) { ". 6") y = ; cos x + ex ; x(3 sin x + cosx) { # ' #.7. H (23.47) 3 ' .' Zxy = y(x) = sin(x ; s)(2 sin s ; 6 cos s + 2es)ds0(. $' (23.7)).289 1. ! n{ ! "! "?2. @ " 3" n{ !? 9?3. ! <! ! "& y "< !? 4 $ $ $ " ! "& 3"& "< !?4.
"! " " DaA b]? ? ".5. !!! 9&" DaA b]? ?$ $? # 3 3 ! 4?6. 9&" $" ! " < 3 ! n{ !?7. 9 " < 3 !?8. .!! < 3! n{ ! " ? ?$?9. @ 3! <? 4 $ ? ! ! !!! 9 < !?10. 3 < 3! n{ !?11. 9& < 3"& ?12. "$!! $ < 3 ! n{ ! 3"& !"& C#?13. 9 < 3 ! n{ ! 3" 1?14.
;< 7 ! 1". $3" ! ?15. 0# " e1x ::: e xn x x " 1o # 3 " e A e cos ixA e x sin ix :16. 4 $ e x xe x ::: xr e x (j = 1 m) ?17. $ "! &$? !?18. ! !3 # ! &$ ! <! 3 ! n{o ! !" 1?19. ! 3! < 3 ! $ "& "& &$ !?2< .20. ! ! "! $? 2< 9 $ <! ! $3 .njj290ijijj21. # "$3 $ < ! 3$ < !??# 1 3 ! ! !" 1.22. ! "& 3"& . 33!? ? .291 24 ( '** ,(" + !&#' % " //0'% ( $ . #% ' #0 20.
J, " "$ $ "#( t, ## .@, 8< F1(t y z y0 z 0) = 0: F2(t y z y0 z 0) = 0"$ % ( $ /#0y = y(t) z = z (t). ? $ (8>dy1 = f (t y ::: y )>>11n>< dt(24:10) >>dy>>: n = fn(t y1 ::: yn)dt $ (, $( ' % $%, ( ( '( /. ? # y = fy1 ::: yng f (t y) = ff1(t y) ::: fn(t y)g (24:10) 3 $' ## #:dy = f (t y):(24:1)dt,# / $ ( (24:10) $ #( /(. ? " '$' '(. "( . 2 + "' (24.1) $ "' ( f (t y), ..
3D = f(t y) : /#0 fj (t y) , j = 1 ng:292@, "' 8>dy pp>< 1 = y1 + t y2dt p>dy2>: dt = y1y2 3D = f(t y1 y2) : ;1 < t < +1 y1 0 y2 0:g8 ( (24.1) $ #-/#0y = fy1(t) ::: yn(t)g # t. 8 '( " ' #-/#0. $Z . .$ ( " #) #-/#0 # :y_ (t) = fy_1(t) ::: y_ n(t)g yU(t) = fyU1(t) ::: yUn(t)g :::y(k)(t) = fy1(k) (t) ::: yn(k)(t)g($' y_ = dtd y(t)), { 8b9>>Zb<Z=y(t)dt = >: y1(t)dt ::: yn (t)dt> :aaaZb,# "$, " //0' ' #-/#0 # , //0' ' #3 # ( /#0. 8#/#0 $ ( # t = t0 ( 3 ), ( # ( 3 ) #3 # ( /#0.
4 #-/#0( # % " //0:) (x(t) y(t)) = x_ (t) y_ (t)") ( (t)y(t)) = _ (t)y(t) + (t)y_ (t) ( (t) { # /#0),01x(t)) @ (t) A = x_ (t) (t) 2;(tx) (t) _ (t) ( (t) 6= 0 { # /#0),2930n1X) (x(t) y(t)) @ xj (t)yj (t)A = (x_ (t)y(t)) + (x(t)y_ (t)):j =1= #-/#0 " ( (, , //0 % 3 .@,0t1Zd B@ y(s)dsCA = y(s) = y(t):dt= 0#as=t b b Z Z y(t)dt ky(t)kdt a a0n11=2X2 ky(t)k = @ jyj (t)j A ky(t)k = max jyj (t)j:j =1j =1n? $ 0 A(t) (aij (t))n1 ' A_ (t) = (_aij (t))n1 , ## ', " ' /(A(t)y(t)) = A_ (t)y(t) + A(t)y_ (t)(A(t)C ) = A_ (t)C (C ; 0 #):( # (24.1). ' D { "' (24.1).#$ 24.1.
//0'% ( (24.1) $# Ca b] $ #-/#0y(t) = fy1(t) ::: yn(t)g " . (:1) (t y(t)) 2 D (8t 2 Ca b])62) #-/#0 y = y(t) //0 $# Ca b]63) % t 2 Ca b] 3dy(t) f (t y(t)):dt; 3#% (a b) (a b] Ca b) Ca +1) (;1 a] (;1 +1):? y = y(t) { (24.1) $# Ca b], 3 # (t y(t)) 2 Rn+1 , # t " $# Ca b], "$ Rn+1 # : F # $ ! 294 (24.1).
Rn % y1 ::: yn $ . #0 '( #( /$ $ ( (24.1). V, '( #( # $, ". 8' y = y(t) #3 " $' '( #( (24.1).2 $, //0'% ( $. H" ' ## , $' ' .' (t0 y1 ::: yn) (t0 y) { /# # " D: J, . %3 y = y(t) (24.1), . ' y(t0 ) = y $ $) (24.1).
? ## $ #:dy = f (t y) y(t ) = y:(24:2)0dt? #( , " % '% #% (24.1) ( , # % $ $ ' # (t0 y) (. . 24.1).;. 24.18 ## $ B (24.2) ? B# #% //0'% ( $ (( f (t y) ff1(t y1 ::: yn) ::: fn(t y1 ::: yn)g: 3$. * fj (t y1 ::: yn) f (t y) @fj =@y1 ::: @fj=@yn (j = 1 n) D Rn+1 .
/! 295(t0 y) (t0 y1 ::: yn ), ( D Ct0 ; h6 t0 + h] , $) (24.2) ) , ) ., #'( %#: . y = y(t) ' ( # # t = t0 (h > 0 { ). B , ( ( %#. % ($ (24.2) 3 ' 3 ' . , ., 3 "' . @, $ B8< y_1 = py1 y1(0) = 0: y_2 = y1y2 y2(0) = 0 : (y1 y2) (0 0) (y1 y2) = fy1(t) 0g y1(t) {/#0 81>< t2 t 0y1(t) = > 4: 0 t < 0:@ "Z , 1 B $' (: $ @f@y1 @y@ (py1) = 2p1y $ '( # (t y1 y2) = (0 0 0)).11B# #% (, $' #3 ". (, ". .
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