1. Интегралы ФНП Диф_ур (853736), страница 20
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H (24.1) $ y = '(t) ##(-"' $ B (24.2) (: $ B ' y(t0) = '(t0)).#$ 24.2. 8#-/#0y = y(t C ) fy1(t C1 ::: Cn) ::: yn(t C1 ::: Cn)g$. n $'% % C1 ::: Cn $ ) (24.1), . ":1) "% % $% % C1 ::: Cn /#0y(t C ) (24.1) ##-"' $# Ca b]62) ## " " ' # (t0 y) 2 D (D { "' (24.1)), . $ C1 = C1 ::: Cn =296Cn % Cj #, /#0 y(t C ) = fy1(t C1 ::: Cn) :::yn(t C1 ::: Cn)g $ B (24.2) '(#( (t0 y).A 2) $, "# (8>y1(t0 C1 ::: Cn) = y1><y(t0 C ) = y , > (24:3)>: yn(t0 C1 ::: Cn) = yn(' $% C1 ::: Cn) % " .J, ". " , "#$ "# ( C = C . F " ", , , " D $ B. 24.1. #$', #-/#0y fy1(t) y2(t)g = fC2eC t =2 CC1 te;C t =2g(24:4)2 ".
8< y_1 = y12 y2(24:5): y_2 = t;1y2 ; y1y22 :12128 $ # #-/#0 (24.4):y_1 = (C2eC t =2) = C2eC t =2tC1 = C1C2eC t =2t! C C C111 ;C t =2;Ct=2;Ct=2+ te(;C1t) = C e;C t =2(1 ; C1 t2):=C ey_2 = C te2228 (24.5):y12 y2 = C22eC t CC1 te;C t =2 = C1C2teC t =2 t;1y2 ; y1y22 =2 C !2;1 C1 ;C t =2Ct=2= t C te; C2 e C1 t2e;C t = CC1 e;C t =2(1 ; C1t2):222& ( $ $% , , # #-/#0 (24.4) (24.5) 3. F $, #-/#0 (24.4)2971121211122211221122121122122 (24.5) " $# Ca b] 3.
# t = 0: ' $' # (t0 y10 y20) 2 D $ $" D = f(t y) : t 6= 0 y1 y2 2 Rg (24.5).#3, "# ( (24.3) ,8 (24.3) " C2eC t =2 = y1C1 t e;C t =2 = y:2C2 0A3 , (, y1 y2C1t0 = y1 y2 , C1 = C1 = t :0 , C2 = C2 = y1e;C t =2 = y1e;(y y =t )t =2 = y1 e;t y y =2:,# "$, #-/#0 y(t) = fC2eC t =2 C1C2; teC t =2g (24.5) ' y1(t0) = y1y2(t0) = y2: F $, #-/#0 (24.4) ". (24.5).".( ( (24.1) # 3,## # : ' ".
(24.1), $ F (t y C ) = 0 W(t y) = 0 , $ #% ". ## /#0, $ .21 021 021 01 20200 1 212112$ $ //0'% ( # , ' # $# , " //0 ( # //0' #. %. 24.2. ' (y_1 = y2 y_2 = ;y1 :(24:6)2984//0 (24.6) t6 yU1 = y_2: $ y_2 $ ;y1 '$ (24.6). 8 $' yU1 = ;y1 , yU1 + y1 = 0 # ' ( /#0 y1 = y1(t): ? %## 2 + 1 = 0 $% %# 12 = i: J, ". y1 = C1 cos t + C2 sin t:#'# y2 = y_1 (. (24.6)), # (24.6) / y2 = ;C1 sin t+C2 cos t:,# "$,y = fC1 cos t + C2 sin t6 ;C1 sin t + C2 cos tg: 24.3. ' (x_ = y2 + sin t y_ = 21 xy :(24:7)4//0 (24.7), " 'xU = 2yy_ + cos t: =$ (24.7) % 2yy_ = x .
8 $' " 'xU = x + cos t , xU ; x = cos t:F ( #. ? ". , , ". " ' x = C1et + C2e;t ; 12 cos t:=$ (24.7) %y2 = x_ ; sin t = C1et ; C2e;t ; 21 sin t:&',8>t;t 1>< x = C1e + C2e ; 2 cos t>1>: y2 = C1et ; C2e;t ; sin t:2299J, $ .8 " ( % #"0(. &' , " %. "$ % # //0' ' #( /#0 # ( . ? # , , ( #"0( ( . % #"0(, $ % , 3 ( ( . 24.4. ' (x_ = y y_ = x:(24:8)F 3 " " ' # .
# " ( % #"0(. &# " (24.8), #"0 d(x + y) = x + y , d(x + y) = dtdtx+y# x + y = C1et: 8 $ (24.8) , . #"0 d(x ; y) = y ; x , d(x ; y) = ;dtdtx;y$ #( % x ; y = C2e;t: =#, "#%x + y = C1etx ; y = C2e;t$ #% # % (24.8):x = 21 (C1et + C2e;t ) y = 12 (C1et ; C2e;t):& W(t y1 ::: yn) = C W { # /#0, C {, $ (24.1), # y1 = y1(t) ::: yn = yn (t) ". 3003. H (24.1) $ # /#0 W(t y1 ::: yn), % . $ % y = y(t) (24.1).V, #"0 (24.1) $ ' ( ( . ? ( n % #"0(, n % 8>>< W1(t y1 ::: yn) = C1 (24:9)>>: Wn(t y1 ::: yn) = Cn:? ' @ W 1 ::: @ W1 @y1@yn @ Wi 4 = @y @ Wnj@Wn @y1 ::: @yn " Q D (24.9) $ ".( (Q) (24.1).
F $, "#% ( (24.9) $ ". //0'( (24.1) ( /. J, ' $ /#0( W1 ::: Wn: ? #" = j@ Wi=@yj j " Q /#0( W1 ::: Wn " /#0' $( Q: 8 ( (24.9) 3 "'$ ' y1 = y1(t) ::: yn = yn(t) $. ( -(" '** ,("+ !? //0'( (24.1) ' f (t y) ( y, . ( $ (( ( //0'% (. ,# 8>dy1 = a (t)y + ::: + a (t)y + b (t)>>1111nn1>< dt (24:10)>>> dyn = a (t)y + ::: + a (t)y + b (t)>:n11nnnndt301 aij (t) bj (t) { $ /#0, yj = yj (t) { $ /#0(i j = 1 n). '$' #- "$, (24.10) 3 $' .( ##( /:dy = A(t)y + b(t)(24:11)dt010101yb(t)a(t):::a(t)1nB 1 CB 1 CB 11Cy = BB@ CCA b(t) = BB@ CCA A(t) = BB@ CCA :ynbn(t)an1(t):::ann(t)2 " '$' . $' (24.11).
# $( #-/#0 y = y(t) ( $' 0 A(t)) $ (24.11). ,#"$, (24.11) { //0' n- #.8#-/#0 b(t) $ ' (24.11).? b(t) 0 (.. # b1(t) ::: bn(t) 0), (24.11) $ 6 (.. b(t) 6 0) (24.11) $ ( (. ? (24.11) "' ', . ( z_ = A(t)z:d ; A(t) $ $' (24.11) ## L dt#: Ly = b(t). =$ ( . I "$'$ Cnk C6 b] n-% #-/#0( y(t) = fy1(t) :::yn(t)g, % $# Ca b] $ y_ (t) :::y(k) (t) k- # # ' ( # n Cnk Ca6 b] # , $ ## , ##% #-/#0% ').= . 3.1) ? 0 A(t) $# Ca b] (..
aij (t) Ca b]), L ( $ Cn1Ca b] CnCa b] % $#Ca b] #-/#0(:L : Cn1Ca b] ! CnCa b]:2) L , ..L(C1y(t) + C2z (t)) = C1Ly(t) + C2Lz (t)302 $'% C1 C2 $'% y(t) z (t) Cn1Ca b]: ( # $ , //0#' /#0 3 0, ( # $ , dtd (;A(t)) ( , $ ( % L = dtd ++ (;A(t)).? $ Y0 "$' ( ( ( Ly y_ ; A(t)y = 0 c ( 0( A(t), $ ( 1 2 $ 3 #, Y0 { ( .B# #% //0'% (, "', ## $' Y0 ## /#0( "$ "$.
2 , ' Y0 #' ( $ ." ' (t0 y) = (t0 y1 ::: yn ) { /#( ( # Rn+1 . ' $dy = A(t)y + b(t) y(t ) = y :(24:12)0dt= . 3. 24.1. (24.12) A(t) b(t) Ca b], (t0 y) 2 Rn+1 , $) (24.12) )y = y(t): 0 ) Ca b].,# "$, (( //0'% ( $' '( $ "': .
, ' //0'( . 8 (% #. 24.5. #$', $ B8< y_1 = y12y2 y1(0) = y1(24:13): y_2 = 0y2(0) = 1303 ( % t y1 y2) , % t 2 (;1 +1) y 6= 0.=$ (24.13) %, y2 = y2(t) 1: " y_1 = y12 y1(0) = y1:$ $' , " 'dy1 = dt6 Zy dy1 = Zt dt , ; 1 + 1 = t , y = y1 :12y12y1 y11 ; y1t0y y111=#, $ B (24.13) . :y = fy1 y2g = fy1(1 ; y1 t);1 1g:(24:14)F , # ## B. # $ t = y1;1 (y1 6= 0), ## " " ' $ y1 6= 0, (24.14) 3 .' ( ;1 < t < +1, # ## $ # t = y1 .1% - F ' $ #% //0'% (.
@ #-/#0 "". . "$.#$ 24.3. & #-/#0(y1(t) = fy11(t) ::: yn1(t)g ::: yn(t) = fy1n(t) ::: ynn(t)g$ ( $( $# Ca b], . 1 ::: n, , #, % t 2 Ca b] 3y (t) + ::: + nyn(t) 0:3041 1(24:15)? 3 3 (24.15), 1 ::: n { , '# , # 1 = 2 = ::: = n = 0 #/#0( y1(t) ::: yn(t) $ ( $( $# Ca b].J, # 3 (24.15) # n # 38101 010>y(t):::y(t)0y(t)+:::+y(t)01nn 1n>CC BB 1 CC BBCC< 1 11BB 11BCBCBCA :=,>@A@A @>: 1yn1(t) + ::: + nynn(t) 0yn1(t):::ynn(t)0n(24:16)#$ 24.4. ' y11(t):::y1n(t) W (t) W Cy1 (t) ::: yn(t)] = det(y1(t) ::: yn(t)) = y (t):::y (t) n1nn"0 # #-/#0 y1(t) ::: yn(t) $ 8# ( #) #/#0( y1(t) ::: yn(t): 24.2 (B4#$ C$ $DD 6$$$).
- y1(t) ::: yn(t) Ca b], W (t) W Cy1 (t) ::: yn(t)] ( , .. W (t) 0 (8t 2 Ca b]).". ,# ## #-/#0 y1(t) ::: yn(t) ($ $# Ca b], . 1 ::: n, , #, 3 (24.15). @ $, # ( (24.16) "#% ( % t 2 Ca b] ' ( 1::: n) 6= (0 ::: 0).
F $3 ' , # ' (24.16) ". ' % t 2 Ca b]. $', #$( ' # /#0( y1(t) ::: yn(t). , #$." # 24.1. ? # W (t) W Cy1(t) ::: yn(t)] ". ' % " ( # t = t0 2 Ca b], #/#0( y1(t) ::: yn(t) ( $ $# Ca b] (, #, $#).B# # , $ 3 W Cy1(t) ::: yn(t)] 0 . , #-/#0( y1(t) ::: yn(t) ( $305 $# Ca b]: @, #-/#00 10 21ty1(t) = @ t A y2(t) = @ tt2 A ( $ " $# Ca b] (#3 !), % # 2 W (t) = tt tt2 3 $# Ca b]. # ( ( dy = A(t)y(24:17)dt ( $# Ca b] 0( 24.2 3 "'. 24.3. * y1 (t) ::: yn(t) { ) (24.17) Ca b] A(t). /! (:1) ) y1(t) ::: yn(t) Ca b] ! !, ! W (t) W Cy1(t) ::: yn(t)] e Ca b]B2) ) y1 (t) ::: yn(t) Ca b] ! !, ! W (t) W Cy1 (t) ::: yn(t)] ( Ca b].4#3, , 3 1). 4' ,# ## W (t) 6= 0 % " ( # t = t0 $# Ca b], $ 24.1 #, /#0 y1(t) ::: yn(t) ( $ $# Ca b].
4#3 "%'.' y1(t) ::: yn(t) (24.17) ( $ $# Ca b]. 3, . # t = t0 2 Ca b]#, W (t0) = 0. , "0 y1(t0) ::: yn(t0) W (t0) ( $, .. . 1 ::: n, , # y (t0) + ::: + nyn(t0) = 0:1 1(24:18) #-/#0 '(t) = 1y1 (t) + ::: + nyn(t): 8 ( Y0 ( ( (24.17) /#0 #$( //0'( .306=$ (24.18) , ' '(t0) = 0: @ # 3 ' ' y = y(t) 0 (24.17). 8 #-/#0 y 0 y = '(t) $# Ca b],.. '(t) 0 (8t 2 Ca b]): &',y (t) + ::: + nyn (t) 0 (8t 2 Ca b]):#'# $' 1 ::: n , $, y1(t) ::: yn(t) ( $ $# Ca b], 3 "'. J, W (t0 ) = 0 3, W (t) 6= 0 % t 2 Ca b]. , #$.=$ ( # .
( # W (t) ( y1(t) ::: yn(t) ( //0'( (24.17) ( $# Ca b] 0( A(t):3) # W (t) ". ' % " ( #t = t0 $# Ca b], W (t) 0 (8t 2 Ca b])64) # W (t) % " ( # t = t0$# Ca b], $# Ca b].F ( , '$' /( E1 1Rt SpA(s)dsW Cy1 (t) ::: yn(t)] = W Cy1 (t0) ::: yn(t0)]e t y1(t) ::: yn(t) { ( (24.17) ( $# Ca b] 0( A(t), t = t0 { $' /# # $#.
H$ SpA(t) "$ 0 A(t),.. % aii (t), .% ( :SpA(t) a11(t) + ::: + ann(t):0 "( (24.17). ' #n. 8 . 3 .#$ 24.5. 20W(t) = (y1 (t) ::: yn(t))307(24:19)"0 #( n ( $% $# Ca b]( y1(t) ::: yn(t) (24.17), $ ) (24.17) ( $# Ca b]).J, #'# #3( "0 yj (t) 0 W(t) (24.17), /' 0 ( //0' _ t) = A(t)W(t) , (y_1(t) ::: y_ n(t)) = A(t)(y1(t) ::: yn(t)) (24:20)W(# # n # y_1(t) = A(t)y1(t) ::: y_ n(t) = A(t)yn(t):& .' /'( 0 ( 3 $' ". ( (24.17). & #3 .3. 24.4. " (24.17) Ca b] A(t) ( ( () ) W(t).".
' F = (f1 ::: fn) { $' 0 det F 6= 0: 8$' # '% # "0fj ( 0 n $ B:y_ = A(t)y y(t0) = f16 :::6 y_ = A(t)y y(t0) = fn(24:21) t0 { $' /# # $# Ca b]. ,# ## 0 A(t) $# Ca b], $ (24.21) y = y1(t) ::: y = yn(t) $# Ca b] (. 24.1). & $ % ( 0 (24.19). /'( 0( ( ( $# Ca b]) ( //0'( (24.17). 4(', #3( "0 (24.17) ( $B (24.
21)) det W(t0) = det(y1 (t0) ::: yn(t0)) = det(f1 ::: fn) = det F 6= 0( " 0 F ). J, "0 0 W(t) ($ $# Ca b] (. 24.1). , #$.308 24.5. * (24.17) A(t) Ca b]. /! ) y = yo:o:(t) = W(t)c c1y1(t) + ::: + cnyn(t)(24:22)! W(t) = (y1(t) ::: yn(t)) { ( Ca b]) ) (24.17), c = fc1 ::: cng { .". 4//0 (24.22) t , , _ t) = A(t)W(t), " '0 W(t) W(_ t)c A(t)W(t)c A(t)y(t)y_ (t) = W(.. #-/#0 (24.22), (24.17) " # c = fc1 ::: cng. ? ' y_ = A(t)yy(t0) = y { $' $ B (24.17) (t0 2 Ca b],y 2 Rn ), /#0 (24.22) ' y(t0) = y(, W(t0)c = y , c = c = W;1 (t0)y $ #-/#0 (24.22), c = W;1 (t0)y #$( $ B.
&', (24.22) { ". (24.17). , #$.' W(t) `(t) { /' 0 ( //0'( (24.17). B# $ 3 "(? &$' ' . 3.5) ? W(t) `(t) { /' ( $# Ca b]) 0 ( ( (24.17) ( $#Ca b] 0( A(t), . 0 #, `(t) = W(t)M . det M 6= 0:4(', 24.5 #3( "0 j (t) 0`(t), ' (24.17), j (t) = W(t)mj , mj = fm1j ::: mnj g { #( ( #. 0`(t) 3 "' $ / `(t) = W(t)(m1 ::: mn) = W(t)M:20 M 3, # ## " det M = 0, det `(t) == det W(t) det M = 0 3 "' /'0 `(t):3090 "( ' //0' Ly dy(24:23)dt ; A(t)y = b(t): 24.6. * (24.23) A(t) b(t) Ca b] W(t) { ) Lz = 0. /! ) (24.23) Zt ;1y = yo (t) = W(t)c + W(t) W (s)b(s)ds(24:24)t0! c = ( 1 ::: cn) { , t = t0 { () Ca b].".
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