1. Интегралы ФНП Диф_ур (853736), страница 15
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# # #, # "$, "$' " .(. ? #( " Q % x y . $ @ W=@x@ W=@y Q % " $ % ". ' ( ", # , $ (21:51) # , " .( ( #% W(x y C ) = 0.F /# #$ //0'( .8 ( #% W y ; Cx ; (C ) = 0:J' $ @ W=@y = 1 6= 0 " $231@ W=@x @ W=@y "( /#( $#( "G 3.( "' D /#0 (u) (# ## (u)G# D). J, #, ( $ (21:51) # a C , " " .(. 8 (21:51) (y ; Cx ; (C ) = 0 , ( y = Cx + (C );x ; 0 (C ) = 0x = ;0 (C ):=# C $ ( p $ (y = px + (p)6x = ;0 (p) 3 # , $ F (x y) = 0:F $, # y = xp + (p) x = ;0 (p) " .( ( '% #% y = xC + (C ) B(21.45).
//0' . 4 % ' . '$ '$' 3. , . 21.18. @( ".( //0' 323y+6xdy(21:52)x dx = 2y2 + 3x2y : y (21.52) //0'( /(2xy2 + 3x3)dy = (3y3 + 6x2y)dx, /#0 M = 2xy2 +3x3 N = 3y3 +6x2y k = 3: J, (21.51) { . '$' #( % (, " x3. 8 $' "$(2 !2 32 !33yyy42545x + 3 dy = 3 x + 6 x dx:4 $ y=x = z ( dy=dx = z + xdz=dx), 333z+6zdzzdzz + x dx = 2z 2 + 3 , x dx = 2z 2++3z3 :232$ , 2z 2 + 3 dz = dx , Z " 1 + z # dx = ln jxj + ln C ,1z 3 + 3zxz z2 + 3p, ln jz j + 12 ln(z 2 + 3) = ln C1jxj , jz j z 2 + 3 = C1jxj ,p, z z 2 + 3 = Cx (C = C1):vu2uyyt#'( : x 3 + x2 = Cx: 21.19. ' $ Bdy = ;1 y(2) = 0:(y2 + 2y ; x + 1) dx(21:53)& $' $ y + 1 = z ( dy=dx = dz=dx), (21.53) dz = ;16 dx = x ; z 2:(z 2 ; x) dx(21:54)dz ( ' x = x(z ): ,# ## ". .
x0 = x x = Cez ( !), ' , 3 ' ". (21.54) x = C (z )ez (z ) {# $ /#0. = (z )ez (21.54),C 0(z )ez + C (z )ez = C (z )ez ; z 2 , C 0 (z )ez = ;z 2 ,Z2 ;z0, C (z ) = ;z e , C (z ) = ; z 2e;z dz:( " $ 6 " 'C (z ) = (z 2 + 2z + 2)e;z + C $ ".( (21.54) x = (z 2 + 2z + 2) + Cez (z = y + 1): $' = 2 y = 0 ( z = 1), 2 = 5 + Ce , C = ;3e;1 233 $x = (z 2 + 2z + 2) ; 3ez;1 = y2 + 4y + 5 ; 3ey : 21.20.
@( ' # y0 ; y = xy2(21:55)%. $ # M (06 1):A (21.55) I. " y2, " 'y;2 y0 ; y;1 = x:& $ y;1 = z ( y;2 y0 = ;z 0), ( ;z 0 ; z = x , z 0 = ;z ; x: 0 $'% % (. $."E( "), (, Z RZR z = e; dx C ; e dxxdx = e;x C ; exxdx == e;x CC ; xex + ex] = Ce;x ; x + 1:&', y = (Ce;x ; x + 1);1 : T ( ' # , %. $ # M (06 1), 3 x = 0 y = 1:I ' 1 = (C + 1);1 , C = 0: J, # '# y = (1 ; x);1 .. "(. 21.21.
@( ".( e;2y dx ; (1 + 2xe;2y )dy = 0:(21:56)J' M = e;2y N = ;(1 + 2xe;2y ): =@M = ;2e;2y @N = ;2e;2y @N @M :@y@x@x @yJ (21.56) % //0% (. # ( % //0%).@% /#0 u = u(x y) # , du e;2y dx ; (1 + 2xe;2y )dy:234, u(x y) 3 ' @u = e;2y @u = ;1 ; 2xe;2y :@x@y= , (, u = e;2y x + '(y) '(y) { # $ /#0. ," ';2e;2y x + '0(y) = ;1 ; 2xe;2y , '0 (y) = ;1 , '(y) = ;y + C1:&', u = xe;2y ; y + C1, $ ".( (21.56) xe;2y ; y = C: 21.22. ' y = xy0 ; y02(21:57) #$' " .A (21.57) B.
='$ # (, $% ' $(, p = dy=dx: , (21.56) y = xp ; p2 :@ ( . ' x y p. 4//0 . , (x 6= const)dy = p + x dp ; 2p dp ,dy = pdx + xdp ; 2pdp , dxdxdx2" x = 2px=2pdp, p = p + (x ; 2p) dx , 4 dp=dx = 0 , p = C = const($' , x = C (21.57)).=#, (21.57) , ((y = xp ; p2y = xp ; p2x = 2p6p = C:=# $' p, 2xy = 4 y = x ; 2 ( { $' ).235= ' # y = x2=4 " .( ( '% #% y = Cx ; C 2: 4(', # C $ ((.
(21.51))(y = Cx ; C 20 = x ; 2C 6 y = x2=4: #'# y = x2=4 " . ( '% #%, " (21.57)(. 20.1). :224 y = x =4y = Cx ; C 26y = x2=4 { " . 1. "! 3 ! 3 ? 3 "& y ? "$!! , ! !? @ 9! $?2. 3 ! <! (<) ! 3 ?dy = f (x) dy = f (y)?3. <! ! dxdx4. 3" ! "! ! !! "? 4 $ ! "&? = "& !3! <! & !? & "$3?5. "! " ($: A-")? $? ? ".6. 3" ! "! "? <! !? =# 1 ! <?!ady1x + b1 y + c17.
<! ! dx = f a x + b y + c ?2228. 3" ! "! "? ! & ! <! $ <?4 $ $ 1 " $ " ! 3 !? @ "93" &" "3" &" ! <?9. 4 $ 3 ! C#? ? " 9 <! 3 !.10. ! "! ! I? <!?23611. ! "! ! "& &? "! 9 & ? 9&" $" , $ 3 !!! "& &.12. "$!! 9 ! "& &. ? .13. @ #3? $"& "& # !3? "$!! #3,! 3 x?14. 4 $ $! ! ! 3"& F (x y ) = 0 F (y y ) = 0? ? ".15. $3 9 " $ ! ! y = f (x y )? ? < !.000237 22'** ,( + ! !(0" '!. & 0 /- &'/.+ !, '+A< ) ', " = !? //0' F (x y y0 ::: y(n)) = 0(22:1)3 $ $( /#0 y = y(x) # (n > 1) $ )! .
H$ # /y(n) = f (x y y0 ::: y(n;1))(22:2)$( ' ( $( y(n) (x): B, # % # //0'( / d2y + a(x)dxdy + b(x)dx2 = 0 % # //0 $( /#0 . # # # #( #. ( # $ #% (. "( . @ # #. I/' # (22.2), % #$ 3 (22.1), $ ' ( $(.' D { "' /#0 f (x y y1 ::: yn;1)(D Rn+1 ): ? $ #3 "' (22.2).#$ 22.1. L#0 y = y(x) $ ) (22.2) $# Ca6 b], . :2381) % x 2 Ca6 b] # (x y(x) y0 (x) ::: y(n;1)(x)) 3" D62) /#0 y = y(x) //0 n $ $# Ca6 b] % x 2 Ca6 b] 3y(n) (x) f (x y(x) y0 (x) ::: y(n;1)(x)):; ( (a6 b) % Ca6 b) (a6 b] (. $ 20.1).
@, /#0y = 2 cos3x ; 3 sin 3x y00 = ;9y ( (a6 b) = (;16 +1) ( #( #$( /#0 .B# //0' #,/# y = y(x) $ '( #( (22.2). = ' # $ # "' R2 . H" ( ' # , $' ' . 8 # $' ' . 8 n{ # (n > 1) #% ( 3 "' n. =: '(x0 y0 y10 ::: yn0;1) { /# # " D: J # y = y(x) (22.2), . 'y(x0 ) = y0 y0 (x0) = y10 ::: y(n;1) (x0) = yn0;1(22:3)$ ( $( B) (22.2).@, # myU = F (x y y_ ) 3 . "( ( $# @' , $# 3 y = y(t), $' '# ' #, ' #' y_ (t0) = y10:8 , ## . ' #, ' (22.3), . "$.
22.1. * f (x y y1 ::: yn;1) @f ::: @f D: /! @f@y @y1@yn;1 (x0 y0 y10 ::: yn0;1) ( D, h > 0 , $) (22.2),(22.3) ) Cx0 ; h6 x0 + h] ) .239& ". #'( %# ( :. $' ' #( ( # # x = x0 (h > 0 { ).A ( ' . $ (22.2), (22.3). 23 ' #, , $ (22.2), (22.3) -# . $# Ca6 b] (x0 2 Ca6 b]):( ' # ". (22.2).#$ 22.2. E " ## y = '(x) (22.2) (..
#( $ B) $ ) (22.2). 3 ) " Q D $ /#0 y = W(x C1 ::: Cn) $. n$'% % C1 ::: Cn . . ":1) "% % $% % C1 ::: Cn /#0y = W(x C1 ::: Cn) (22.2) #$# Ca6 b]62) ## " " ' # (x0 y0 y10 ::: yn0;1) 2 Q . $ C1 = C10 ::: Cn = Cn0 % #, /#0 y = W(x C10 ::: Cn0) $ B (22.2), (22.3) ('( #(. 22.1. ', /#0 y = C1ex + C2 ". y00 = y0 " Q = R3 :8 $ y0 = C1ex y00 = C1ex , /#0y = (x C1 C2) = C1ex + C2 "% $% % C1 C2: ' ' (x0 y0 y10) { $'/# # R3 : , 3 (# $ C1 = C10 C2 = C20 %, #% y(x0 ) = y0 y0 (x0) = y10 :(22:4) y = C1ex + C2 y0 = C1ex C1ex + C2 = y0 C1ex = y10:00=$ % C1 = C10 = y10e;x $ C2 = C20 = y0 ; C1ex = y0 ; y10:24000=#, (22.4) , "' C1 =C10 C2 = C20. /#0 y = y10ex;x + y0 ; y10 y00 = y0 ' (22.4).F $, /#0 y = C1ex + C2 ". y00 = y0 " Q = R3 :0# $' ".
( /.#$ 22.3. &F (x y C1 ::: Cn) = 0(22:5)$ ! (22.2) " Q D ". ( Q) "$.; ( . ! 22.1. H" ', (22.5) ". (22.2), ' .:) //0' (22.5) n $ x, , y ' /#0 x (.. y = y(x))6") $ 8>F (x y C1 ::: Cn) = 0>>>< d F (x y C1 ::: Cn) = 0dx>: : :>n>d>: n F (x y C1 ::: Cn) = 0dx# ' C1 ::: Cn: ? " (22.2), (22.5) ". . 22.2 ', (22:6)(x ; C1)2 + (y ; C2)2 = a1 (a = const > 0) ". (y00 )2 = a(1 + y02 )3:4//0 (22.6) $ x , y = y(x): # (22.6),241 8>(x ; C1) + (y ; C2)y0 = 0>< 1 + y02 + (y ; C2)y00 = 0>1>: (x ; C1)2 + (y ; C2)2 = :a=$ % y ; C2 :02 + 102)y0(1+yy0y ; C2 = ; y00 x ; C1 = ;(y ; C2)y = y00 :, ' " ' (1 + y02 )2y02 + (1 + y02 )2 = 1 , (y00 )2 = a(1 + y02)3:(y00 )2(y00 )2a % //0' , $ (22.6) { ".( .,, -( & ? $' ##%-" "$( # % (22.1) ( (22.2)) 3, .
' 3 ' ' . # # 3 #. @3 ## (, # #% 3 "' 3 0 ". 3 ( # # , ( //0' y(n) = f (x)(22:7)# # " 3 3 #. ? /#0 f (x) $# Ca b] ". # % ' :Z;y= f (t1)dt1 + C1 x0 = 0 2 Ca6 b]6(n 1)x0242Zx Zt2;y= f (t1)dt1dt2 + C1x + C26(n 2)0 02Zx Zt3 Zt2x;y=f (t1)dt1dt2dt3 + C1 2 + C2x + C360 0 0: : :n;1n;2Zx Ztn Zt2y=::: f (t1 )dt1dt2:::dtn + C1 (nx; 1)! + C2 (nx; 2)! + ::: + Cn;1x + Cn0 00($' 3( $ 0 = 0 ).8( n{#( ( 3 ' /(n 3)Zx Ztnx0 x0Zt21Zx::: f (t1)dt1dt2:::dtn = (n ; 1)! f (t)(x ; t)n;1dt x0 2 Ca6 b]: (22:8)x0x0J, ' " $' .' % . 22.3.
' y00 = x + cos x:= ' $:2Zy0 = (x + cos x)dx = x2 + sin x + C1613Z 0 x2xA@y = 2 + sin x + C1 dx = 6 ; cos x + C1x + C2:F ' ". % . 22.4. ' y000 = e;x =2:2' :Zy00 = e;t21=2dt1 + C16x0xt0y = Z Z 2 e;t21=2dt1dt2 + C1x + C20 0243Zx Zt3 Zt22x2 =2;t1y=e dt1dt2dt3 + C1 2 + C2x + C3:0 0 0J' "$ (( ) # e;t21=2 3 $ /#0, " $' $ ( . / (22.8), #'( $':2Zx1x;t=22y = 2! e (x ; t) dt + C1 2 + C2x + C3:02,, &( F (x y(k) y(k+1) ::: y(n)) = 0(22:9) 1 k n ; 1: ? "$' $ $z (..
y(k) = z ), . :F (x z z 0 ::: z (n;k)) = 0:# ' $( /#0z = z (x) k 0 ' # % (22.9). ? # z = '(x C1 ::: Cn;k) { ". , ". % (22.9) ' y(k) = '(x C1 ::: Cn;k)# (22.7) # # $3 . 22.5. ' y(5) ; x1 y(4) = 0:2444 $ y(4) = z: , z 0;x;1z = 0 # $ %:dz = dx , Z dz = Z dx , ln jz j = ln jxj + ln C ,1z xzx, jz j = C1jxj , z = C1x , z = Cx (C = C1):J $' z y(4), y(4) = Cx # ' :2Cx3 + C x + C 600+y000 = CxC6y=112260y = Cx4 + C 1x2 + C 2x + C 36 y = Cx5 + C 1x3 + C 2x2 + C 3x + C4:24212062"$ $' #, ".
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