1. Интегралы ФНП Диф_ур (853736), страница 13
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20.6. ', " /#0 y = y(x) $ # ( x = 2 sin ty = 4 cos t (0 t < 2) $ B y0 = ;4xy;1 y(0) = 4.& , ' y(0) = 4 .. . $ t0 t # '# y = y(x) % $ # (064). F $ 3 ' $( . , , . : t0 = 0: ,' 3 (20.3).=0y y0 (x) = yt0 = ;4 sin t = ;2 tg t ;4xy;1 = ;4 2 sin t = ;2 tg t:x0t 2 cos t4 cos t200&', y0 (x) ;4x=y(x) 3 (20.3) . 20.7. ' y = y(x) '( $y0 = x2y + cos x y(0) = 1:(20:8)8' y00 (0): /#0 y = y(x) (20.8), 3y0 (x) x2y(x) + cos x: //0 3 x, "'y00 (x) 2xy(x) + x2y0 (x) ; sin x:3 $' x = 0 '$ ' y(0) = 1: $ $( y0 (0) % $ (20.8): y0 (0) = 0 1 + cos 0 = 1: &',y00 (0) = 2 0 y(0) + 0 y0 (0) ; sin 0 = 0: 20.8. @( F (x y) = 0 # 3 % 0% # (# #) '% #% y0 = x2 + y2 ; 4:@, 0( #( #( y = y(x) $# M (x0 y(x0 )) " . ( y0 (x0) = 0: ? y = y(x) ' # , 3y0 (x) = x2 + y2(x) ; 4 y0 (x) = 0 # x2 + y2 (x) = 4: &', 3 % 0% # "$ #3' x2 + y2 = 4: 20.9. &' //0' (#%ln xy = 1 + Cy(20:9) C { $' ().4//0 (20.9) (, y = y(x)), oy y ; xy0 = Cy0 y2 ; xyy0 = Cxy2y0 xy(1 + Cy)y0 = y2x y2( $' ## " .
#% (20.9)). H" ( //0' (201 (20.9), # $ (8 x>< ln = 1 + Cyy>: xy(1+ Cy)y0 = y2:I '!!xx02xy ln y y = y x ln y y0 = y: 20.10. @ /#0 y = x ln x , ', 3 "' $ By0 = ey (e ; y);1 y(e) = e: ' ey (e ; y);1 #M (e e) $ B ' y(e) = e .
1. ! "! 3"?2. @ $! 9" 3 ! $"& "&?3. $ "! ! 3 !? ?" , n{ !.4. # "! 93 ! 3 ! !? ? .5. ! ! "! < 3 ! ( , )? =# ! y = (1 ; x) 1 9"3 <! y = y2 D0A 2]? ?$? ! ! "! 3?6. @ 9 D? $" 3 ! !?7. ! $3! $ ($ <) ! 3 ! !? ?! $ ".8. 0 ! $ 9 <! 3 !.@ 9 $" " 3 ! !? 3, $ < F (x y C ) = 0 !!! 9 ! y = f (x y)?9. ? ! <! ! y = f (x y)! $3 y(x0) = y0. =# #3, $ $< y = f (x y) y(x0) = y0 < 9 DaA b]? ?$?;000020210. < ! <! $3 $. @ $! " ?? < 3 <!?11.
4 $ "33" " <?12. < "! 9" < 3 !?? p . " y = '(x) 9"3 9"<! ! y = 2 y?13. ! ! "! 9 "& y = G(x C )? H & &! !, 3! 9?14. @ !!! 9! ! y = y(x C ) 3"& "&3 ! y = f (y x)?15. ! ! "! 3 !? 2< ! y = x2 + y2:000203 21 ( = ! ("'** ,(" + !- != ', # B, / .( #0, ' . '( $(20.4), " . B 3 , . # " ".
. 23 , # //0'% ( '$ $' .' % /#0( % (.. #). # % //0'% ( 3 '. @# $ #% ( 3. @" ( # (, # .% #, 3( # F.B# "&# "# //0' " (=$- "@#", 1971 .). 3 ( # , #'# $(.8 ##% #% ( /$#%) $% ' % x y . 8 " $' . //0' //0'( /M (x y)dx + N (x y)dy = 0:(21:1)J' ' % x y , # ## (21.1) 3 ' ## /#0 y = y(x), . (21.1), # /#0 x = x(y), ". . (21.1) 3.23 ' #, ". (21.1) $' y = y(x) { x = x(y): @, xdy ; ydx = 0 #' ( $ 24 y = Cx (C = const)x = 0: x = 0 3 "' $ ( y = Cx ## $ ( C:2044 (21.1) #3 3 ' ' $. ' 3 $' ( $ .% /:y(x0 ) = y0x(y0 ) = x0:H" ' B, $' (21.1) ' ( $ $% yx0 x0y : 8 ' $ $' #:dy = ; M (x y) y(x ) = y 00dxN (x y) {dx = ; N (x y) x(y ) = x :00dyM (x y)A (21.1) #3( # M0(x0 y0) ( " D = D(M ) \ D(N ) $ ( ( M (x0 y0) N (x0 y0) 6= 0). ? # (x0 y0) /#0 M (x y) N (x y) , ( # (21.1) $ ## .
,# # $ .= , 3 #$', dy = f (x y)(21:2)dx x y % , '# " (. 20.3), dx = 1 dy f (x y)/# #% # O /#y = y(x) "% ( (21.2). 8 $"3 0" 3' : % x y (21.2) ( F (x y dy=dx) = 0), " ' '# " y = y(x) ( , $ .%), # # .
x = x(y) ( $ ( # , ,205/#0 x = const). ? 3 (21.2) $ //0'( /f (x y)dx ; dy = 0 # , x y ## % (21.2), # . , # 3 $ # (21.1).,, &( dy = f (x)(21:3)dx # ' 3 #( /#0 y = y(x): ?f (x) $# Ca6 b] , ## $ ' , ". (21.3) " #' %"$% /#0 f (x) $# Ca6 b] :ZZxy = f (x)dx = f (t)dt + Cx0 { $' , x0 2 Ca6 b] { /# #, $ $# Ca6 b]: 21.1. @( $ By0 = 2 sin 3x y() = 1:= , (, Zy = 2 sin 3xdx = ; 23 cos 3x + C: $' x = " ' 1 = 2=3 + C $ = 1=3.:y = ; 32 cos 3x + 13 :206,, &( ' dy = f (y)(21:4)dx ( # $ .
3 , /#0 f (y) # $#Cc6 d] ". '. , (21.4) 3$' #( /dx = 1 :(21:5)dy f (y)J' ' $( /#0 x = x(y) ' $( ( { y. = (21.5), ( ". :Zy dyx = f (y) + C(21:6)y0 { $' , y0 2 Cc6 d] { /# #, y $ $# Cc6 d]: & (21.5) ". (21.4) , # f (y) 6= 0 (8y 2 Cc6 d]):' ' f (y) = 0 #( # y = 2 Cc6 d]: /#0 y = (21.4), , . ,# "$, (21.4) "( (21.6) y 2 Cc6 ) ( 6 d]: B #' y = , f ( ) = 0: 21.2. @( ". dy = y + y2(21:7)dx ' $ B '( #( M0(06 1). A#$' .
( $. ' f (y) = y2 + y ". ' % #%: y = 0 y = ;1: L#0 y = 0 y = y(x) ;1 , , (21.7) ( ;1 < x < +1: , y 6= 0 y 6= ;1, (21.7) dx = 1 :dy y(y + 1)207= y, " '#ZZ "1dy1x = y(y + 1) = y ; y + 1 dy = ln jyj ; ln jy + 1j + C:=#, (21.7) $ 24 x = ln jy=(y + 1)j + C ec y 6= 0 y 6= ;1y 0 y ;1:"% ( , # ## $ f 0 (y) = 2y + 1 % y 2 R: ( ' # $ B ' y(0) = 1: y = 0 y = ;1 , "' # yx = ln y + 1 + C: $' x = 0 y = 1 0 = ln(1=2) + C # = ln 2: J, #$( $ B $ x = ln jy=(y + 1)j + ln 2: A, # #y = 1 /#0 y=(y + 1) 3', ' 3 "' $' ' y.
8 $' y = ex=(2 ; ex): , $ B . 3# (;16 ln 2): =#, #'( . ". :26 ;14 x = ln jy=(y + 1)j + C ec y 6= 0 y =y = 0 y = ;1: $ B /#0xey = 2 ; ex x 2 (;16 ln 2):, ,# $ //0' f (x)dx + g(y)dy = 0208(21:8) # dx /#0, $. '# x, dy{ /#0, $. '# y.' /#0 f () g(y) c% "% @u dy D(f ) D(g): A / du(x y) = @udx+@x@y y (21.8) #( /0x1yZZd B@ f (x)dx + g(y)dyCA = 0x0y0 x0 x 2 D(f ) y0 y 2 D(g): @ /#0, .
//0 a, .e.Zxx0Zyf (x)dx + g(y)dy = C:y0(21:9)2 ".( (21.8). 4(', y = y(x C ) { /#0, (21.9), (21.9), 3. 8$ //0 "% ( 3, (21.8). F $, (21.9) { ".( (21.8). J, ".( $ % ZZf (x)dx + g(y)dy = C: 21.3. @( ' # (2x ; 3)dx + 3ydy = 0%. $ # M0(16 2). ##% $% x .?=ZZ(2x ; 3)dx + (3y)dy = C , x2 ; 3x + 32 y2 = C: $' x = 1 y = 2 " ' 1 ; 3 + 6 = C # = 4: &', # ' # $ x2 ; 3x + (3y2)=2 = 4: A, y(1) = 2 > 0 209vuu ( #( ( /: y = t 2 (4 + 3x ; x2):3B x, .% 4 + 3x ; x2 0 , x2 ; 3x ; 4 0 , ;1 x 4:, -( ? M (x y)dx + N (x y)dy = 0(21:10)/#0 M (x y) N (x y) # $3 3M (x y) = M1(x) M2(y) N (x y) = N1(x) N2(y)#3( $ #% /#0( '# ( ( (x, y), , (21.10) { $ .
. I ', /#0 Mi(x) Nj (x) % "% . (21.10) M1(x)M2(y)dx = ;N1(x)N2 (y)dy(21:11) $ " $ M2(y)N1 (x), #, M2(y)N1 (x) 6= 0. $ M1(x) dx = ; N2(y) dy:N1(x)M2(y)= , o".( (21.11):Z N2(y)Z M1(x)(21:12)N1(x) dx = ; M2(y) dy + C M2(y)N1 (x) 6= 0 ($': x 2 D(M1) \ D(N1 ) y 2 D(N2) \ D(M2 )):? 3 M2(y)N1(x) = 0 (21.12) '$ ' ". (21.11), # ## ' $3. ' #' (24 M2(y) = 0N1(x) = 0210 (.. 2( ) = 0 N1( ) = 0 )., ( #( x = y = (21.11) "3, /#0 x = y = ( # M ( 6 ) # ' $ % x = y = # ## ( # (21.11) $ ## ). x = y = "' " ( 3' ')6 % 3 #' # (21.12).
21.4. ' (x + 2)pydx ; 3xdy = 0:(21:13)= " ?$ " $ xpy #, . I 'x + 2 dx = 3 dy , Z x + 2 dx = 3 Z dy , x + 2 ln jxj = 6py + C:pypyxx ' (, # xpy = 0: . x = 0 y = 0 (21.13) ( $ #$% % # ' # M (06 0)). , " x = 0 (y > 0)". dx = 3x :(21:14)dy (x + 2)pyH $ fx0 = 6=py(x+2)2 ( (21.14) . #3( # ( x = 0 (y > 0) ( #( #), (21.14) #( # (. B). , x = 0(y > 0) 3 "' " (21.14) ($ (21.13)). , " y = 0 (x 6= 0) ". F #3 # dy = (x + 2)py :(21:15)dx3xH $ fy0 = (x + 2)=(6xpy ) ( (21.15) 1 #3( # ( y = 0 (x 6= 0) $ 3"' " (21.15). ' (x0 6 y0) = (x06 0) {211/# # ( (.
J B '( #((x06 0) x0 6= 0 ## ', , (x ; x0) + 2 ln xx = 6py:0F y = 0 (x 6= 0) $ " % (21.13). @ ',p , x + 2 ln jxj = 6 y + C ". #'( :". 2py + Cx+2lnjxj=64x = 0 (y > 0) y = 0 (x 6= 0)6/#0 y = 0 (x 6= 0) { " .Ady = f (ax + by + c)(21:16)dx a b c { , b 6= 0 $(ax + by + c = z # $ . . 4(', //0 #$ $, (, y0 = 1b (z 0 ; a) (21.16) z 0 ; a = f (z ) , dz = bf (z ) + a:bdx x 6= const 3 ' #( /dz = Cbf (z ) + a]dx:F $ . . 21.5. @( ".( y0 = (3x + 2y ; 1)2:4 $ z = 3x + 2y ; 1: , z 0 = 3 + 2y0 $y0 = (z 0 ; 3)=2. % , " 'z 0 ; 3 = z 2 , dz = 2z 2 + 3:2dx212$ , dz = dx , Z dz = Z dx + C , 1 Zdzq2z 2 + 32z 2 + 32 z 2 + ( 3=2)2 = x + C ,pp2z, 2p3 arctg p32 = x + C: z = 3x + 2y ; 1 ".( %- :p2p322p arctg 4 p (3x + 2y ; 1)5 = x + C:2 33' @, /#0 F (x y) $ ( /#0( k (: A{(), % t 2 A R .
(:1) (x y) 2 D(F ) ) (tx ty) 2 D(F )62) F (tx ty) tk F (x y) (8(x y) 2 D(F )):H. # 3 A 3A = R n f0g A = R+ = ft > 0g:p@, /#0 F (x y) = x3 ; y3 ( /#0( k = 3=2 # A " 3 A = R+: 4(', t 2 R+ (8(x y) : x3 ; y3 0):qqF (tx ty) = t3x3 ; t3y3 t3=2 x3 ; y3 t3=2F (x y):? F (x y) { x y, " ( /#0( k '# , # ( ## ) 3 ', k: @,F (x y) = 3x2y2 ; 7xy3 { ( k = 4 ($, # 3 A 3 $' 3A = R).213 ' //0' M (x y)dx + N (x y)dy = 0:(21:17)? /#0 (x y) N (x y) { /#0 ( ( 3 k, (21.17) $ //0' .
F x 6= 0 3 "' #y !y !(21:18)f x dx + g x dy = 0:4(', $ # t x (t = x)" '! y!y !ykkM (x y) M x 1 x x = x M 1 x x f x ! y!y !ykkN (x y) N x 1 x x = x N 1 x x g x :? g(z ) 6= 0 (( g(z ) = 0, ## ( x = 0, a ' '), (21.18) ( $ (21.17)) # dy = F (y=x)(21:19)dx F (z ) ;f (z )=g(z ). & (21.19) $ %y=x = z , " 'dy = z + x dz y = xz ) y0 = z + xz 0 , dxdx (21.19) # dz = F (z ) , xdz = (F (z ) ; z )dx:z + x dx $ . . ?x (F (z ) ; z ) 6= 0 ".( Zdz = Z dx + C:F (z ) ; zx8' $' # %( /#0 y (z = y=x) ".( % (21.17) x (F (z ) ; z ) 6= 0: ?214x = 0 F (z ) ; z = 0 # (21.17), % # ' ".
. 21.6. ' q(21:20)xdy = ( x2 + y2 + y)dx:pJ' M (x y) = x N (x y) = x2 + y2 + y { /#0( ( 3 c k = 1 (' , t 2 A = R+ )., /#0 x = 0 (21.20). x 6= 0 $ " (21.20) x, v10p 2 2u y !2 yuydyx+ytdy = @ x + x A dx , dx = 1 + x + x($' " $# (+), > 0, $# (;), x < 0). &$y = z , y = xz ) y0 = z + xz 0 x $ . :dz = p1 + z 2 + z , xdz = p1 + z 2dx:z + x dx ( x > 0: , , $ , " 'Z dzZp 2p 2p 2 = dxx , ln(z + 1 + z ) = ln x+ln C1 , z + 1 + z = C1x:1+zJ $' z y=x, ".( % p(21.20) ( x > 0): y + x2 + y2 = C1x2 (C1 > 0):? x < 0, ,p ##, ".( p x2 + y2 ; y = C2 (C2 > 0): A3 " x2 + y2 + y 6= 0 q2q222x = C2 x + y + y , x2 + y2 + y = Cx :2=#, "$' $ ". #, ".( (21.20) x 6= 0 3 $' p 2 %x + y2 + y = Cx2 (C > 0): #'( :2p 2 224 x + y + y = Cx (C > 0)x = 0:215J, dy = f a1x + b1y + c1 !dxa2x + b2y + c2 # , ' $ % u = x ; x0v = y ; y0 (x0 y0) { (a1x + b1y + c1 = 0a2x + b2y + c2 = 0:8 , # ( ( .
'), #//0 ( " 0',.. a1 = ka2 b1 = kb2: 8 % 01dy = f @ k(a2x + b2y) + c1 A dxa x+b y22 $ a2 + b2y = z # $ . . 21.7. ' dy = x ; y + 1 :dx x + y ; 3& ((x ; y + 1 = 0x+y;3=0 (x y) = (16 2): 4 $ %u = x ; 1 v = y ; 2: #'# dx = du dy = dv dv = u + 1 ; v ; 2 + 1 , dv = u ; v , dv = 1 ; v=u :du u + 1 + v + 2 ; 3du u + vdu 1 + v=u , $v = z , v = uz , dv = z + u dz ududu 2Z (1 + z )dzZ dudz1;zdz1;2z;zz + u du = 1 + z , u du = 1 + z , 1 ; 2z ; z 2 = u ,216, ; 21 ln j1 ; 2z ; z 2j = ln juj ; 12 ln C , (1 ; 2z ; z 2)u2 = C:8$.' # .' z = (y ; 2)=(x ; 1) u = x ; 1 ".( % x2 ; 2xy ; y2 + 2x + 6y = C1:% A dy = a(x)y + b(x)(21:21)dx a(x) b(x) { $ /#0, $ ( #).
/#0 b()$ (21.21). ? ', (21.21) $ 6 (.. # b(x) 6 0) (21.21) $ . A z 0 = a(x)z $ , . (21.21).4#3 . $' 3. , o3 $ "'( ( . ,# ## $' $ B " #//0 a(x) b(x) (.. $# Ca6 b]). 8( ' #' $', ..$' ( # # x = x0: 21.1. (21.21) a(x) b(x) Ca6 b], (x0 y0) (x0 2 Ca6 b]) $)dy = a(x)y + b(x) y(x ) = y00dx ) Ca6 b] . 0 ) ( 3Rx a(t)dt 2Rs a(t)dtxZ;66y(x) = y(x6 x0 y0) = exb(s)ds775 :(21:22)4y0 + e x0x02170". (21.22) $ x = x0, "' y(x0 ) = y0 .
J /#0 (21.22) ' . 4, /#0 (21.22) $# Ca6 b] 3//0 (# ## /#0 a(x) b(x) Ca6 b]). 4//0 (21.22) , " 'Rx a(t)dtdy(x) = exdx023 RxsRRx a(t)dtxZ;a(t)dta(t)dt; a(x) 664y0 + e x0 b(s)ds775 + ex0 e x0 b(x) =x0= a(x) y(x) + b(x):J, 3 y0 (x) a(x)y(x) + b(x): F $, /#0 (21.22) (21.21). ,#$.7. #'# y0 $', $ $' C $ % , ".
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