1. Интегралы ФНП Диф_ур (853736), страница 12
Текст из файла (страница 12)
0 # /#0 z = xy x2 + y2 ; 4 = 0.& /#0 E 3L(x y ) = xy + (x2 + y2 ; 4) ( 0 #:99L0x = y + 2x = 0 >>=y = ;2x >>=L0y = x + 2y = 0 > , x = ;2y > :L0 = x2 + y2 ; 4 = 0 >x2 + y2 = 4 >8$ # " ( # $' , y2 + x2 = 42(x2 + y2 )# 42 = 1 12 = 12 :p = 1 = 12 y = ;x = 2p = 2 = ; 21 { y = x = 2.p pp p&', # M12( 2 2) M34( 2 2) 0 # /#0 E 3 0 # /#0z = xy x2 + y2 ; 4 = 0:3. 8 $ /#0 z = xy % (% #%p pp pz (0 0) = 0 z ( 2 2) = ;2 z ( 2 2) = 2:@"' % $( 2, ' ;2. &', M = 2, m = ;2.
1. 4 $ $! $ " 1 f (x y) F (x y) = 0? (. . 19.2 3 ! 19.1).2. ! ! C#? (. (19.5) (19.7)).3. 4 $ 3 C# "! "& 1?4. "! 93< 3< $.189 20'** ,( + ! (;< ). ;(!( '** ,(+ ! != '. +< !! ' ! 0 /&'/. ' &A, #% $ /#0 % $# $( //0, $ . $ "# //0' //0' % $%. ? $/#0 $ '# , . $ "# //0' 6 $ /#0 $ % " , . % $%.:dy = f (x y)6 3) F (x y y0 ::: y(n)) = 061) F (x y y0 ) = 062) dx2u2u2u@@@4) M (x y)dx + N (x y)dy = 065) @x2 + @y2 + @z 2 = 4(x y z ):J' 1){4) { "# //0' , 5) { % $%.2 " ' "# .#$ 20.1. * //0' $ #'( # %.( $( (//0) $( /#0.8 % % 1),2) 4) { //0' #6 3) { //0' n{ #.
8 # . #mxU = F (t x x_ )2ddx _ = dt U = dtx2 m = const : F 3'( # m ( ( F = F (t x x_ )' OX , x = x(t) { "0 '( # 190 t: A#$ # 3 $ /$# $# @' .&#' % " //0'% (, % , $ . ,#, , (mxU = F1(x y x_ y_ t)myU = F2(x y x_ y_ t) . # 3 $# @' , . 3 '( # m # 0 ( ( FG = fF1 F2g. J' $ /#0x(t) y(t) "0( ( '( # t. #$ , "# //0' # 3 #% #%, % $ ' #'. @ $ ( #.'"( , "( .
! + .'" //0' dy = f (x y)(20:1)dx #, $ ' $( dy=dx: J'x { $ ( ), y = y(x) { $ /#0. F 3 3 $' //0'( /M (x y)dx + N (x y)dy = 0(20:2) "$' f (x y) = ;M (x y)=N (x y): = / $ ##% %.#$ 20.2. 3 (20.1) $ 3 D (x y), #% ' f (x y) (.. 3 "' ).191@, "' dy = px + 2qy ; 1dx 3 D = f(x y) : x 0 y 1g.#$ 20.3. M, /#0 y = y(x) ) (20.1) $# Ca b], .:1) #a (x y(x)) 2 D % x 2 Ca b]62) /#0 y = y(x) //0 $# Ca b] %x 2 Ca b] 3dy(x) f (x y(x)):(20:3)dx= ## , (20.1) ( //0' ) $ /#0, ". . 3 (. (20.3)). @ # . # #% #%% , # 3' , 20.3. 20.1. = $# Ca b] (a b) % C b) (a b].
# a = ;1 b = +1. 20.3 % % ( ' ' "$# Ca b], . 3# (a b) Ca b) (a b]).@, /#0 y = 41 x2 //0' y0 = py 3# C0 +1) (' 1) 2) 20.3).M/# y = y(x), .. 3 = f(x y) : x 2 Ca b] y = y(x)g # R2 $ ! //0' (20.1), "' //0' " " ' //0' ".192;. 20.1 #( (20.1). ? y = y(x) ' # //0' (20.1), #3( ( # (x y(x)) #' , ( #//0 #( k = f (x y(x)) (. 3 (20.3)).
# ( y = y(x) //0' (20.1), $ ' ' # . , (20.1) " 3 ' # f16 f (x y)g # (x y) " D . & $'( # ( y) 2 D , $ "' D . # f16 f (x y)g: 3 #( $#, .% % #) $ ! (. . 20.1). J, ( $ (20.1). " (20.1) . ='$ '# (, 3 "3' " ' # . //0' . $ #% " { $# {" 3.= ( //0' y0 = 1, 3$', " 3 ( y = x + , { $' . M/# % ( ( ' #) , ' "# # . F $ #' O . 4 ##( (, $' % " #,$ # % . @, $ # (26 3)% ' "#" y = x +1: 8 , ' y(2) = 3 ## y0 = 1: "". #$ $' //0' (20.1), $ $193dy = f (x y) y(x ) = y(20:4)00dx ( $( B) //0' (20.1). # (x0 y0) $ .V, # 3 3' " D (20.1).= $' ' y(x0 ) = y0 $ ( % % /#0 f (x y) (. 3 B)) ' (20.1).#$ 20.4.
5 ) (20.1) $ y = '(x) ##(-"' ##( $ B(: $ B ' y(x0 ) = '(x0)).3 ) (20.1) " Q D $/#0 y = W(x C ) $. $'( ( , . . ":1) " $ ( /#0y = W(x C ) (20.1) # $#C b] (#, (x W(x C )) 2 Q 2 Ca b]), ..dW(x C ) f (x W(x C )) 8x 2 Ca b]6dx2) ## " " $ B (20.4) '( #((x0 y0) 2 Q . C = C 0 #, /#0y = W(x C 0) ( $ B.J, 2) $, "# W(x0 C ) = y0 (x0 y0) { $' # " Q, %" = 0. 20.1. ', /#0 y = Ce3x ".
y0 = 3y ( " Q = R2).A 1) 20.4 :d (Ce3x) 3 (Ce3x) (8x 2 (;1 +1)):dxF 3 " $ ( : '' (x0 y0) { $' ( /#) # #194R2 : , 3 ( = 0 #, " ' y(x0) = y0. =Ce3x = y0 , C = C 0 = y0e;3x :J, # $ = 0 ., /#0 y = y0e3(x;x ) $ B y0 = 3y y(x0 ) = y0. =#, 1) 2) 20.4 ". , $ y = Ce3x {". y0 = 3y:# ( (20.1) /#0 y = y(x), $( . ? $' (20.1) # 3 x y ( "#, .. 3. $( y0 ), , ( (20.1). A .#$ 20.5. 3 ! " Q D //0' (20.1) $ F (x y C ) = 0 (C { $' )(20:5) . ".
Q "$.; ( ( F (x y) = 0 (20.1).='$ % /#0(, 3 ' .( # , (20.5) ". (20.1). ! 20.1. ) //0 (20.5) x,, ' /#0 :@F + @F dy = 0:@x @y dx") # (20.5) $ % (8>< F (x y C ) = 0@F @F dy(20:6)>: @x + @y dx = 000# . ? " % //0' (20.1) ( # ), (20.5) { ".( .195 20.2.
', x2 ; y2 ; Cx = 0 ". x2 + y2 ; 2xy y0 = 0:8 F (x y C ) x2 ; y2 ; Cx: & (20.6):(2x ; 2y y0 ; C = 0x2 ; y2 ; Cx = 0:=# $ % . 4 $ % :C = 2x ; 2y y06 x2 ; y2 ; (2x ; 2y y0 )x = 0 , ;x2 ; y2 +2xy y0 = 0 ,, x2 + y2 ; 2xy y0 = 0: % //0' ,x2 ; y2 ; Cx = 0 { ".( . $".( (20.5) #3 $ ' " $ B (20.1) (, #, ). 20.3. @( $ Bx2 + y2 ; 2xy y0 = 0 y(1) = 2:8'$ ". x2 ; y2 ; Cx = 0 . x = 1 ( y = 2), " '1 ; 4 ; C = 0 , C = ;3:&', x2 ; y2 +3x = 0 { ( ,$ .( %( $ B "$.( ' # "3 " . $ B (20.4). ##% % /#0 f (x y) $" $(? $?. * f (x y) ! Y = f(x y) : jx ; x0j a jy ; y0j bg (a > 0 b > 0) ( (x0 y0).
/! $) (20.4) , , ) y = y(x): 0 ) Cx0 ; h6 x0 + h] ! h = min(a Mb ) M = (xymaxjf (x y)j:)2196F ' . $B (20.4). @ /#0 f (x y) ( $. @3 " ' /#0 f (x y) , " $ (20.4) " $$(. 3$. * f (x y) 0fy (x y) D. /! (x0 y0), ( D, Cx0 ; h6 x0 + h] , $) (20.4) (x0 y0) ) y = y(x) ) .M# $, #( # #(x0 y0) . ' # (20.1), %.$ # (x0 y0), # ' # (. . 20.2).;.
20.2 ( : . y = y(x) ' ' ( # # x = x0 (h > 0 { ). " . A "' , .($ Bp03 .'. @, $ B y = y y(0) = 0 y 0 ( ;1 < x < +1p0 $ fy = 1=2 y ( . '( # (x0 y0) = (06 0): 8 , #, '$ ' . 4 #$( $ B, # y 0 ., , . # ( 2y = x0 =4x x<0:0#$ 20.6. y = '(x) (20.1) $ ), #3 # (x0 '(x0)) '(197#( y = '(x) #( (20.1), .. . ( #(( , . ) '# y = y(x) (20.1), %.
$ # (x0 '(x0)), . '( #( y = '(x).= . 3: /#0 f (x y) " D $ fy0 . D, $ # #% # (x y) 2 D, %#% fy0 = 1 " (20.1) "''# # # y = '(x), % #% #% fpy0 = 1: 20.4. @( " y0 = 2 y:J' f (x y) = 2py fy0 = 1=py: H $ fy0 . D = f(x y) : y 0g # # (x y) 3.% "0 y 0: fy0 jy=0 = 1: J, y = 0 3p0"' " y = 2 y: , " #.L#0 y = '(x) 0 , p0 y = 2 y: B , #3( p# (x0 '(x0)) = (x06 0)( y = '(x) = 0 $ B( y0 = 2 y y(x0 ) = 0 2$% : y 0 y = (0x ;xx0<) x x: x0 &',0y 0 { " . 4 % "% ( , # ## D % # (x y) ## (x 0), #% fy0 = 1: =#, #'( : y = 0:A#3 .
$# . "% (. @ . .#$ 20.7. B y = '(x) $ ! y = y(x C ) ( { , $' ), #3( ( # (x '(x)) # y = '(x) #, #(( , ( $ #% ( y = y(x C ) #3 $##( y = '(x) # "# 3 #% y = y(x C ):B , $' " . ( #%y = y(x C ) 3 (, # $ (8 y = y(x C )><@y(x C ) :>:0 =@C198# #$', ( # " .(. 20.1. 3! y = y(x C ) ! ! (20.1) ) ! .4(', y = '(x) " .
( y = y(x C ) '% #% (20.1), #3( ( # (x0 '(x0)) # #( '( #( y = y(x C ) $'0 (x0) = y0 (x0 C ) '0 (x0) = f (x0 y(x0 C )) f (x0 '(x0)): F $, y = '(x) { (20.1). B , "(# (x0 '(x0)) $ B (20.4) ( y0 = '(x0)) , #((, : y = '(x) y = y(x C ) y = y(x C ) { (20.1), # . #(py = '(x) # (x0 '(x0)):@, y0 = 2 y (.
20.3) (y = (x ; C )2 (x > C ) '% #% " . y = '(x) 0 " ., ## " p#$ 20.4, " y0 = 2 y:$ ' (20.1). B# " #$ , $ ( " D (f16 f (x y)g: 8 " D . # , #3( # (x y)#% f (x y) = k (k = const):(20:7),# # $ //0' (20.1), (20.7) { $#. (20.7) #$, #3( # (x y) $# ' # (20.1) 3 f16 kg = f16 f (x y)g: # $#(20.7), .% $ $ ( k, $"$ #3( $# .
( f16 kg " (' ##( # (x0 y0) 2 D) ' # , # $#( (20.7) # f16 kg: # "$ #, ", , "3 #$ '( #( (20.1).199 20.5. 2 $# ' ' # $ B y0 = x2 + y2 y(0) = 0:A $# (20.7) x2 +y2 = k:F ' ( #0#% #3( 0 # (060) (# (060) #3 $#(, .($ k = 0). =$"$ #3( $# f16 kg.@, $# x2 + y2 = 1 "$ 450 3' OX . 4 ' # (060)# #3( $# x2 + y2 = k # #$ ( $#,$"3 . f16 kg: # "$ # " "3 $"3 '( #( $('( $ (. . 20.3).;. 20.3 . #'# .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
