1. Интегралы ФНП Диф_ур (853736), страница 10
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16.8. @( $ /#0 z (x y) $( exyz ; xyz = 0:&, /#0 z (x y) " 3, //0 3:exyz d(xyz ) ; d(xyz ) = 0158# (exyz ; 1)d(xyz ) = 0:8 #%, 16.3, Fz0 = (exyz ; 1)xy 6= 0', exyz ; 1 6= 0 ( xy 6= 0). d(xyz ) = 0yzdx + xzdy + xydz = 0:83 ( //0dz = ; xz dx ; yz dy % $@z = ; z @z = ; z :@xx @yy. ' /#0 F (x y z ) 16.3, $ #( # .
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(16.7) (16.8)).3. ! y(x) F (x y) = 0 (c. 16.1).4. ! ! y(x) F (x y) = 0 (c. 16.2).5. ! z(x y) F (x y z) = 0: (. 16.2).6. ! ! z(x y) F (x y z) = 0:7. ?3 F (x y z) = 0 ! !" x(y z) y(x z)(! F (x y z) ! ! & !).@ @x @y ?@y @x !@z = @z @u + @z @v + @z A @z = @z @u + @z @v + @z @w :1. @x@u @x @v @x @x @y @u @y @v @y @w @yn ! !F@x@y7. @y @x = ; Fy ; FFx = 1:xy0000160 17/ ( &!'( !(0" '!' /#0 z (x y) #( # # (x y) $@z @z (17:1)@x@y % "$%zx0 zy0 :H $ ". /#0 x y, #, ', ' $:@ @z ! @ @z ! @ @z ! @ @z ! :@x @x@y @x@x @y@y @y? #, $ $! /#0 z (x y) "$ :@ 2z z 00 6 @ 2z z 00 6 @ 2z z 00 6 @ 2z z 00 :xxxyyxyy@x2@y@x@x@y@y2; $ ! .
$ (17.1) $ $ ! . 17.1. @( $ 2- # /#0:z = x3 + xy2 ; 5xy3 + y5:@% $ 1- #:zx0 = 3x2 + y2 ; 5y36 zy0 = 2xy ; 15xy2 + 5y4$, "$ #3( $ % $ x y,:00 = 6x6 zxy00 = 2y ; 15y26 zyx00 = 2y ; 15y26 zyy00 = 2x ; 30xy + 20y3:zxx" zxy00 = zyx00 . H' $3 $, ( .161H $, "$ //0 $ , $ ) $.B % zxy00 zyx00 .& % $% # $: 1) . # //0( 000 zxyy000 )6 2) . ' (, zxxy000 , zxyx000 , zyxx000 ).# //0 (, zxxy000 , zxyx000 , zxyy000 , zyxx000 /# 17.2. @( $ zxxy0 z = x2y3.' %:0zx = 2xy3 zy0 = 3x2y2 600 = 2y3 zxy00 = 6xy2 zyx00 = 6xy2 (zyy00 ")6zxx000 = (zxx00 )0y = 6y2 zxyx000 = (zxy00 )0x = 6y26 zxyy000 = (zxy00 )0y = 12xyzxxy000 = (zyx00 )0x = 6y2:zyxxJ' $, .
' # //0, :000 = zxyx000 = zyxx000 = 6y2zxxy000 , .( ( ( $( zxyy% # //0( x y. , #3 ## zxy00 = zyx00 , $' 17.1 17.1. 17.1 ( 34 54 $6#4). ) , ) , , ." /#0 % % z = z (x y):1) & #3 % % $%II #.' $ zxy00 zyx00 # (x y). #3, ( #. '(x) = z (x y + 4y) ; z (x y)A = '(x + 4x) ; '(x) = z (x + 4x y + 4y) ; z (x y + 4y) + z (x y):162 E 3 /#0 '(x) $# Cx x + 4x] , 4x < 0, $# Cx + 4x x]:A = '0 (x + 14x)4x = (zx0 (x + 14x y + 4y) ; zx0 (x + 14x y))4x0 < 1 < 1: ' zx0 ## /#0 y $# Cy y + 4y] , 4y < 0 $# Cy + 4y y] , E 3,A = zxy00 (x + 14x y + 24y)4y4x:(17:2)4 /#0 (y) = z (x + 4x y) ; z (x y) $, (y + 4y) ; (y) = A: 3 E 3, ' ae000A = (y + 34y)4y = zy (x + 4x y + 34y) ; zy (x y + 34y) 4y == zyx00 (x + 44x y + 34y)4x4y6 0 < 3 < 1 0 < 4 < 1:& (17.2), % # zxy00 (x + 14x y + 24y) = zyx00 (x + 44x y + 34y):% # 4x ! 0 4y ! 0 ' % $% zxy00 zyx00 # (x y) $# , ( # zxy00 = zyx00 :2) B# ' $ $' # # ', .
.8 #% 000 = zyxx000 # ## zxy00 = zyx00 ) (zxy00 )0x = (zyx00 )0x ) zxyx000 = zyxx000 6zxyx000 6000 = (zy0 )00xy = (zy0 )00yx = zyyx000 = (zxy00 )0y = (zyx00 )0y = zyxy000 # ## zxyy000 = zyyxzxyy .. , #$.163 3 p 17.123 ' . @, /#0 z = x + y :zx0 = 2x zy0 = 3p1y2 (y 6= 0)6zxy00 = 0 zyx00 = 0 (y 6= 0)6 zxy00 = zyx00 , # # Ox, #% y = 0, $ zy0 , , ', $ $ zyx00 . % $% % # 17.1"". /#0 "', , # .33 ' /#0 z (x y) //0 # (x y) x y .
dx dy. F .$ . /#0 4z , ( ' # ,## 3 $, $ //0 /#0 3 $ $@z dx + @z dy:dz = @x(17:3)@y //0 % # (//0 $e ! . ? /#0 z (x y) //0 #( # # (x y), dz /#0( x y. B , dz $ #3 dx dy.J/# dx dy ( % ). , dz "/#0( '# x y. 4, /#0 //0 # (x y) . dx dy(, . , # $ . /#04z //0 dz ). F . $ . 4dz , ( ' # //0d(dz ). F ( //0 $ ! /#0 z (x y) # (x y) "$ d2z (: d z ).164,# "$,d2z = d(dz ) % dx dy:#3, //0 # 3 $ $ #.
# (17.3) //0 2 1 (. #0 P14) dx dy, ae: @z! @z ! @z !@z2d z = d @x dx + @y dy = dx d @x + dy d @y =(17:4) ! ! ! ! ! !@@z@@z@@z@@z dy := dx @x @x dx + @y @x dy + dy @x @y dx + @y @y22@z @z? $ @y@x @x@y# (x y) , ', , ( $' # :222@z@z@22d z = @x2 dx + 2 @x@y dxdy + @yz2 dy2:(17:5)L (17.5) #' / $ # % . F "' 3 '$ #( / :@!2@2d z = @x dx + @y dy z:(17:6)? #"# ( (17.6) $ # #'( ", $ $%@ 2 @ 2 @ 2 z ' $ @ 2z @ 2z @ 2z , $ (17.6)@x2 @y2 @x@y@x2 @y2 @x@y (17.5).L (17.6) 3 ' (, ( $, .@ @ ' ## "$ //0'% ? @x@y, $' ( #% z @z @z , $ @x@y165!@@z dx + @z dydx+dyz=@x@y@x@y@@ dx + @ dy!2 z = @ dx + @ dy! @z dx + @z dy! =@x@y@x@y@x@y!!@@z@z@@z@z= @x @x dx + @y dy dx + @y @x dx + @y dy dy =0 210 2122@z@z@z@z= @ @x2 dx + @x@y dyA dx + @ @y@x dx + @y2 dyA dy , , # / (17.5).
@ $3 / (17.6) $ d2z .? d2z //0( /#0( x y ## (x y), $3 ! :d3z = d(d2z ) % dx dy:8"., n-! ## //0 //0 (n ; 1)- #:dnz = d(dn;1 z ) % dx dy $' $ n- # 3 /(:@!n@nd z = @x dx + @y dy z:(17:7)& # ' , 3( /#0 3 x y / n 2 3 #$'(., , 3 /#0 z (x(t) y(t)) = z(t) t. 8 //0 dnz ## ( ' . 4dn;1z $ .
dt t, dt ## ,166 //0 3% dx = x0 (t)dt dy = y0 (t) ". ($. t). / //0 1- #@z dx + @z dydz = @x@y 3 3 d2z ' ! @z@z2d z = d(dz) = d @x dx + @y dy = @z ! @z !@z@z d(dy)= d @x dx + @x d(dx) + d @y dy + @y(17:8)( /( (17.4), #( d2z = d2z { //02- # ( /#0) d2z 6= d2z.F $, //0 II # ".
( ". /(17.7) 3( /#0 n > 2 ". % ,', //0 # n 2 " (.# x(t) y(t) ( /#0 t, x(t) = x0 + ht y(t) = y0 + kt(17:9) x0 y0 h t { , % //0 dx = hdt dy = kdt $ t (), d(dx) = 0 d(dy) = 0, / (17.8) . (17.4) d2z = d2z. ( 3 dnz = dnz n 3,.. //0 % # " (:dnz = dnz (n = 1 2 3:::):(17:10) //0 % # % ( /#0 "', , # .167 !"". / ,( /#0 % %.' /#0 z (x y) $ (n + 1)- # # ' #( -# #(x0 y0).
$ # ' # :9x = x0 + ht =(17:11)y = y + kt 0 h k { , t { .@ ( /#0 z (x y) 3 ' ## 3 /#0 :z (x0 + ht y0 + kt) = z(t):(17:12) / ,( /#0 # # t = 0:0 (0) z00 (0) 2(n)zzn+R z(t) = z(0) + 1! t + 2! t + ::: + n(0)t(17:13)n+1! Rn+1 { ( , .( (n+1)zt) tn+1 0 < < 1Rn+1 = (n + (1)! / E 3 Rn+1 = o(tn) t ! 0(17:14)(17:15) / .J / (17.13) t dt z(0) '.=$ (17.12), (17.11), ae:z(dt) = z (x0 + hdt y0 + kdt) = z (x0 + dx y0 + dy)z(0) = z (x0 y0)z(dt) ; z(0) = z (x0 + dx y0 + dy) ; z (x0 y0) = 4z:J' 4z 3 ' ## . /#0 z (x y),$ . , dx dy ( jdtj,168 jdxj, jdyj " ' , " # (x0 + dx y0 + dy) %' -# #(x0 y0).)4,z(n)(0)dtn = dnz(0)a # ## / //0 (.(17.9) (17.10)),dnz(0) = dnzjt=0= dnz (x0 y0) (n = 1 2 :::)dn+1z(dt) = dn+1z (x0 + hdt y0 + kdt) = dn+1z (x0 + dx y0 + dy):8 $' "$( $ (17.13) (17.14) /,( /#0 z (x y):2n z (x y )dz(xdz(xd0 y0 )0 y0 )4z = 1! + 2! + ::: + n!0 0 + Rn+1(17:16) ( / E 3n+1 (x + dx y + dy )dRn+1 = 0 (n + 1)!0 0 < < 1:(17:17)4 / pp = dx2 + dy2 .
,# ## dx = hdt dy = kdt, = h2 + k2jdtj ,',Rn+1 = o(dtn) dt ! 0 () Rn+1 = o(n ) ! 0: (17:18) 3 / .L ,( (17.16) 3 . /#0 $ //0. 8 3 $, //0 I # ( ' . /#0. & % '% % "# ( # ! 0 (R2 = o() ! 0). 4" #3 . //0 " # # . 4z , " " # # (R3 = o(2), R4 = o(3) ! 0 ..).L ,( (17.16) # /#0 " .1694 ( / ,( $ ' ( 0 .? 3' dx = x ; x0 dy = y ; y0, 4z = z (x y) ; z (x0 y0), $ (17.7) @!n@nd z = @x (x ; x0) + @y (y ; y0) z (n = 1 2:::) / ,( (17.16) # :!@@z (x y) = z (x0 y0) + @x (x ; x0) + @y (y ; y0) z (x0 y0) + :::+@!n@1(17:19)+ n! @x (x ; x0) + @y (y ; y0) z (x0 y0) + Rn+1:8 # / ,( $ ( z (x y) $( (x ; x0) (y ; y0).L ,( % "3% %, # '$ # (.
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$ # M0 # M . , #4z < 0, 4z > 0. F ( .4z '$ /$0 ( # /#0.#$ 18.1. ' /#0 #'#% %z = u(M ) #( # # M0 4z = u(M ) ; u(M0) { ., # /#0 . $ # M0 # M .? 9_ (M0) : 8M 2 _(M0) 4z < 0 ( 4z > 0), # M0$ ( ) /#0 z = u(M ), 171$ /#0 ( # z = u(M0 ) $ () /#0.M0 { # max u(M ) () u(M0) = max u(M )M0 { # min u(M ) () u(M0) = min u(M )#$ 18.2. ,# # /#0 $ , # { /#0.M0 { # extr u(M ) () u(M0) = extr u(M )" #4 # /#0 y = f (x) # x0"% f 0 (x0 ) = 0 ..4 /#0 #'#% % $. 18.1.
, ( ." /#0 % % f (x y).' # # (x0 y0): /#0 f (x y0 ): ? /# % z = f (x y) #' y = y0 , ,x0 #( # /#0 f (x y0 ) (. . 18.2).;. 18.21728 "% # /#0 df (x y0 ) = 0 6 9dx x=x# , $fx0 (x0 y0) = 0 6 9:(18:1)0; , /#0 f (x0 y), aefy0 (x0 y0) = 0 6 9:(18:2)=$ #$( , (18.1) (18.2) "% # /#0 f (x y) # (x0 y0):J, /#0 f (x y) //0 # (x0 y0), $ fx0 fy0 . ( # , ', fx0 (x0 y0) = 0 fy0 (x0 y0) = 0, #, (//0dz (x0 y0) = 0:(18:3),# "$, "% # //0( /#0 #( # //0 ( #.#$ 18.3. 8 # $ " /#0, #% "% #,$ , $ %, #% $ , { # /#0.,# 3 ## /#0 "% # /#0 #'#% % .
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