Методические указания к выполнению курсовой работы (Обыкновенные дифференциальные уравнения)
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32«!"»$%...2'(!(1. *I. 3! ./"&.). *!&!2II. -!!0!!*44 '!1. 8!01,&!02 0.!x0 ) .0$ 1,0&' && 40(x,.y(x)2(x2x0 ) :Ck ( x x 0 )ky( x ) =k =02. 8!=44'3. >!4.5../*44 '!6. -y(x),./,0. 1, 240'.(x x 0 ) .x,4./.(x x 0 )207.y(x)1=!&-*44 '!(x x 0 ) ,./y(x).&.!(x x 0 )*44 '0 !20&&&0C 0 , C1 , C 2!. ..8. >!C 0 , C1 ....C n1!!n-(9. 8!02 0!10. 8!!n).1.
7.!(x x 0 )2.2 !. 3!C 0 , C1 , C 2 , C 3 ,42!!4-. 1, 2.«!"»$%.III. -..3!0!$ 1!1. 8!244y(x)'.2a k (x x 0 )k , 2y=ak =k =02. D0!),3. >!0!4.406. 8!./.>!!01. >an .*44 '!&,!. 3. >*44 '!ak(! n -!&!11!2' 0 ,&.!,"!001. 4.(x x 0 )2«&&. 3 & n+1,!0!0a n +1 .*44 '!*44 '2.&!x0 ) :y( k ) ( x 0 )k!a 0 , a 1 ... a n*44 '!00144'y(x). >'&5.
Dn(x2»$%.'I...4(2. 4&!,./ n-2!, .*&./ n-200:y1 = y 2y(n )= f ( x, y, y , y ..y( n 1)y 2 = y3)..............yn 1 = yny n = f ( x, y1, y 2 ....y n 1, y n )II.!0!&./ n-2y( x 0 ) = y0y1 ( x 0 ) = y0y ( x 0 ) = y0y 2 ( x 0 ) = y0y ( x 0 ) = y0y3 ( x 0 ) = y0..................................y( nIII. 81)( x 0 ) = y0( n!1)y n ( x 0 ) = y 0( n0!Y = F( x , Y),&:1)./ n-2&4:Y( x 0 ) = Y0 , 2y1y1y2y0y2y2y3y0Y = ....ynY = ....ynF( x , Y) = ....yn1ynIV. -1yn!0Y0 = y 0 .....f ( x , y, y1, y 2 , y3..y n[x 0 , x n ] ,&y0( n1)' &&.$ 1!1. >!40:"$! :)1 i+1x i +1 = x i + hYi +1 = Yi + h FiFi = F( x i , Yi )2. >3. - 0 !4.i=0!0 !&!!&' (!2 4«!2 h=i = n,y(x).xnx0n).y (x )".»$%.n.V.
-..5!0&c&' &$ 1(1- 0 0.n&[x 0 , x n ] ,)!1. . >!40:"$! :)1 i+1x i +1 = x i + hYi +1 = Yi + h Fi +0.5Fi +0.5 = F( x i +0.5 , Yi +0.5 ),h2h= Yi + F( x i , Yi )2x i + 0 .5 = x i +Yi +0.52. >i=0!3. - 0 !4.VI. -&!0 !!&&' &!2 4-2 h=i = n,' (&0$ 1!.ny (x )y(x)x0.)..(2- 0 0.nxn&[x 0 , x n ] ,)!1. .
>!40:"$! :)1 i+1x i +1 = x i + h~(F( x i , Yi ) + Fi +1 )2~= F( x i +1 , Yi +1 )Yi +1 = Yi + h~Fi +1~Yi +1 = Yi + h F( x i , Yi )2. >3. - 0 !4.i=0!0 !&!!&' (!2 4«!2 h=i = n,y(x).xnx0n).y (x )".»$%..VII. n..6!0[x 0 , x n ] ,-0 2 - 0' &&.$ 1!1. . >!40:"$! :)1 i+1x i +1 = x i + hYi +1 = Yi + YiYi =1(k 1(i ) + 2 k (2i ) + 2 k 3(i ) + k (4i ) )6k 1(i ) = h F( x i , Yi )k (2i )k 1(i )h= h F( x i + , Yi +)22k 3(i ) = h F( x i +k (i )h, Yi + 2 )22k (4i ) = h F( x i + h , Yi + k 3(i ) )2.
>3. - 0 !4.i=0!0 !&!!&' (!2 4«!2 h=i = n,y(x).xnx0n).y (x )".»$%..'..7(3. * "7 6. *$ 1!1. >!2.&0403. 8&40!!u n ( x ) , n=1..m.0!60'!,0-' &.m: y( x ) =&Cn u n (x ) .n =14. 8!=5.!6. 80! 407. >!40 ' &)./9. >!,010. 8./.2!2&&, m& ./0('-. 5.0.0 1 40!&!!12.y(x).'0 !C1, C 2 ... C m .C1, C 2 ... C m .!11. >'R(x, C1 , C 2 ...
C m )' 1'! 044. 3, 408. 82,!40'&y(x)'2 42.&&.«!".»$%&.'*..8(4. 4,0$ 16.6.0!, 2! n = 4)2. D!8!0yyiyi = i +12h3.!1yi =&!,0& 40'& 40-'xiy(x)y(x)4yi +1 2 yi + yih!nh (00!,:120.020&0(n-1)-2 0(n+1)-yi .!6. >!7. - 0 !001020.T&.yi .&!'0.2 4«&yi .000&xi .00 !!5. .8.!!1. -4.h -!2!&"»&.$%.*..9!9: 2 x 2 y +(3x8!: x 0 = 1,>2x 2 ) y ( x + 1) y = 0 y(1) = 0,y( x 0 ) = y0 = 0, y ( x 0 ) = y 0 = 144y =(1'!2 0y (1) = 1012(3x 2 x ) y + ( x + 1) yf ( x, y, y ) =22x 244444144444301001!:2(3x 2x ) y + ( x + 1) y2x 2f ( x ,y,y )U.
. 400f ( x , y, y )'&!.V0&( x 0 , y0 , y0 ) = (1, 0, 1) ,( x x 0 ) = ( x 1) .2)!97!:y = C0 ( x 1)0 + C1 ( x 1)1 + C2 ( x 1)2 + C3 ( x 1)3 + C 4 ( x 1) 4 + C5 ( x 1)5 + ... == C0 + C1 ( x 1) + C2 ( x 1) 2 + C3 ( x 1)3 + C 4 ( x 1) 4 + C5 ( x 1)5 + ...U 2y = C1 ( x 1)0 + 2C 2 ( x 1)1 + 3C3 ( x 1)2 + 4C4 ( x 1)3 + 5C5 ( x 1) 4 + ...
== C1 + 2C 2 ( x 1) + 3C3 ( x 1) 2 + 4C4 ( x 1)3 + 5C5 ( x 1) 4 + ...y = 2C 2 ( x 1) 0 + 6C 3 ( x 1)1 + 12C 4 ( x 1) 2 + 20C 5 ( x 1) 3 + ... == 2C 2 + 6C 3 ( x 1)1 + 12C 4 ( x 1) 2 + 20C 5 ( x 1) 3 + ...>*44 '44'!2 0*44 'y :g1 ( x ) = 2 x 2*44 'y :g 2 ( x ) = 3x 2x 2g3 ( x ) = ( x + 1)g4 (x) = 0*44 'y :*44 '*44 '!0 4:( x 1) : g 4 ( x ) = 0 ( x 1) 0 .g1( x ) , g 2 ( x ) , g3 ( x )g 4 (x )!0 0U &:*44 ':g(x ) =«g(k ) (x 0 )(x x 0 )kk!k =0!"»$%( x 1) ,...102 !!*44 './! 0 1M = M (xU(x1=& 1!M0(xx0 ):x 0 )0 .0!x0 ),*44 '3x 2*44 ',0g1 ( x ) =g 2 (x) = 3x - 2x 2 g 2 (1) = 1g 2 (x) = 3 - 4x g 2 (1) = -1g 2 (x) = -4 g 2 (1) = -4g 2 (x) = 0 g 2 (1) = 0g 2 (x) =244( x 1) 0 + ( x 1)1 + ( x 1) 2 =0!1!2!= 2 + 4( x 1) + 2( x 1) 2114( x 1) 0 +( x 1)1 +( x 1) 2 =0!1!2!= 1 ( x 1) 2( x 1) 2g3 (x) = -(x - 1) g3 (1) = -1g3 (x) = -1 g3 (1) = -1g3 (x) = 0 g 3 (1) = 001=x , .
. 3x 2 = 0 x 0 + 0 x + 3 x 2g1 ( x ) = 2 x 2 g1 (1) = 2g1 ( x ) = 4x g1 (1) = 4g1 (x) = 4 g1 (1) = 4g1 (x) = 0 g1 (1) = 0*44 '!g 3 (x ) =,21( x 1)1 = 2 ( x 1)( x 1) 0 +0!1!y, y , y!0:{ 2 + 4( x 1) + 2( x 1) 2 } { 2C 2 + 6C3 ( x 1) + 12C4 ( x 1) 2 + 20C5 ( x 1)3 + ... } ++ { 1 ( x 1) 2( x 1)2 } { C1 + 2C 2 ( x 1) + 3C3 ( x 1)2 + 4C 4 ( x 1)3 + 5C5 ( x 1) 4 + ... } ++ { 2 ( x 1) } { C0 + C1 ( x 1) + C 2 ( x 1) 2 + C3 ( x 1)3 + C4 ( x 1)4 + C5 ( x 1)5 + ... } = 0«!"»$%,...11-,( x 1) ,2( x 1)*44 '&&0:!!4C2 + C1 2C0=0( x 1)112C3 + 8C2 C1 + 2C 2 C0 2C1=0( x 1) 224C4 + 24C3 + 4C2 + 3C3 2C2 2C1 C1 2C 2=0( x 1)340C5 + 48C4 + 12C3 + 4C4 3C3 4C2 2C3 C2=0( x 1)D!004-x20& c 6-1:2C0 + C1 + 4C2 = 0C0 3C1 + 10C2 + 12C3 = 03C1 + 27C3 + 24C4 = 05C2 + 7C3 + 52C4 + 40C5 = 0D*44 'C0C1!0&:y(1) = 0C 0 + C1 (1 1) + C 2 (1 1) 2 + C 3 (1 1) 3 + ...
= 0 ,y (1) = 1C1 + 2C 2 (1 1) + 3C3 (1 1) 2 + 4C 4 (1 1)3 + ... = 1 ,U 2*44 '4C2 = 10 3 1 + 10C2 + 12C3 = 010C2 + 12C3 = 33 1 + 27C3 + 24C4 = 027C3 + 24C4 = 3*44 '!,C2C35C2 + 7C3 + 52C4 + 40C5 = 0,0: C2 =1,4C3 =-11.24:y = ( x 1)«, C1 = 1 .:5C2 + 7C3 + 52C4 + 40C5 = 0D!2 0 + 1 + 4C2 = 0D, C0 = 0 .!111( x 1)2 +( x 1)3 + ...424!"»$%...12")0V017a k (x x 0 )k , 2: y=!ak =k =0[!&: y(1) = 00a0 =y (1) = 1y\ &44'y>y (1) =!2 0x0 =10y(1) 0= =00! 1y (1) 1= =11!1a1 =: y =,21 y(1) + y(1) 3 1 y (1) + 2 1 y (1)22 1( x + 1) y (3x 2x 2 ) y2x 2y(1) = 0 , y (1) = 1 :1 0 + 0 3 1 1 + 2 12 1==2xy + y 3xy + 2x 2 y=2x 212a2 =y ,\ &y =44xy + y 3xy + 2x 2 y2x 2====3xy + 4 xy + 2x 2 y )2 x 23yD!y + xy + y3xy + 4 xy + 2 x 2 y3yxy + y 3xy + 2 x 2 y2x 2y2x2y2x 2x3+yy+ 22x 2x3y+yy3y+ 2+ 22x 2x2x2x03y 2 y++y2xx23y+y2xyy23xyyx2x3x+3yxy(1)13a3 =1:2=y (1) 11 4 11==3!624-:y = ( x 1)«111( x 1)2 +( x 1)3 + ...424!"=2y=x2y(1) = 0 , y (1) = 1 , y (1) =,+,4 x ( xy + y 3xy + 2 x 2 y )4x 4y (1) y (1) 3y (1) 3y (1)y(1)+y(1)++2 12 12 2 1 2 12 2 12120 1 1 3 3 ( 1 2)1 0 0 11+( )= + + +=42 2 2 222 1 1y (1) =y(1)=( y + xy + yx0 =1y>y (1)12==2!2x:y'y( k ) ( x 0 )k!»$%=14.*..13!9: y + y x sin y = 08!: x 0 = 0,&& 2-2-0y ( 0) = 0'.!y ( x 0 ) = y0 = 02 02-2!!&44&: y = y x +sin y&: y ( x ) = z( x )>U 2y(0) = 1,y( x 0 ) = y0 = 1,440(2y ( x ) = z ( x ) = z x + sin yy =z00:8!z = z x + sin y: y ( 0) = 10z ( 0) = 080010& 4.Y = F( x, Y), Y(0) = Y0 , 2Y=-y01yyz, Y =, F( x , Y) =, Y0 ==z00zzz x + sin y02 !.!>240 ' &«!!"!»$%'!.)..14'6_ 2 h=1 0= 0 .52_ 2 h=1 0= 0 .220x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0z0=10z0z0 x 0 + sin y00==0 0 + sin 100.84151x1 = x 0 + 0.5 = 0 + 0.5 = 0.5Y1 =y1z1= Y0 + 0.5 F0 ( x 0 , Y0 ) =F1 ( x1, Y1 ) =z1z1 x1 + sin y1011+ 0.5=0.84150.420700.42070.4207=0.4207 0.5 + sin 10.6311=2x 2 = x1 + 0.5 = 0.5 + 0.5 = 1Y2 =y2z2= Y1 + 0.5 F1 ( x1, Y1 ) =- 0 !10.42071.2104=+ 0.50.42070.63110.7363&' :'012xyy (x) = z00.51111.210400.42070.73630x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0z0=10z0z0 x 0 + sin y0=00 0 + sin 1«!=00.8415"»$%...151x1 = x 0 + 0.2 = 0 + 0.2 = 0.2y1Y1 =z1= Y0 + 0.2 F0 ( x 0 , Y0 ) =F1 ( x1, Y1) =Dz1z1 x1 + sin y1=011+ 0.2=0.84150.168300.16830.1683=0.1683 0.2 + sin 10.8078!2- 0 !&' :'012345\0=!1.&xyy (x) = z00.20.40.60.81111.03371.09961.19471.314000.16830.32990.47530.59650.6871&[0, 1].«40!"' & y(x)»$%y (x) ,..").16'6! (1-6,6!'6)_ 2 h=0x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0z0=10z00=z0 x 0 + sin y00 0 + sin 1=00.8415x1 / 2 = x 0 + 0.25 = 0 + 0.25 = 0.25y1 / 2Y1 / 2 =z1 / 2F1 / 2 ( x1 / 2 , Y1 / 2 ) =%7011+ 0.25=0.84150.21040= Y0 + 0.25 F0 ( x 0 , Y0 ) =z1 / 20.21040.2104=0.2104 0.25 + sin 10.7889=z1 / 2 x1 / 2 + sin y1 / 21x1 = x 0 + 0.5 = 0 + 0.5 = 0.5Y1 =y10.21041.10521+ 0.5=0.78890.39440= Y0 + 0.5 F1 / 2 ( x1 / 2 , Y1 / 2 ) =z1F1 ( x1, Y1 ) =z1z1 x1 + sin y1=0.39440.3944=0.3944 0.5 + sin 1.10520.6963x 3 / 2 = x1 + 0.25 = 0.5 + 0.25 = 0.75Y3 / 2 =y3 / 2z3 / 2= Y1 + 0.25 F1( x1, Y1 ) =F3 / 2 ( x 3 / 2 , Y3 / 2 ) =0.39441.20381.1052+ 0.25=0.69630.56850.3944z3 / 2z3 / 2 x 3 / 2 + sin y3 / 2=0.56850.5685=0.5685 0.75 + sin 1.20380.50702x 2 = x1 + 0.5 = 0.5 + 0.5 = 1Y2 =y2z2= Y1 + 0.5 F3 / 2 ( x 3 / 2 , Y3 / 2 ) =«1.10520.56851.3894=+ 0.50.39440.50700.6479!"»$%1 0= 0 .52...17- 0 !&' :xyy (x) = z00.5111.10521.389400.39440.6479'012_ 2 h=0x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0z0=10z0=z0 x 0 + sin y000 0 + sin 1=00.8415x1 / 2 = x 0 + 0.1 = 0 + 0.1 = 0.1Y1 / 2 =y1 / 2z1 / 2= Y0 + 0.1 F0 ( x 0 , Y0 ) =F1 / 2 ( x1 / 2 , Y1 / 2 ) =011+ 0.1=0.84150.084150z1 / 2z1 / 2 x1 / 2 + sin y1 / 20.084150.08415=0.08415 0.1 + sin 10.08331=1x1 = x 0 + 0.2 = 0 + 0.2 = 0.2Y1 =y1z1= Y0 + 0.2 F1/ 2 ( x1 / 2, Y1/ 2 ) =F1 ( x1, Y1 ) =z1z1 x1 + sin y1=10.084151.0168+ 0 .2=00.083310.16670.16670.1667=0.1667 0.2 + sin 1.01680.8171x 3 / 2 = x1 + 0.1 = 0.2 + 0.1 = 0.3Y3 / 2 =y3 / 2z3 / 2= Y1 + 0.1 F1( x1, Y1 ) =F3 / 2 ( x 3 / 2 , Y3 / 2 ) =1.01680.16671.0335+ 0.1=0.16670.81710.2483z3 / 2z3 / 2 x 3 / 2 + sin y3 / 2!"#$ "#=#%«!0.24830.2483=0.2483 0.3 + sin 1.03350.7846."»$%1 0= 0 .22...18- 0 !&' :'012345\0=!1-xyy (x) = z00.20.40.60.8111.01681.06651.14611.25121.375800.16670.32350.46190.57420.6552&40&' & y(x)[0, 1].«!"»$%y (x) ,.)..19'6–, (2-6,6!'6)_ 2 h=1 0= 0 .520x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0=z010z0=z0 x 0 + sin y000 0 + sin 1=00.8415~y011~Y1 = ~1 = Y0 + 0.5 F0 ( x 0 , Y0 ) =+ 0.5=00.8415z10.4207x1 = x 0 + 0.5 = 0 + 0.5 = 0.5~z0.42070.4207~~==F1 ( x1, Y1 ) = 1~z1 x1 + sin ~y10.4207 0.5 + sin 10.63111x1 = x 0 + 0.5 = 0 + 0.5 = 0.5Y1 =y1z1= Y0 +F1 ( x1, Y1 ) =[]10.5~~+ 0.25F0 ( x 0 , Y0 ) + F1 ( x1, Y1 ) =02z1z1 x1 + sin y100.4207+0.84150.6311=1.10520.36810.36810.3681=0.3681 0.5 + sin 1.10520.7095=~y0.36811.28931.1052~Y2 = ~2 = Y1 + 0.5 F1( x1, Y1 ) =+ 0.5=0.70950.72290.3681z2x 2 = x1 + 0.5 = 0.5 + 0.5 = 1~z~~F2 ( x 2 , Y2 ) = 2~z x + sin ~y22=20.72290.7229=0.7229 1 + sin 1.28930.23772x 2 = x1 + 0.5 = 0.5 + 0.5 = 1Y2 =y2z2= Y1 +[]1.10520.5~~+ 0.25F1(x1, Y1) + F2 (x 2 , Y2 ) =0.36812«!"0.36810.7229+0.70950.2377»$%=1.37790.6049...20- 0 !&' :xyy (x) = z00.5111.10521.377900.36810.6049'012_ 2 h=0x 0 = 0 Y0 =F0 ( x 0 , Y0 ) =y0=z010z00=z0 x 0 + sin y00 0 + sin 1=00.8415~y011~Y1 = ~1 = Y0 + 0.2 F0 ( x 0 , Y0 ) =+ 0.2=00.8415z10.1683x1 = x 0 + 0.2 = 0 + 0.2 = 0.2~z0.16830.1683~~=F1 ( x1, Y1) = 1~=z1 x1 + sin ~y10.1683 0.2 + sin 10.80781x1 = x 0 + 0.2 = 0 + 0.2 = 0.2Y1 =y1z1= Y0 +F1 ( x1, Y1 ) =[]10.2~~+ 0.1F0 ( x 0 , Y0 ) + F1 ( x1, Y1 ) =02z10.16490.1649=0.1649 0.2 + sin 1.01680.8175=z1 x1 + sin y100.1683+0.84150.8078~y0.16491.04981.0168~Y2 = ~2 = Y1 + 0.2 F1 ( x1, Y1 ) =+ 0.2=0.81750.32840.1649z2x 2 = x1 + 0.2 = 0.2 + 0.2 = 0.4~z~~F2 ( x 2 , Y2 ) = 2~z x + sin ~y222!"#=0.32840.3284=0.3284 0.4 + sin 1.04980.7360$ "##%«!."»$%=1.01680.16491 0= 0 .22...21- 0 !&' :'012345\0=!1&-xyy (x) = z00.20.40.60.8111.01681.06621.14521.24931.372900.16490.32030.45730.56860.6491&[0, 1].-«!40"' & y(x)»$%y (x) ,...221)1 _ 2 h=1 0= 0 .520x 0 = 0 Y0 =y0z0=10x = x0 = 0Y=K10 =1y= Y0 =0zk 10yk 10z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .500 0 + sin 10=0.4207x = x 0 + 0.25 = 0 + 0.25 = 0.25Y=K 02=11y1 01= Y0 + K10 =+=0 2 0.42070.2104z2k 02 yk 02z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .50.21040.2104 0.25 + sin 1=0.10520.3944x = x 0 + 0.25 = 0 + 0.25 = 0.25Y=K 30=11.0526y1 0.10521= Y0 + K 02 =+=0 2 0.39440.1972z2k 30yk 30z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .50.19720.1972 0.25 + sin 1.0526=0.09860.4097x = x 0 + 0.5 = 0 + 0.5 = 0.5Y=K 04=10.09861.0986y= y0 + K 30 =+=00.40970.4097zk 04 yk 04 z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .50.40970.4097 0.5 + sin 1.0986=0.20490.34280.09860.20490.10210.10521 01+2+2+=Y0 = (K10 + 2 K02 + 2 K30 + K04 ) =0.40970.34280.39530.39446 0.42076«!"»$%...231x1 = x 0 + 0.5 = 0 + 0.5 = 0.5Y1 =y1z1= Y0 + Y0 =10.10211.1021+=00.39530.3953x = x1 = 0.5y1.1021= Y1 =Y=z0.3953K11 =k11yk11z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .50.3953=0.3953 0.5 + sin 1.10210.19760.3472x = x1 + 0.25 = 0.5 + 0.25 = 0.751.1021 1 0.19761.2009y1=+= Y1 + K11 =Y=0.3953 2 0.34720.5689z2K12=k12 yk12 y= 0.5 F( x, Y) = 0.5zz x + sin y= 0.50.56890.2845=0.5689 0.75 + sin 1.20090.2528x = x1 + 0.25 = 0.5 + 0.25 = 0.751.1021 1 0.28451.2443y1=+= Y1 + K12 =Y=0.3953 2 0.25280.5217z2K13=k13 yk13 y= 0.5 F( x , Y) = 0.5zz x + sin y= 0.50.52170.2609=0.5217 0.75 + sin 1.24430.2779x = x1 + 0.5 = 0.5 + 0.5 = 1y1.10210.26091.3629=+= Y1 + K13 =Y=z0.39530.27790.6733K14 =k14 yk14 z= 0.5 F( x , Y) = 0.5zz x + sin y= 0 .50.67330.6733 1 + sin 1.3629=0.33660.15260.28450.26090.33660.270811 0.1976=++2+2Y1 = (K11 + 2 K12 + 2 K13 + K14 ) =0.25280.27790.15260.260266 0.3472«!"»$%...242x 2 = x1 + 0.5 = 0.5 + 0.5 = 1y1.10210.27081.3729+=Y2 = 2 = Y1 + Y1 =z20.39530.26020.6555- 0 !&'0x0' :y1y (x) = zK0x00.250.250.5y11.00001.05261.0986y (x) = zx0.50.750.751y1.10211.20091.24431.3629y (x) = z00.21040.19720.4097Y10.51.10210.39530.39530.56890.52170.6733Y211.37290.6555«!"»$%ky00.10520.09860.20490.1021ky0.19770.28450.26090.33660.2708kz0.42070.39440.40970.34290.3453kz0.34720.25280.27790.15260.2602...25_ 2 h=1 0= 0 .220x 0 = 0 Y0 =y0z0=10x = x0 = 0Y=K10=1y= Y0 =0zk10yk10z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .200 0 + sin 1=00.1683x = x 0 + 0.1 = 0 + 0.1 = 0.1Y=K 02=11y1 01= Y0 + K10 =+=0 2 0.16830.0841z2k 02 yk 02 z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .20.08410.0841 0.1 + sin 1=0.01680.1667x = x 0 + 0.1 = 0 + 0.1 = 0.1Y=K 30=11.0084y1 0.01681= Y0 + K 02 =+=0 2 0.16670.0833z2k 30yk 30z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .20.08330.0833 0.1 + sin 1.0084=0.01670.1675x = x 0 + 0.2 = 0 + 0.2 = 0.2Y=K 04=10.01671.0167y= y0 + K 30 =+=00.16750.1675zk 04 yk 04z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .20.16750.1675 0.2 + sin 1.0167=0.03350.16340.01670.03350.01670.01681 01+2+2+=Y0 = (K10 + 2 K02 + 2 K30 + K04 ) =0.16750.16340.16670.16676 0.16836«!"»$%...261x1 = x 0 + 0.2 = 0 + 0.2 = 0.2Y1 =y1z110.01671.0167+=00.16670.1667= Y0 + Y0 =x = x1 = 0.2y1.0167= Y1 =Y=z0.1667K11=k11yk11z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .20.1670.1667 0.2 + sin 1.0167=0.03330.1634x = x1 + 0.1 = 0.2 + 0.1 = 0.31.01671.0334y1 0.03331=+= Y1 + K11 =Y=0.1667 2 0.16340.2484z2K12=k12 yk12 y= 0.2 F( x , Y) = 0.2zz x + sin y= 0.20.24840.0497=0.2484 0.3 + sin 1.03340.1569x = x1 + 0.1 = 0.2 + 0.1 = 0.31.01671.0416y1 0.04971=+= Y1 + K12 =Y=0.1667 2 0.15690.2451z2K13=k13 yk13 y= 0.2 F( x, Y) = 0.2zz x + sin y= 0.20.24510.0490=0.2451 0.3 + sin 1.04160.1579x = x1 + 0.2 = 0.2 + 0.2 = 0.4y1.01670.04901.0658=+= Y1 + K13 =Y=z0.16670.15790.3246K14=k14 yk14z= 0.2 F( x , Y) = 0.2zz x + sin y= 0 .20.32460.3246 0.4 + sin 1.0658=0.06490.14910.04970.04900.06490.04931 0.03331=++2+2Y1 = (K11 + 2 K12 + 2 K13 + K14 ) =0.15690.15790.14910.15706 0.16346!"#$ "##%«!."»$%...27- 0 !&'0x0' :y1y (x) = zK0x00.10.10.2y111.00841.0167y (x) = zx0.20.30.30.4y1.01671.03341.04161.0658y (x) = zy1.0661.03341.04161.0658y (x) = z00.08410.08330.1675Y10.21.01670.16670.16670.24840.24510.3246Y20.41.0660.3237x0.40.50.50.60.32370.39830.39280.4632Y30.61.14500.4622x0.60.70.70.8y1.14501.19121.19751.2487y (x) = zx0.80.90.91y1.24921.30671.31161.3722y (x) = z0.46220.52560.51830.5759Y40.81.24920.57490.57490.62380.61530.6574Y511.37290.6564«!"»$%ky00.01680.01670.03350.0167ky0.03330.04970.04900.06490.0493ky0.06470.07970.07860.09260.0790ky0.09240.10510.10370.11520.1042ky0.11500.12480.12310.13150.1237kz0.16830.16670.16750.16340.1667kz0.16340.15690.15790.14910.1570kz0.14920.13830.13950.12650.1385kz0.12670.11220.11370.09760.1127kz0.09780.08080.08260.06460.0815...28\0=!1-&[0, 1].-0 2 - 0«!40"' & y(x)»$%y (x) ,.*..29!9(3: y +(1 + x 2 ) y = 1V40>'40y( 1) = 0,: u n (x ) = x'2n 2y(1) = 0(1 x 2 ),n = 1,2,3:2u1( x ) = (1 x )u 2 ( x ) = x 2 (1 x 2 )u 3 ( x ) = x 4 (1 x 2 )22u1( 1) = (1 ( 1) ) = 020&40u1(1) = (1 (1) ) = 0u 2 ( 1) = ( 1) (1 ( 1) ) = 0u 2 (1) = (1)2 (1 (1)2 ) = 0u 3 ( 1) = ( 1)4 (1 ( 1) 2 ) = 0u 3 (1) = (1) 4 (1 (1)2 ) = 0b040V0!2' &1.:3y=n =1C n u n ( x ) = C1 (1 x 2 ) + C 2 x 2 (1 x 2 ) + C 3 x 4 (1 x 2 )y = C1 (1 x 2 ) + C 2 ( x 2U 2' &:2x 4 ) + C 3 (x 4x6)4C 2 x 3 + 4C 3 x 3: y = 2C1 x + 2C 2 xy = 2C1 + 2C 2 12C 2 x 2 + 12C 3 x 2030C 3 x 4012L[ y] = 2C1 + 2C 2 12C 2 x + 12C 3 x26C 3 x 5!44'!42 0:2230C 3 x + (1 + x ) (C1 (1 x ) ++ C 2 (x 2>0x 4 ) + C 3 (x 4x 6 ))2:L[ y] = 2C1 + 2C 2 12C 2 x 2 + 12C 3 x 2= C1 ( 2 + 1 x 4 ) + C 2 (2 12x 2 + x 2= C1 ( 1 x 4 ) + C2 (2 11x 2840' 130C 3 x 4 + C1 (1 x 4 ) + C 2 ( x 2x 6 ) + C3 (12x 2 30x 4 + x 4x 6 ) + C3 (12x 2 29x 4x8 ) =x8 ):«!"»$%x 6 ) + C 3 (x 4x8 ) =...30R(x, C1 , C 2 , C 3 ) = L[ y] f ( x )R(x, C1 , C 2 , C 3 ) = C1 ( 1 x 4 ) + C 2 (2 11x 2>(-1, 1)0(x 6 ) + C 3 (12 x 204029 x 4x 8 ) ( 1)' &)':x1 = -0.5x2 = 0x 3 = 0.4800 1 40'':R(-0.5, C1, C2 , C3 ) == C1 ( 1 ( 0.5)4 ) + C2 (2 11 ( 0.5)2( 0.5)6 ) + C3 (12 ( 0.5)2 29 ( 0.5)4 ( 0.5)8 ) + 1 = 0R(0, C1, C2 , C3 ) == C1 ( 1 0 4 ) + C 2 (2 11 0 20 6 ) + C 3 (12 0 229 0 408 ) + 1 = 0R(0.4, C1, C2 , C3 ) == C1 ( 1 0.4 4 ) + C 2 (2 11 0.4 2>0.4 6 ) + C 3 (12 0.4 2*44 ',29 0.4 400.4 8 ) + 1 = 00:1.0625 C1 0.7656 C 2 + 1.1836 C 3 + 1 = 0C1 + 2 C 2 + 1 = 01.0256 C1 + 0.2359 C 2 + 1.1769 C 3 + 1 = 0-0010:1.0625 C1 0.7656 C 2 + 1.1836 C 3 = 1C1 + 2 C 2 = 11.0256 C1 + 0.2359 C 2 + 1.1769 C 3 = 11.0625=211.0256=0.7656 1.183620.235910 = 1.25351.17691.06251 1.183611.025610 = 0.04331 1.17691=111.06253=11.02560.7656 1.183620.23590.765620.23590 = 1.1671.176911 = 0.039510«!"»$%..=1.1670= 0.9311.25352=0.0433= 0.03451.25353=0.0395= 0.03151.2535!: yC1 =1C2 =C3 =D.31>0.931 (1 x 2 ) 0.0345 ( x 20& 40'0x 4 ) 0.0315 ( x 4'x6):y( 0.5) = 0.931 (1 ( 0.5)2 ) 0.0345 (( 0.5)2 ( 0.5)4 ) 0.0315 (( 0.5)4 ( 0.5)6 ) = 0.6903y(0) = 0.931 (1 02 ) 0.0345 (02 04 ) 0.0315 (04 06 ) = 0.931y(0.4) = 0.931 (1 0.4)2 ) 0.0345 (0.42 0.44 ) 0.0315 (0.44 0.46 ) = 0.7767\ 2 4,0,*:: (-1, 0),2( 1, 0) ,': (-0.5, 0.6903),( 0, 0.9310),( 0.4, 0.7767).«!"»$%.*..32!9: y + (1 + x 2 ) y = 1-!y( 1) = 0,U1'x1 = -0.5h=& 40-yi +1 yi2h1'x 3 = 0.5y(x)yi =yi +1 2 yi + yihh>x2 = 0x 3 = 0.5[ 21!0x1 = -0.5x i (i22 y 2 + y10 .5y4'!0:x1, x 2 , x 3 :2 y1 + y 00 .5y34) 0+ (1 + x i2 ) yi = 10y20, i1442yi .20yi +1 2 yi + yix4 = 1xi& 40 ' y(x)40 :81 ( 1)= 0 .54:x2 = 0Dyi =y(1) = 0[-1, 1]&x 0 = -1(42+ (1 + 0 2 ) y 2 = 12y 3 + y 20 .520+ (1 + ( 0.5) 2 ) y 1 = 1+ (1 + 0.5 2 ) y 3 = 1& : x 0 = -1 y 0 = 0x 4 = 1 y4 = 0«!"»$%!,!0...330020&:y0 = 0y 2 2 y1 + y020.52 y 2 + y1y3y40.522 y3 + y 20.52y4 = 0y0 = 0+ (1 + ( 0.5) 2 ) y1 = 12+ (1 + 0 ) y 2 = 14 (y 22 y1 ) + 1.25 y1 = 14 (y 32 y 2 + y1 ) + y 2 = 14 ( 2 y 3 + y 2 ) + 1.25 y 3 = 12+ (1 + 0.5 ) y3 = 1y4 = 0y0 = 06.75y1 + 4 y 2 = 14 y1 7 y 2 + 4 y 3 = 16.75 y 3 = 14y 2y4 = 06.75y1 + 4 y 2 = 1D!4 y1 7 y 2 + 4 y 3 = 1:4y 2-0016.75 y 3 = 10& 40x1 , x 2 , x 3 :=26.754040744 = 102.93756.756.7510=40114 = 99.56256.750:y1 =74.25= 0.721102.9375y2 =99.5625= 0.967102.9375y3 =74.25= 0.721102.937513«==!14011744 = 74.256.756.754140741 = 74.251"»$%'.- 0 !..34&' :01234\0&-xy-1-0.500.5100.7210.9670.7210&2,[-1, 1].«!"»$%.
