Методические указания к практическим занятиям по обыкновенным дифференциальным уравнениям (1012977), страница 4
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6 4 2L%5714± 04 222:1=1-5252=1-5255(4;1 ), 5: y = (C1 + C 2 x ) e: y = (C1 + C 2 x )e1/ 2x(71x21"121=2==122=124) k = 2..".#$%. 474.3.% : y + 4 y + 5y = 0$G$%)$:1.52. ( 27 552:7 5+4 +5= 0:D = 16 4 5 1 = 41, 23. 6 4 214±2=44± 42=214±2 i= 2±i2=1= 2+i2= 2 i:= 2+i - 54 5- 4 D45,7= 2,4 5- 4 D45,7= 2,=12= 2 i - 5=1%7: y = C1 cos(1 x ) e2x+ C 2 sin(1 x ) e1: y = C1 cos x e"2x+ C 2 sin x e2x= 2+i2= 2 i2x4.4.% : y (6) + 18 y ( 4) + 81y = 0$G$%)$:1.52.
( 27 5522(4=0( 2:7 52+ 18426+ 18:+ 81) = 0+ 18=02+ 81 = 01, 2=± 0 =04+ 812=0.".#$%( 26:t 1,2 =%LL%72=t,4+ 81 = 0 - C 5:4 81=18 ± 0= 92t1 =2= 93, 4=±9 =± 91 = ±3it2 =2= 95,6=±9 =± 91 = ±3i4) k = 2.:=0 -52=0 -55(4;1 ), 5(73= 3i - 54 5- 4 D5,7= 0,=35= 3i - 54 5- 4 D5,7= 0,=35(4;1 ), 5(7) k = 2.44= 3i - 54 5- 4 D5,7= 0,=36= 3i - 54 5- 4 D5,7= 0,=3(;1 ), 554(7) k = 2.4:y = (C1 x + C 2 ) e 0 x1"t 2 + 18 t + 81 = 0 - 57155225L+ 1818 ± 18 2273. 6 4 24.
48=02=0+ (C 3 + C 4 x ) cos(3 x ) e 0 x3= 3i3+ (C 5 + C 6 x ) sin(3 x ) e 0 x= 3i: y = C1 x + C 2 + (C 3 + C 4 x ) cos(3x ) + (C 5 + C 6 x ) sin(3x )4= 3i6= 3i.".#$%4.2. )%$'%&$ %$% !%&$. 49= " 1%%&0) + Q..)H)$%" 0) (=!$ $>$%)$ 4.6.45 G)::a n y (n) + a n 1y (n31)+ .... + a 2 y + a 1 y + a 0 y = f ( x )– a n , a n 1 .....a 1 , a 04.2.1.7$G$%)$0$"0 # !) H)) =! )L# >F%&( = " 1%%&(+?1. ( 2'D7472;, : 434C142.( x ) = C1 y1 ( x ) + C 2 y 2 ( x ) + ...
+ C n y n ( x )y2.(:4:4:5y( x ) = C1 ( x ) y1 ( x ) + C 2 ( x ) y 2 ( x ) + ... + C n ( x ) y n ( x )3.:C1 y1 ( x ) + C 2 y 2 ( x ) + ... + C n y n ( x ) = 0C1 y1 ( x ) + C 2 y 2 ( x ) + ... + C n y n ( x ) = 0C1 y1" ( x ) + C 2 y 2 " ( x ) + ... + C n y n " ( x ) = 0......C1 y1( nL3 C4. ( 25.1)(x ) + C 2 y 2 ( n7 5,1)( x ) + ... + C n y n ( n,:1)(x) =:;f (x)anC1 , C 2 ....C n .: C1 , C 2 ....C n .C j (x) , 43C j (x ) :~C j ( x ) = C j ( x )dx + C j6. %47D2: 4.
2. %72..".#$%. 504.5.% : y + 2y + y =xex$G$%)$:1. ( 2D2;: y + 2 y + y = 07+ 2 +1 = 0D = 4 4 1 1= 02± 0= 12=1, 2+=(yx= C1 e{+ C2 1x2e 3x2 x)ey1 ( x )2. 64= 12= 1xy11y2 ( x )724:4:5xy = C1 ( x ) e{+ C 2 (x ) 1x2e 3xy1 ( x )y2 ( x )3.:xC1 eC1 (ex+ C2 x ex) + C 2 (x e4. ( 2=0x) =xC1 e:4+ C2 x exxx+ C 2 (e=0x eC1 + C 2 x = 0x)=exC1 + C 2 (1 x ) =x: C1 = x C 23%:x C2 + C2x C2 =:C1 ( x ) =5. %x:C1 e4.xe1x51dx = x + C€12: 4. 2:C2 =1,x3C1 = 1:C 2 (x ) =1dx = ln x + C€ 2x1x:.".#$%y = ( x + C€1 ) ex€ ) x e+ (ln x + C2xy = C€1 e x + C€ 2 x e x x e x + ln x x e x 144424443 1444424444327: y=( x+"1)ex.
5122+ (ln x +2)x ex4.6.% : y + 4y =1sin 2 x$G$%)$:1. ( 2D2;: y + 4 y = 07+4=01, 2=±4 = ±2 i1= 2i2=2i= C1 cos(2 x ) + C 2 sin( 2 x )14243123yy1 ( x )2. 647y2 ( x )24:y = C1 ( x ) cos(2 x ) + C 2 ( x ) sin( 2 x )14243123y1 ( x )3.y2 ( x ):C1 cos(2x ) + C 2 sin( 2x ) = 0C1 (cos(2x )) + C 2 (sin(2x )) =4. ( 21sin 2 x:C1 cos(2x ) + C 2 sin( 2x ) = 02C1 sin( 2x ) + 2C 2 cos(2x ) =:43: C1 =1sin 2xC 2 sin(2x )cos(2x )4:5:.".#$%%:2C 2 sin 2 (2 x )1+ 2C 2 cos(2 x ) =cos(2 x )sin(2 x )2C 2 sin 2 (2x ) + 2C 2 cos 2 (2 x )1=cos(2 x )sin(2 x )C 2 = ctg(2 x ) ,4.:C1 ( x ) =5. %2C 21=cos(2 x ) sin(2x )C1 = 1351dx = x + C€12.
52:C 2 ( x ) = ctg (2 x )dx =1€ln sin(2 x ) + C22: 4. 2:€ ) cos(2 x ) + (0.5 ln sin(2x ) + C€ ) sin(2x )y =( x+C12€ sin(2x ) x cos(2x ) + 0.5 ln sin(2 x ) sin(2x ) y = C€1 cos(2x ) + C144442424443 144444442444444432"72€ ) cos(2x ) + (0.5 ln sin(2x ) + C€ ) sin(2x ): y =( x+C122.".4.2.2.$"7#$%=,5 32 ! * "%4!$G$%)1 >17)cos( x )&$ ef ( x ) = # Pm ( x ) ''$'(sin( x ) $%(4•CD5C=$H) >F% ' =! # ' * "FM4:44344x)cos( x )&'$ '$'(sin( x ) $%7323 74.x,:,m.C :D51 $' ) f ( x )f (x ) = 5 ,C)cos( x )&$ e: Pm ( x ) ''$'(sin( x ) $%7Pm ( x ) = p o + p1 x + p 2 x 2 + p 3 x 3 + ...
+ p m x m 4•x. 53':(C0x: f ( x ) = 5{ e{D=pof ( x ) = x cos(3x ) ,C=10x: f ( x ) = {x cos(3x ) e{14243D= p1xf ( x ) = 6 e 3x ,CD=13x: f ( x ) = 6{x e{= p1xf ( x ) = ( x + 10x 2 ) sin( x ) e2x,CD2xf ( x ) = ( x + 10x 2 ) sin( x ) e{14243 123= p1x + p 2 x 2+?1. ( 22.'D5 D33(7));, : 43(C1<2y( x ) = C1 y1 ( x ) + C 2 y 2 ( x ) + ... + C n y n ( x )34744.::.".m -#$%54xx3Pm ( x ) ,3 7.
543Dx,m=0-5 G4cos( x ) ,4G5 43x3. 3 4444.5 D4:,35:5C./ Dx574m -4m443 44s73G5 4;1C :4523 744;1,x m ), :: 5 7y74:5D:45 G: 4.1 ,( +i )5s = 0.= x s [Q m ( x ) cos( x ) + R m ( x ) sin( x )] e3 7:7 5 37,3 44( +i ),77x,,34x4,:3 44 .-5 G4x3-5 G4x3s - 435,7 5 3Q m ( x ), R m ( x ) -m -4( 740,C1C1xs - 4C).3 44-5 G53 44,3Q m ( x ) = b k + b k +1 x + b k + 2 x 2 + ... + b k + m x m x:4: 4= x s Q m (x) ey7C13 44; 3 443 44= 0,4D,: m4)G5 433D= 0.,43=0sin( x ) ,-5 Gsin( x ) ,cos( x )5 57G5 4cos( x ))4sin( x )3 4443 44m.".5.#$%4: 75 G:5y 5 D34: 4G4245.3. %C44.%;: 46. 6 42C1 373 C5 G24G;4C2573;32« 4»,444D7;:y IV + 4 y = 022(1, 2=0+ 4) = 0=03, 4= ±2 i= C1 + C 2 x + C 3 cos(2x ) + C 4 sin(2x )y2.54775;45:13:5x=0= 0 m =123:e 2x=2=0 m=0C71-4bk .5 G.;124C :$G$%)$:+4..% : y IV + 4 y = 5x + e 2 x43x4.7.1.
( 25.5 5 «( 2477 5;1:,+ # y7: y=y, 47bk .723,345:.557,5.5. %D335.4. ( 242.5 G77,75.2. %6bk5 G:5.1. %4. 5532»,.".#$%3. ' 5 5 53:,3 445. 56,5 D33 44 .13:5x=0= 0 m =11 3 4423:e 2x=2=0 m=02 3 444.5 D%7y713 4432:: 4=0: y7712S=2= x 2 ( b 0 + b1 x ) e 0 x:=2=0 m=0= x S (b 2 ) e 2 x2+i = 2+i 0 = 25:= x S ( b 0 + b1 x ) e 0 x7y72= 0 m =1+i = 0+i 0 = 057: y776. 6 42S=0= (b 2 ) e 2 x5C1 3: y=y2+ y71+ y7: y = C1 + C 2 x + C 3 cos(2 x ) + C 4 sin( 2 x ) + x 2 (b 0 + b1 x ) + b 2 e 2 x"4.8.% : y + 4 y = cos(2x ) + 4 x 23x sin(2x )$G$%)$:1.
( 2;1y + 4y = 02+4=01, 2y= ±2 i= C1 cos(2 x ) + C 2 sin( 2 x )D7;:2–.".#$%2.543:cos(2 x )=0=2 m=023:4x 2=0=0 m=233:3x sin( 2 x )=0= 2 m =135124)5 D%71:13. ! 44y773cos(2 x )3:3x sin( 2 x )3:4x 23 4423=0:7: y71 3 44=0=0 m=22 3 4472:= 2 m =112= x [(b 0 + b1 x ) cos(2 x ) + (b 2 + b 3 x ) sin(2x )]:=0=0 m=2= x S (b 4 + b 5 x + b 6 x 2 ) e 0 x2: y776. 6 4"= 2 m =1S =1+i = 0+i 0 = 05=0= x S [(b 0 + b1 x ) cos(2 x ) + (b 2 + b 3 x ) sin( 2 x )] e 0 x7y7:: 4+ i = 0 + i 2 = 2i54. 5725S=0= b 4 + b5 x + b 6 x 2C1 32: y=y+ y71+ y72:y = C1 cos(2 x ) + C 2 sin( 2 x ) + x [(b 0 + b1 x ) cos(2 x ) + (b 2 + b 3 x ) sin( 2 x )] + b 4 + b 5 x + b 6 x 2.".#$%. 584.9.% : y + 4 y = 4x 2x$G$%)$:1. ( 2;1D7;:y + 4y = 02+4=01, 2= ±2 i= C1 cos(2 x ) + C 2 sin( 2 x )y2.543:4x 2=0=0 m=223:x=0= 0 m =131524x 2:x347K3 442y7:4=03=0:=0 m=2: 471 3 442:=0 m=2= x S (b 0 + b1 x + b 2 x 2 ) e 0 x+i = 0+i 0 = 05:13.
! 444.77: y75.:5 G:= b 0 + b1 x + b 2 x 25 Gy7= b 0 + b1 x + b 2 x 2y7= b1 + 2 b 2 xy7= 2 b2y7+ 4y 7= 4x 2S=07x2 b 2 + 4 (b 0 + b1 x + b 2 x 2 ) = 4x 2x324.".#$%%5 G454:2 b2 + 4 b0 = 0x0:4x4 b1 = 1x24 b2 = 47:b0 =1b1 =124b2 = 15: y77%21 x + x245 :y7=1y7=1 +2 x4y7=2y7+ 4y 72+4 ( 11 x + x242= 4x 224x 2x4x 22+ y7xxx6. 6 4y=yx1 x + x 2 ) = 4x 242 2 x + 4x 2 = 4x 2"1=:1: y = C1 cos(2x ) + C 2 sin(2x )121 x + x24x.
5947.".#$%5.P(Dr = ( x , y, z ) T -55-. 60-545, 45, t-D4754;14()4F( t , r, r& ) , 3D757 5,, r& = ( x& , y& , z& ) T -4.:D:d 2rm 2 = F( t , r, r&)dt45DCG54:5D5,, CM:d2xm 2 = X( t , x , y, z, x& , y& , z& )dtd2ym 2 = Y( t , x, y, z, x& , y& , z& )dtd 2zm 2 = Z( t , x, y, z, x& , y& , z& )dt(=!$ $>$%)$ 5.1.,:5445 D:-x ( t ), y( t ), z( t )-x& , y& , z&-4:=!$ $>$%)$ 5.2. %:4,.5:G5,55744=!$ $>$%)$ 5.3. ( 2555 4474:::@5::t4: 5):5 D5C 1 ;Dx ( t ), y( t ), z( t ) ,75 :4555:l5D, 54.:x& , y& , z& ,..5 43553477:5.".#$%. 61x& = uy& = vz& = wmu& = X( t , x, y, z, u, v, w )mv& = Y( t , x , y, z, u , v, w )& = Z( t , x , y, z, u , v, w )mwC;,74:l1-3 45 ,: 2.=!$ $>$%)$ 5.4.:: 24:,1:1-345 ,.:y1 = f1 ( x , y1 , y 2 ,...., y n )y 2 = f 2 ( x , y1 , y 2 ,...., y n )...........y n = f n ( x , y1 , y 2 ,...., y n )%5$!$(4"n-l.C1)): y( x ) = y1 ( x ), y = y 2 ( x ), y = y 3 ( x ), ...., y ( nC : 7D7= !1 + + % !0 >F% ' ) "$0$% : y ( n ) = f ( x , y, y , y ,...y ( n' 35: 41)= y n (x) .:y1 = y 2y2 = y3...........y n = f ( x , y1 , y 2 ,...., y n )44C::5.1.
) "$0& >)%$'%&(Dn-35% !5%&(4(545 ;75,5 ;7).= " 1%%&0) + Q..)H)$%" 0).".#$%. 62=!$ $>$%)$ 5.5.45 G() n -3 45::x& 1 = a 11x1 + a 12 x 2 + ... + a 1n x nx& 2 = a 21 x1 + a 22 x 2 + ... + a 2n x n........x& n = a n1 x1 + a n 2 x 2 + ... + a nn x n6;dx i, i = 1..n - 4dtt , x& i =K4$": 4:5::4: ;x 1 ( t ), x 2 ( t ) ....x n ( t )5::x 1 ( t ), x 2 ( t ) ....x n ( t )5: x ( t ), y( t ), z( t ), u ( t )) +>M*$%)145 n5n -3 45 5,2n -3 4DC5 .5.1.% :x& = 2x + 3yy& = 4 y + x$G$%)$:::4%3:;4%246 +5= 0 -/6 4&y& 6 y& + 5 y = 05 :57 5.:D = 62=: &y& 4 y& = 2( y& 4 y) + 3y47( 21D2-3 4x = y& 4 y; x:x : x& = &y& 4 y&577( 2561624 5 = 16=1527 5 32=6 + 16=52–4: y = C1e t + C 2 e 5 t,4..
..".#$%Dx = y& 4 y ,54. 63; y& = C1e t + 5C 2 e 5 t ,:3 :x = C1e t + 5C 2 e 5 t 4(C1e t + C 2 e 5 t ) = 3C1e t + C 2 e 5t14424431442443y&":yx = 3C1 e t + C 2 e 5 ty = C1 e t + C 2 e 5t;+"a 11 a 12aa 22: A = 21.... ....a n1 a n 21.2.5n5:1,7 52 ... n ./.... a 1n.... a 2n.... ........ a nn(B )E) = 0: det(A57 5 3– G3C: 7A.3. %:5 D53 41,33.1.3 5A3.2.C: (Aj3.3.
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