Сборник задач по высшей математике 2 том
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HbIM HHTerpHpOBaHHeM npaBoil: 'faCTH.CHa'faJIanOHH3HM CTeneHb KocHHyca:
16cos3 2x = 8(1 +cos4x)cos2x = 8cos2x+4· (2 cos4xcos2x) =
= 8 cos 2x + 4· (cos2x + cos6x) = 12 cos2x + 4 cos6x.
IIocJIe nepBoro HHTerpHpOBaHH~ nOJIY'faeM |
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y" = 6 sin 2x + ~ sin 6x + eX - |
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x + C1 , |
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(6.6) |
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nOCJIe BToporo |
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y' = -3 cos 2x - ~ cos6x + eX - |
~2 + C1 x + C2, |
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(6.7) |
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a nOCJIe TpeTbero - |
06TIIee perneHHe |
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3 . 2 |
1 . 6 |
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X |
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x |
3 |
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C x 2 |
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+ |
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C |
(6.8) |
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Y = -2 sm x - |
54 sm x + e |
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(; + |
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12 |
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2 X + |
3· |
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IIo,n;cTaBJI~~ B (6.6)-(6.8) |
3Ha'feHH~ |
x = 0, |
y |
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= -1, y' = -~, y" |
= 3, |
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nOJIy'fHMCOOTBeTCTBeHHO: 3 = 1 + C1 , |
-~ = -3 - ~ + 1 + C2 , |
-1 = 1 + C3 , |
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oTKy,n;a C1 = 2, C2 |
= 2, C3 = -2. IIo,n;cTaBJI~~ 9TH 3Ha'feHH~ B BblpaJKeHHe |
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,LI;J1~ y, nOJIY'faeMTpe6yeMoe 'faCTHOeperneHHe |
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3 . 2 |
1 . 6 |
x |
+ e |
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x 3 |
+ x |
2 |
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x- |
2 |
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Y = - - sm x - - sm |
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- - |
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3aMe"l,a'ltue. IIpoH3Be,n;eHH~ npOH3BOJIbHbIX nOCTO~HHbIX Ha KOHKpeTHble 'fHCJIa TaK:lKe MO:lKHO C'fHTaTb npOH3BOJIbHbIMH nOCTO~HHbIMH. HanpHMep,
B (6.8) |
- 2 |
C1 x 2 |
MO:lKHO nHcaTb C1x |
BMeCTO -2-' |
Pewum'b c.//,eoY1O'l4ue oug)(pepe'lt'll,ua'//''b'lt'ble ypa6'1te'lt'IJ.R 6'blCWUX nopSlo-x;o6, a maM, zoe UMe10mCSl 'lta"l,a'//''b'lt'ble YC.//,06'IJ.R, 'ltaiJ.mu coom6emcm6Y1O'l4ue "I,acm- 'It'ble pewe'lt'IJ.R:
2.6.36. |
yIV = cos2x. |
2.6.37. |
y(9) = ebx. |
2.6.38. |
ylll = 6x2. |
2.6.39. |
ylll = 4cos3 x-x. |
2.6.40.ylll = cos X cos 2x cos 5x.
2.6.41. ylll = x 2 + 3x - 1, y(O) = 1, y'(O) = 2, y"(O) = 3.
2.6.42.yV = sin~, y(O) = y'(O) = 1, y"(O) = 8, ylll(O) = 6, yIV(O) = -2.
2.6.43.ylll = (x !42)5' y(O) = 0, y'(O) = 2, y"(O) = -~.
2.6.44.PernHTb ,n;Hq,q,epeHIJ;HaJIbHOe ypaBHeHHe 'feTBepToronop~,n;Ka
2yIV =3W.
Q ~ho ypaBHeHHe HMeeT BH,n; F(x, ylll, yIV) = 0, n09TOMY em nop~,n;OK MO:lKHO CHH3HTb Ha TpH e,n;HHHIJ;bI npH nOMOTIIH 3aMeHbI ylll = p, yIV = p'. IIpHXO,n;HM K ypaBHeHHIO 2p' = 3 vp. PacCMOTPHM ,n;Ba CJIy'fM:1) ecJIH p = 0,
90
T. e. y'" |
= 0 TO Y = CI X 2 + C2 X + C3 - |
He o6m:ee pemeHHe; 2) eCJIH p of. 0, |
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TO ~3 dp |
= dx. OTclO)J,a nOJIyqaeM p~ |
= x + CI , HJIH y'" = ±(x + cd3/ 2 . |
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nOCJIe)J,oBaTeJIbHble HHTerpHpOBaHHa )J,alOT |
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2( |
C)Q. |
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Y = ±5 x + |
I 2 |
+ 2, |
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y' = ±3~(X+ CI)~ + C2 x + C3 , |
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8 |
9 |
- |
2 |
+ C3 x + C4 • |
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Y = ±315(x |
+ Cd 2" + C2 x |
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TaKHM 06pa30M, 06m:ee pemeHHe HMeeT BH)J,
y = ±3~5v(x + Cd 9 + C2 X 2 + C3 x + C4 ,
K KOTOPOMY npHcoe)J,HHHM nOJIyqeHHOe paHee He o6m:ee pemeHHe y = CI X 2 +
+ C2 x + C3 • |
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Pewum'b aaHHbte aurj)(jjepeHv,ua.t!'bHbte ypaflHeH'U.R 6btCWUX nopSlaX:06:
2.6.45. |
(1 + X2 )ylll + 2xy" = x 3 • |
2.6.46. |
x 4 y'" |
+ 2x3 y" = 1. |
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2.6.47. |
x 4 y IV + 2X3 ylll = 1. |
2.6.48. |
ylll + y" tg x |
= sin 2x. |
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2.6.49. |
xylll - y" = x 2 ex . |
2.6.50. |
yIV - |
2(y'" - |
1) ctg x = O. |
2.6.51. HathH qacTHoe pemeHHe )J,H<p<pepeHIIHaJIbHOrO ypaBHeHHa
xy'" - y" + x 2 - 2 = 0,
y)J,oBJIeTBopalOm:ee HaqaJIbHbIM YCJIOBHaM y(l) = 2, y'(l) = -~,
y"(l) = -3.
Q .n;aHHoe ypaBHemt:e HMeeT BH)J, F(x, y", y"') = 0, T. e. He cO)J,epX{HT aBHO y
H y'. IIo9TOMY nOJIOlKHM y" = p, y'" = p'. IIoJIyqaeM JIHHeiiHOe OTHOCHTeJIbHO HeH3BecTHoii <PYHKIIHH p = p(x) ypaBHeHHe
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p 2 |
p - |
x = x-x. |
Ero 06m:ee pemeHHe HMeeT BH)J, (npoBepbTe!) p = x ( CI - X - ~). HaM OCTa-
eTca pemHTb npocTeiimee )J,H<p<pepeHIIHaJIbHOe ypaBHeHHe y" = CI X - x2 - 2.
ITO)J,CTaBHB H3 HaqaJIbHbIX YCJIOBHii x = 1, y" = -3, = O. lIHTerpHpya nOJIyqalOm:eeca paBeHCTBO y" = _x2 - 2, HMeeM y' = - ~3 - 2x + C2 .
CHoBa nO)J,CTaBJIaa HaqaJIbHble YCJIOBHa x = 1, |
y' |
= -~, HaxO)J,HM C2 = 2. |
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4 |
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+ 2x + C3 • |
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3HaqHT, y' = - ~ - 2x + 2. |
OTclO)J,a y = - f2 - |
x 2 |
HaKoHeII, |
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1 |
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+ C3 , |
nOJIyqHM C3 |
13 |
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yqHTbIBM, qTO y(l) = 2, T. e. 2 = -12 - 1 + 2 |
= 12' H, |
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CJIe)J,oBaTeJIbHO, y" - |
- 12 - |
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+ x + |
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2.6.52. PemHTb )J,H<P<PepeUIIHaJIbHOe ypaBHeHHe y"(l + 2lny') = 1.
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a B 9TOM ypaBHeHlIH flBHO OTCYTCTBYIOT H apryMeHT x, H HCKOMM <PYHKlI,Hfl. IT09TOMY ero MO:lKHO OTHeCTH H K THny F(y, y', y") = 0, a, 3HaqHT, MO:lKHO
nOJIO:lKHTh y' = p = p(y), y" = p. p', H K THny F(x,y',y") |
= 0, |
a, 3HaQHT, |
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MO:lKHO nOJIO:lKHTh y' = p = p(x), y" = p'. |
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1) B nepBoM CJIYQae npHXO,1I,HM K ypaBHeHHIO pp' (1 + 2ln p) |
= 1. 3,1I,eCh |
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p |
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= dy' a ypaBHeHHe nOCJIe pa3,1I,eJIeHHfl nepeMeHHhIX MO:lKeT |
hITh 3allHCaHO |
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BH,1I,e p(l + 2lnp) dp = dy. OTCIO,1I,a Jp(l + 2lnp) dp = Jdy, T. e. (nocJIe |
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HHTerpHpOBaHHfl no QacTflM) p2lnp = y + C1 , HJIH (y')2In y' = y + C1 • ITo- |
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JIYQHJIH ypaBHeHHe, He pa3peIlIeHHoe OTHOCHTeJIhHO y' (H He pa3peIlIeHHoe OTHOCHTeJIhHO y'). Ero npOHHTerpHpOBaTh HeJIh3fl.
2) Bo BTOPOM CJIYQae npHXO,1I,HM K ypaBHeHHIO p'(l + 2lnp) = 1. 3,1I,eCh
dp
p' = dx' a ypaBHeHHe MO:lKeT 6hITh 3allHCaHO B BH,1I,e dp( 1+ 2ln p) = dx. OTCIO-
,1I,a J(1+2Inp) dp = Jdx, T.e. 2plnp-p = x+C2, HJIH 2y'lny'-y' = x+C2.
8TO ypaBHeHHe TaK:lKe HeJIh3fl npOHHTerpHpOBaTh.
3) Pe3YJIhTaThI npe,1I,hI)J.yIIIHX ,1I,eiiCTBHii MO:lKHO 06'he,1I,HHHThH nOJIYQHTh napaMeTpHQeCKYIO <POPMY 06IIIero peIlIeHHfl HCXO,1I,HOrO ypaBHeHHfl. ITOJIo- :lKHM y' = t. Tor,1I,a
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y + C1 = t2 ln t, |
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{ |
= 2t In t - |
t |
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x + C2 |
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- 06IIIee peIlIeHHe ,1I,aHHoro ypaBHeHHfl B napaMeTpHQeCKOii <popMe. |
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Pew,um'b |
aurj)(pepeuv,uaJl,'b'H,'bl.e ypa6ueuUSl: |
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2.6.53. |
(yll)2 + (y"')2 = 1. |
2.6.54. |
y'ylll = 3(y")2. |
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2.6.55. |
xy'" + y" = X + 1. |
2.6.56. |
(yll)2 _ 2y'y" + 3 = O. |
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2.6.57. |
(yll)2 - y'y'" = (~r |
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2.6.58. |
ylll = 3yy', yeO) = y'(O) = 1, y"(O) = ~. |
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2.6.59. |
y" + 2y"ln y' = 1, yeO) = 1, y' (0) = -1. |
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3 |
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2.6.60. |
yllly2 - 3yy'y" + 2(y')3 + ~(yyll - (y')2) = Y2' |
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PeUJ,umb aurj)(jjepeHv,UallbH'bI.e ypa6HeH'U.R.:
2.6.61.(1 + X2)y" + (y')2 + 1 = O.
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y' |
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2.6.62. |
xy" = y' In x' |
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2.6.63. |
xy'" + y" = 1 + x. |
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2.6.64. |
(1 + x 2)y" - 2xy' = 0, y(O) = 0, y'(O) = 3. |
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2.6.65. |
y' |
+ lnx, |
1 |
y"(1 + lnx) + x = 2 |
y(I) = 2'y'(I) = l. |
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2.6.66. |
y" = ~ (1 + In ~), |
y (1) = |
~, y' (1) = l. |
2.6.67. |
yy" + (y')2 _ (y')3In y = O. |
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2.6.68. |
y'" = (y")3. |
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2.6.69. |
(x + I)y" - (x + 2)y' + x + 2 = O. |
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2.6.70. |
3y'y" = y + (y')3 + 1, y(O) = -2, y'(O) = O. |
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2.6.71. |
y2 + (y')2 _ 2yy" = 0, y(O) = y'(O) = l. |
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2.6.72. |
2yy" - 3(y')2 = 4y2, y(O) = 1, y' (0) = O. |
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2.6.73. |
y' = X(y")2 + (y")2. |
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KOHTponbHble Bonpocbl III 60nee CnOJKHble 3aACIHIIIR
2.6.74.IIo'ieMYo6~ee peWeHlIe )J.H<p<pepeHIJ;MaJIhHoro ypaBHeHMH BToporo
nOpH.II.Ka CO)J,ep:lKMT POBHO )J,Be nocToHHHhle? KaKYIO POJIh MrpaIOT OHM B CTpyKType 06~ero peweHMH?
2.6.75.MorYT JIM 'iepe3TO'iKY (xo,Yo) nJIOCKOCTM Oxy npOXO)J,MTh nHTh pa3JIM'iHhIX 'iacTHhIXpeweHMtl: )J,M<p<pepeHIJ;MaJIhHOro ypaBHeHMH TpeThero nopH)J,Ka? BTOpOro nOpH)J,Ka?
Pew,umb aurj)(jjepeHv,uallbH'bI.e ypa6HeHU.R.:
2.6.76. y'" (1 + (y')2) - 3y' (y")2 = O.
2.6.77.2yy" + (y")2 + (y')4 = O.
2.6.78.yy" + (y')2 = y2ln y.
2.6.79.yy" = (y')2 + y' Vr.y2"+----,-(y....,,')"'2.
2.6.80.1 + (y')2 = 2yy", y(I) = y'(I) = l.
2.6.81.y' = xy" + y" _ (y")2.
2.6.82.(x - I)y'" + 2y' = x + 1.
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2X2 |
2.6.83. |
X(y')2 y" |
= (y')2 + lx4. |
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3 |
2.6.84. |
';1 - x 2y" + JI - (y')2 = O. |
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93
§7. lU1HEViHbiE AVltlltllEPEHU.VlAl1bHbIE YPABHEHVlH BblCWErO nOPHAKA
np~BaplilTenbHble CB~eHIIISI
~JIU7-l.eti'H,'bIM 'HeoiJ'HOpoiJ'H'bIM iJug)(pepe'H'Il,ua.l/,'b'H'bIM ypaB'He'HUeM Bmopo~o nopS!iJ'lCa
Ha3bIBaeTCH YPaBHeHHe BH,lla
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y" + p(x)y' + q(x)y = f(x), |
(7.1) |
r,lle <PYHKIIHH p(x), q(x) H |
f(x) HenpepbIBHbI Ha HeKoTopoM OTpe3Ke [a, b]. |
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IIpH 3THX YCJIOBHHX cYIIIecTByeT e,llHHCTBeHHoe pellIeHHe ypaBHeHHH (7.1), Y,llo- |
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BJIeTBOpHIOIIIee 3a,llaHHbIM |
Ha'la.JIbHbIM YCJIOBHHM: y(xo) = Yo, y' (xo) = |
Yo npH |
Xo E [a,b].
«lIYHKIIHH f(x) Ha3bIBaeTCH npaBoti 'I.aCm'b1O ypaB'He'HUS! (7.1), a COOTBeTCTBYIOIIIee ypaBHeHHe Ha3bIBaeTCjI TaK)Ke JIHHeitHbIM ,llH<p<pepeHIIHa.JIbHbIM ypaBHeHHeM BToporo nOpH,llKa C npaBoit '1aCTbIO.IIpH f(x) == 0 rrpHXO,llHM K JIHHeitHoMY O,llHoPO,llHOMY ,llH<p<pepeHIIHa.JIbHOMY ypaBHeHHIO BTOpOro nOpH,llKa (HJIH ypaBHeHHIO 6e3 npaBoit '1aCTH)
y" + p(x)y' + q(x)y = O. |
(7.2) |
11111HeMHo He3aBIIIClilMbie (jlYHK4111111
~«lIYHKIIHH Y1 (x) H Y2(X) Ha3bIBalOTCH .l/,U'Heti'HO 'He3aBUCUM'bI.MU Ha OTpe3Ke [a, b],
eCJIH TO)K,lleCTBO
(7.3)
HMeeT MeCTO TOr,lla H TOJIbKO TOr,lla, KOr,lla C1 = C2 = O.
ECJIH)Ke CYIIIeCTBYIOT TaKHe '1HCJIaC1 H C2, H3 KOTOPbIX XOTH 6bI O,llHO OTJIH'IHO OT HyJIH, 'ITO,IIJIH Bcex x E [a, b] HMeeT MeCTO TO)K,lleCTBO (7.3), TO <PYHKIIHH Y1 (x)
H Y2(X) Ha3bIBaIOTCH .l/,U'Heti'Ho 3aBUCUM'bI.MU Ha OTpe3Ke [a, b]. $:
.n;aHHbIe onpe,lleJIeHHH PaBHOCHJIbHbI CJIe.rryIOIIIHM:
<PYHKIIHH Y1 (x) H Y2 (x) Ha3bIBalOTCH JIHHeitHo He3aBHCHMbIMH (3aBHCHMbIMH) Ha OTpe3Ke [a, b], eCJIH
Y1 (x) |
Y1(X) |
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x E [a,b]. |
-(-) 1= const |
( Y2(X) |
== const , |
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Y2 x |
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o JIHHeitHoit 3aBHCHMOCTH HJIH He3aBHCHMOCTH <PYHKIIHit Y1 (x) H Y2 (x) MO)KHO |
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CY,llHTb no Onpe,lleJIHTeJIIO |
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Y2(X )1 |
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y~(x) |
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KOTOPbIit Ha3bIBaeTCH onpeiJe.l/,Ume.l/,eM Bpo'HC'lCo~O (HJIH npOCTO BpO'HC'lCUa'HOM). |
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TeopeMa 2.3. Ecnlll Y1(X) III |
Y2(X) nlllHef.iHo 3aBlllCIIIMbi Ha OTpe3Ke [a,b]. TO |
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W[Y1, Y2] = 0 Ami BCex x 1113 [a, b]. |
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94
TeopeMa 2.4. Ecnlll Yl(X) III Y2(X) nlllHeCiHO He3aBlllCillMbie Ha OTpe3Ke [a,b] peweHIll" AIllcj>cj>epeHlIlllanbHoro ypaBHeHIll" (7.2), TO onpeAenlllTenb BpOHCKoro 3TIIIX cj>YHKlIIilCi OTnlll'leHOT Hyn" BO Bcex TO'lKaXOTpe3Ka [a, b].
CTpyKTypa o6w.ero peweHMSI T1MHeMHoro
AM4J4JepeH4MaTlbHOrO ypaBHeHMSI
TeopeMa 2.5. 061l.1ee peweHllle Yoo nlllHeCiHorO OAHopOAHoro AIllcj>cj>epeHlIlllanbHoro ypaBHeHIll" (7.2) IIIMeeT BillA
rAe Yl(X), Y2(X) - nlllHeCiHO He3aBIIICillMbie peweHIll" 3TOrO ypaBHeHIII".
TaKHM 06pa30M, MH TOro, '1To6bI nOJIy'lHTb o6nree peIlleHHe O.D;HOpO.D;HOro
ypaBHeHHH (7.2), .D;OCTaTO'lHO Hail:TH JII06bIe .D;Ba JIHHeil:Ho He3aBHCHMbIX '1aCTHbIX peIlleHHH 3Toro ypaBHeHHH (B 3TOM CJIy'laerOBopHT, 'ITOOHH o6pa3yIOT rjJyxiJaMex-
ypaBHeHHH
B HeKoTopbIX CJIY'laHXy.D;aeTCH TeM HJIH HHbIM cnoco6oM Hail:TH TOJIbKO O.D;HO
qacTHoe peIlleHHe Yl (x). Tor.D;a .D;pyroe '1aCTHOepeIlleHHe Y2 (x) MO)KHO Hail:uI no |
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<p0pMYJIe |
[ |
x |
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Y2(X) = Yl (x) . ! + .exp - |
jp(x) dX] dx, |
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Yl (x) |
xo |
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r,ll,e Xo E [a,b]. |
06a peIlleHHH Yl(X) H Y2(X) npH 3TOM JIHHeil:Ho He3aBHCHMbI. |
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TeopeMa 2.6. 061l.1ee peweHllle YOH nlllHeCiHoro HeOAHopOAHoro AIllcj>cj>epeHlIlilanbHoro ypaBHeHIll" (7.1) npeACTaBmleTC" B BIllAe CYMMbl
YOH = Yoo + Y'i'
rAe Yoo - 061l.1ee peweHllle COOTBeTCTBYlOll.IerO OAHopOAHoro ypaBHeHIll" (7.2), a
Y'i - HeKOTopoe 'laCTHOepeweHllle HeOAHopOAHoro ypaBHeHIll" (7.1).
nMHeiiiHble OAHOpoAHble AM4J4JepeH4MaTlbHble ypaBHeHMSI BTOpOrO nopSiAKa c nOCTOSlHHblMM K034J4JM4MeHTaMM
~PaCCMOTpHM JIHHeil:Hoe O.D;HOpO.D;HOe .D;H<p<pepeHD;HaJIbHOe ypaBHeHHe (JIO,lJ,Y)
BTOporo nOpH.D;Ka C nOCTOHHHbIMH K03<P<PHD;HeHTaMH |
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Y" +PY' + qy = O. |
(7.4) |
KBa,ll,paTHOe ypaBHeHHe |
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k 2 + pk + q = 0 |
(7.5) |
Ra3bIBaeTcH xapafCmepUCmU"I.eCfCUM ypaBxexueM MH ypaBHeHHH (7.4). |
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95
,Il;JUI COCTaBJIeHHH 06rn;ero pellIeHHH Yoo .D;Hq,q,epeHII,HaJIbHOro ypaBHeHHH (7.4)
He06xo.D;HMo HaitTH KOpHH kl H k2 cooTBeTcTBYIOrn;ero XapaKTepHCTH'IeCKoroypaBHeHHH (7.5) H rrpHMeHHTb CJIe.n;yIOllIJ'1OTeopeMY:
TeopeMa 2.7. nYCTb kl M k2 - KOpHM xapaKTepMCTM'leCKOrO ypaBHeHMft Allft ypaBHeHMft (7.4). TorAa 06Ll\ee peweHMe ypaBHeHMft (7.4) HaXOAMTCft no OAHOIiI M3 clleAYIOLl\MX Tpex <PoPMYll:
1) ECJ1M kl M k2 - AelilcTBMTellbHble M kl #- k2, TO
Yoo = ek1X(Cl + C2 X)i
3) eCllM kl,2 = (} ± (Ji - KOMnlleKCHo-COnpft>KeHHble KOPHM, TO
Yoo = eOX(Cl cos(Jx + C2 sin (Jx).
llMHeMHbie HeoAHOpOAHble AM~~epeH4ManbHble ypaBHeHMSI
(llHAY) BTOpOrO nopSiAKa C nOCTOSlHHblMM
K03~~M4IIIeHTaMM
IIOCKOJIbKY 06rn;ee pellIeHHe Yoo JIHHeitHoro O.D;HOPO.D;HOro ypaBHeHHH (7.4) JIerKO HaxO.D;HTCH rro TeopeMe 2.7, TO B CHJIY TeopeMbI 2.6 .D;JIH HaxolK.D;eHHH 06rn;ero
pellIeHHH JIHHeitHoro HeO.D;Hopo.D;HOrO YPaBHeHHH |
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y" +py' +qy = f(x) |
(7.6) |
OCTaeTCH HaitTH KaKoe-HH6Y.D;b O.D;HO ero '1acTHoepellIeHHe y... B Tex CJIY'IaHX,KOr.D;a rrpaBaH '1acTbf (x) HMeeT CrreU;HaJIbHblit BH.D;, '1acTHoepellIeHHe y.. HeO.D;Hopo.D;HQ-
ro ypaBHeHHH HaxO.D;HTCH MemoiJO.M neonpeiJeAenfl'btX 7Co3gupuquenmoB. 9TOT MeTO.D;
Ha3bIBaeTCH TaKlKe MeTO.D;OM rro.D;60pa '1acTHoropellIeHHH Heo.D;HOpO.D;HOro ypaBHeHHH H CBO.D;HTCH K CJIe.n;yIOIlI;HM .D;BYM CJIY'IaHM.
CJlyqa§: 1. f(x) = eOx Pn(X), r.D;e
a) ECJIH (} He HBJIHeTCH KopHeM ypaBHeHHH (7.5), TO '1aCTHOepellIeHHe y.. MOlKHO HCKaTb B BH.D;e
y.. = eOXQn(X},
r.D;e Qn(X) - MHOrO'lJIeHCTerreHH n C HeH3BecTHblMH K03q,q,HU;HeHTaMH.
6} ECJIH (} - KopeHb YPaBHeHHH (7.5) KpaTHOCTH k, TO '1aCTHoepemeHHe y..
MOlKHO HCKaTb B BH.D;e
y.. = XkeoxQn(X).
B '1acTHOCTH,eCJIH f(x) = Pn(X), T. e. (} = 0, TO y.. HMeeTCH B BH.D;e y.. = Qn(X)
(eCJIH (} = 0 He HBJIHeTCH KopHeM XapaKTepHCTH'IeCKoroYPaBHeHHH) HJIH B BH-
.D;e y .. = xk . Qn(X} (eCJIH (} = 0 - KOpeHb KpaTHOCTH k XapaKTepHCTH'IeCKOrO YPaBHeHHH).
96
CnY'Iail 2. f(x) = eQ",[pn(x)cos,Bx + Qm(x)sin,Bx], r,ll;e Pn(X) H Qm(X) -
MHoraqJIeHbI CTerreHH n H m, COOTBeTCTBeHHO. IIoJIolKHM N = max(n, m).
a)ECJIH a ±,Bi He HBJIHIOTCH KOpHHMH ypaBHeHHH (7.5), TO
6)ECJIH a ±,Bi - KOpHH ypaaHeHHH (7.5) KpaTHOCTH k, TO
B qacTHOCTH, eCJIH f(x) |
= acos,Bx + bsin,Bx, T. e. a = m = n = 0, TO qacTHoe |
pemeHHe Hrn;eTCH B BH,lI;e Y'l |
= A cos,Bx + B sin,Bx (eCJIH qHCJIa ±,Bi He HBJIHIOTCH |
KOpHHMH XapaKTepHCTHqeCKOrO ypaaHeHHH) HJIH B BH,lI;e Y'l = (Acos,Bx+Bsin,Bx)·x
(eCJIH qHCJIa ±,Bi - KOpHH XapaKTepHCTHqeCKOrO ypaaHeHHH).
TeopeMa 2.B. Ecnlll Y'll III Y'l2 - 'laCTHblepeWeHlIIlI COOTBeTCTBeHHO ypaBHeHllliil
y" + py' + qy = /1 (x)
III
y" + py' + qy = /2(x),
TO cPYHKlIIIIlI Y'l = Y'll + Y'l2 - 'laCTHOepeweHllle ypaBHeHlIIlI
y" + py' + qy = /1 (x) + /2(x).
MeTOA Bapllla4111111 npOlll3BOnbHbiX nOCTOJlHHblX
AflJl onpE!AeneHIIIJI '"IaCTHOrOpeWeHIIIJI HeoAHOpoAHOrO
ypaBHeHIIIJI
B 06rn;eM CJIyqae, B TOM qHCJIe TOr,ll;a, KOr,ll;a rrpaaaH qacTb .ITH,I1;Y HMeeT BH,lI;, He rrpe)J;yCMOTpeHHblil: rrpe)J;bl)J;yrn;HM rrYHKTOM, )J;JIH OTblCKaHHH qaCTHOro pemeHHH HCIIOJIb3YIOT MemoiJ 6apuaquu (T. e. H3MeHeHHH) npOU360A'bH"'X nOCmOJlHH"'X (HJIH
MemoiJ JIaapa'H:JICa). CYTb era B CJIe)J;yIOrn;eM. IIycTb Yl(X) H Y2(X) - <pYH,lI;aMeHTaJIbHaH CHCTeMa pemeHHil: O,ll;Hopo,ll;HOrO ypaaHeHHH (7.4). Tor,ll;a qacTHoe pemeHHe MOlKHO rrpe,ll;CTaaHTb B BH,lI;e
Pa3YMeeTCH, MeTO,ll; BapHarr;HH rrpOH3BOJIbHbIX rrOCTOHHHbIX MOlKHO rrpHMeHHTb H B CJIyqae, KOr,ll;a rrpaaaH qacTb .ITH,IJ;Y HMeeT BH,lI;, paccMoTpeHHblil: B rrpe,ll;bI)J;yrn;eM IIYHKTe.
4 CooPH•• _ . no ""oweA NareNarr.... 2 "YJlC |
97 |
YpaBHeHllle 3iiinepa
~,II;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe BTOpOrO nOpH,!I;Ka BH,!I;a
x 2y" +pxy' +qy = 0
Ha3b1BaeTCH ypa6He'HUeM 91:t.n,epa.
TIO,!l;CTaHoBKoit y = e t OHO CBO,!l;HTCH K .nO,II;Y C nOCTOHHHbIMH K03<P<PHIJ;HeH-
TaMH.
0606Iu;eHHOe ,!I;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe 8itJIepa BTOpOro nOpH,!I;Ka
(ax + b)2y + p(ax + b)y' + qy = 0
npHBO,!l;HTCH K .nO,II;Y C nOCTOHHHbIMH K03<P<PHIJ;HeHTaMH nO,!l;CTaHoBKoit ax+b = et.
11111HeiiiHbie AIII«t>«t>epeH4I11anbHble ypaBHeHIIISI C nOCTOSlHHblMIil K03«t>«t>1II4I11eHTaMIil nopSiAKa Bblwe BToporo
.nHHeitHble ,!I;H<p<pepeHIJ;HaJIbHble ypaBHeHHH C nOCTOHHHbIMH K03<P<PHIJ;HeHTaMH
nOpH,!I;Ka n ~ 2
peIIIaIOTCH aHaJIOrHqHO ypaBHeHHHM BToporo nOpH,!I;Ka, onHpaHCb Ha cooTBeTcTBYIOIu;He onpe,!l;eJIeHHH H TeopeMbI. B qaCTHOCTH:
~ 1. CHcTeMa <PYHKIJ;Hit YI = YI(X), |
Y2 = Y2(X), ... , Yn |
= Yn(X) Ha3b1BaeTCH |
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.n,u'He1:t'Ho 3a6ucuMo1:t Ha oTpe3Ke [a, b], |
eCJIH cYIu;ecTBYIOT nOCTOHHHble CI, C2,. ... , |
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Cn, He Bce paBHble HYJIIO TaKHe, qTO HMeeT MeCTO TO)K,!I;eCTBO |
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CIYI + C2Y2 + ... + CnYn == 0, |
x E [a, b]. |
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ECJIH )Ke 3TO TOlK,!I;eCTBO HMeeT MeCTO TOJIbKO npH CI = C2 |
... = Cn = 0, TO |
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,!I;aHHble <PYHKIJ;HH .n,UHe1:tHo He3a6UCU,M,'bI. Ha OTpe3Ke [a, b]. |
$ |
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~ BpOHC7CuaH CHCTeMbI <PYHKIJ;Hit YI, Y2, ... , Yn HMeeT BH,!I; |
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YI |
Y2 |
Yn |
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, |
, |
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YI |
Y2 |
Yn |
(n-l)
Yn
~2. Xapa1l:mepucmu"I,ec1I:oe ypa6HeHue, cooTBeTcTBYIOIu;ee O,!l;HOPO,!l;HOMY ,!I;H<p<pe-
peHIJ;HaJIbHOMY ypaBHeHHIO n-ro nOpH,!I;Ka
(7.7)
HMeeT BH,!I;
(7.8)
98
TeopeMbI 2.3-2.6, C<P0PMYJIHpOBaHHble AJIH YPaBHeHHit BTOpOro IIopH,n;Ka, TaKlKe uMeiOT MeCTO H IIpH JIi060M n> 2. ~
3. MeTo,n; BapHaIJ;HH IIPOH3BOJIbHblX IIOCTOHHHbIX AJIH Heo,n;Hopo,n;Horo JIHHeitHGrO ,ll;H<p<pepeHIJ;HaJIbHOrO YPaBHeHHH TpeTbero IIOpH,ll;Ka C IIOCTOHHHbIMH K03<P<PHIJ;H-
ylll + py" +qy' + ry = f(x) |
(7.9) |
COCTOHT B CJIe.n;yiOIIJ;eM: o6IIJ;ee peIIIeHHe ypaBHeHHH (7.9) |
HMeeT BH,ll; |
r,ll;e Yl, Y2, Y3 - <PYH,ll;aMeHTaJIbHaH CHCTeMa peIIIeHHit O,ll;HOpO,ll;HOrO ypaBHeHHH
y"' + py" + qy' + ry = 0,
a <PYHKIJ;HH C1(x), C 2(x), C3(X) - HaXO,ll;HTCH H3 CHCTeMbI ,ll;H<p<pepeHIJ;HaJIbHblX
YPaBHeHHit IIepBoro IIOpH,ll;Ka
{C~Yl + C~Y2 + C~Y3 = 0,
C~y~ + C~y~ + C~y~ = 0,
C~y~' + C~y~ + C3Y~ = f(x).
2.7.1.YCTaHoBMTb JIMHeilHYIO 3aBMCMMOCTb MJIM He3aBMCMMOCTb ,1I,aHHbIX nap <PYHKU;Mil Ha 06JIaCTHX MX onpe,1I,eJIeHMH.
a) x, cos x; |
6) x, 2x; |
B) tgx, ctgx. |
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a a) WYHKU;MM Y1(X) = x M Y2(X) = cos x onpe,1I,eJIeHbI Ha Bceil npHMOil, T.e. |
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npIl x E (-00, +00). TO:>K,1I,eCTBO C1 x + C2 cos X == 0 MMeeT MeCTO TOJIbKO npM
C1 = C2 = O. B caMOM ,1I,eJIe, eCJIM npe,1I,nOJIO:>KMTb npOTMBHoe, T. e. qTO 9TO TO:>K,1I,eCTBO I1MeeT MeCTO, HanpIlMep, npIl C2 =F 0, TO nOCJIe ero ,1I,M<p<pepeHU;M-
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== |
0 |
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C1 |
' |
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pOBaHIIH nOJIyqIlM HOBoe TO:>K,1I,eCTBO C1 ~ C2 sm x |
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,OTKY,1I,a sm x == |
C |
2 |
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X E (-00,+00), qTO HeBepHO. ECJIM:>Ke npe,1I,nOJIO:>KMTb, qTO C2 |
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= 0, |
TO |
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nOJIyqMM C1 x == 0, qTO TaK:>Ke HeB03MO:>KHO npIl C1 =F |
O. TaKMM 06pa30M, |
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TO:>K,1I,eCTBO C1 x + C2 cos x |
== 0 MMeeT MeCTO TOJIbKO npIl C1 = C2 |
= 0, II, |
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CTaJIO 6bITb, <PYHKU;MII X |
II |
cos x |
JIMHeilHO He3aBIICMMbI Ha ,1I,eilCTBMTeJIbHOil |
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npHMoil. |
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3aMeTIIM, qTO co; X |
t; |
const |
II CO~X |
t; const, |
T. e. <PYHKU;III1 |
x |
II cos x |
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Y,1I,OBJIeTBOPHIOT II ,1I,pyrOMY onpe,1I,eJIeHIIIO JIMHeilHOil He3aBIICIIMOCTII. |
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6) lIMeeM ~ == 2 npM |
x =F |
0 (TO:>K,1I,eCTBO MO:>KHO ,1I,oonpe,1I,eJIIITb no·He- |
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npepbIBHocTII M npM x = 0), TI09TOMY <PYHKU;I1M Y1 = 2x if Y2 = X - |
JIIIHeilHO |
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3aBIICIIMbI. |
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B) BBII)J.y nepMO,1I,MqHOCTII <PYHKU;Mil Y1 |
= tg X II Y2 |
= ctg x C nepIlO,1I,OM |
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T = 71' ,1I,OCTaTOQHO YCTaHOBMTb I1X JIIIHeilHYIO He3aBMCMMOCTb B MHTepBaJIe |
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x E (0,71') (x =F !i ). lIMeeM Yy1 |
= ttgX = sin: x = tg2 X t; const x |
E (0,71'), |
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C gx |
cos x |
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99 |
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