Сборник задач по высшей математике 2 том

Jp-l

,dy

,naJIee, TaK KaK Y = dx' TO

KOHTPOll bHASI PA60TA

BapMaHT 1

1. PernUTb .n;u<p<pepeHU;UaJIbHble ypaBHeHUjI:

3. PernUTb ypaBHeHue: (x + y)y' - 1 = O.

80

4. PeIlIHTb yprumeHHe: (y3 + cos x) dx + (eY + 3xy2) dy = O.

5. 3a 30 ,n;Hei% pacnaJIOCb 50% nepBOHa'laJIbHOrOKOJIH'IecTBapa,n;H». Qepe3 CKOJIbKO BpeMeHH OCTaHeTC» 1% OT ero nepBOHa'laJIbHOrOKOJIH'IeCTBa,ecJIH CKOPOCTb pacna,n;a Pa,n;H» nponOpll;HOHaJIbHa ero KOJIH'IeCTBYB paccMaTpHBaeMbIi% MOMeHT?

BapMaHT 2

1. PeIlIHTb ,n;Hq,q,epeHll;HaJIbHbIe ypaBHeHH»:

Y

5. ABTOM06HJIb ,n;BIDKeTC» np»MOJIHHei%Ho co CKOPOCTbIO 30 M/C. 3a KaKoe BpeM» H Ha KaKOM paCCTO»HHH OH 6y,n;eT OCTaHOBJIeH TopM03aMH, eCJIH coIIpOTHBJIeHHe ,n;BIDKeHHIO IIOCJIe Ha'laJIaTopMOlKeHH» paBHO 0,3 ero Beca (g =

= 10 M/C2 )?

BapMaHT 3

1. PeIlIHTb ,LI;Hq,q,epeHll;HaJIbHbIe ypaBHeHH»:

a) ydx + ctgxdy,= 0, y (2;) = -2; 6) y'2x- y + 32x - y = O.

H TOpM03»lll;ei% CHJIbl conpOTHBJIeHH» B03,n;yxa, npoIIOpll;HOHaJIbHoi% CKOPOCTH (K09q,q,Hll;HeHT npoIIOpll;HOHaJIbHOCTH k). Hai%TH 3aKOH H3MeHeHH» CKOpOCTH v IIa,n;eHH» TeJIa, eCJIH B MOMeHT BpeMeHH t = to v = Vo = O.

BapMaHT 4

1. PeIlIHTb ,n;Hq,q,epeHll;HaJIbHbIe ypaBHeHH»:

a) (1 + y2) dx - .jXdy = 0; 6) y' + ycosx = cos x, y(O) = 2.

81

2.PemHTb ypaBHeHHe: 3x2y' = y2 + 8xy + 4x2.

3.PemHTb ypaBHeHHe: xy' + y = xy2.

4. PemHTb ypaBHeHHe: (sin 2x - 2 cos(x + y)) dx - 2 cos(x + y) dy = O.

5. TeJIo ,n;BIDKeTCH IIPHMOJIHHeilHO C yCKopeHHeM, IIPOIIOPIJ;HOHaJIbHbIM IIPOH3Be,n;eHHIO CKOPOCTH ,n;BIDKeHHH v Ha BpeMH t. YCTaHoBHTb 3aBHCHMOCTb Me)K,D;y CKOPOCTbIO H BpeMeHeM, eCJIH IIpH t = 0 v = Vo.

BaplilaHT 5

1. PemHTb ,n;Hq,q,epeHIJ;HaJIbHbIe ypaBHeHHH:

5. TeMIIepaTypa BbIHYToro H3 IIe'!HxJIe6a B Te'!eHHe20 MHHyT IIOHIDKaeT-

CH OT 100° ,n;o 60°. TeMIIepaTypa B03,D;yxa 20°. Qepe3 CKOJIbKO BpeMeHH OT Ha'!aJIaOXJIa)K,n;eHHH TeMIIepaTypa xJIe6a 6y,n;eT 30°? (YKa3aHHe: CKOPOCTb OXJIa)K,n;eHHH TeJIa IIPOIIOPIJ;HOHaJIbHa pa3HocTH TeMIIepaTYP TeJIa H cpe,n;bI.)

§6. Lt1HTErpLt1POBAHLt1E ALt1(J)(J)EPEHU.Lt1AJlbHbIX YPABHEHLt1~ BbICWLt1X nOPHAKOB

AIII(jJ(jJepeH4I11anbHble ypaBHeHIIIH BToporo nopHAKa. OCHOBHble nOHHTIilH. TeopeMa cyw.eCTBOBaHIIIH

III ~IIIHCTBeHHOCTIil

CBH3b1BaIOrn;ee Me:lK.rr.y co6oit He3aBHCHMYIO nepeMeHHYIO, HeH3BeCTHYIO <PYHKIJ;HIO

y(x), a TaK:lKe ee nepBble )l:Be npOH3BO)l:Hble y'(x) H y"(x), Ha3bIBaeTC:iI aurjjrjjepe'H,-

TO rOBOp:ilT, 'ITOOHO pa3pellieHO OTHOCHTeJIbHO BTOPOit npOH3BO)l:Hoit. MbI 6Y)l:eM HMeTb )l:eJIO TOJIbKO C TaKHMH ypaBHeHH:ilMH.

~3a)l:a'laOTblCKaHH:iI peIIIeHH:iI ypaBHeHH:iI (6.2), y)l:OBJIeTBOp:ilIOrn;ero 3a)l:aHHblM

82

~Pew,e1tUeM yprumeHHx (6.2) Ha3bIBaeTCX BCXKax ¢YHKU;HX Y = rp(x), KOTopax

npH nO,D;CTaHOBKe BMeCTe C y' H y" B 9TO ypaBHeHHe o6paru;aeT ero B TO:lK,D;eCTBO. fpa¢HK ¢YHKU;HH y = rp(x) B 9TOM CJIY'IaeHa3bIBaeTCX u1tmezpa.J!b1toit -X;pU60it. ~

~0614UM pew,e1tUeM ypaBHeHHx (6.2) Ha3bIBaeTCX ¢YHKU;HX y = rp(x, C 1 , C2),

3aBHcxru;ax OT ,D;BYX rrpOH3BOJIbHbiX rrOCTOXHHblX C 1 H C2 H TaKax, 'ITO:

~ JIro6ax ¢YHKU;HX y = rp(x, C?, C~), rroJIyqaroru;axcx H3 o6ru;ero pemeHHX YPaBrrpH KOHKpeTHblx 3Ha'leHHXXrrOCTOXHHblX C1 H C2, Ha3bIBaeTCX 'l.acm-

1tUM pew,e1tUeM 9Toro ypaBHeHHx. ~

,I1;.nx ,D;H¢¢epeHU;HaJIbHOrO YPaBHeHHX BToporo rrOpX,D;Ka (6.2) HMeeT MeCTO TeopeMa cyru;ecTBoBaHHx H e,D;HHCTBeHHOCTH pemeHHx, aHaJIOrH'IHaXcooTBeTcTByroru;eil: TeopeMe ,D;JIX ypaBHeHHiI: rrepBoro rrOpX,D;Ka.

ypaBHeHHX (6.2) rrpe,D;CTaBJIXeT co6oil: ceMeil:cTBo KPHBbIX, 3aBHcxru;HX OT ,D;BYX rrpoH3BOJIbHbiX rrOCTOXHHblX C 1 H C2. 3a,D;aqa KomH B TaKOM cJIyqae COCTOHT B orrpe,D;e- JIeHHH HHTerpaJIbHoil: KPHBOil: y = y(x), rrpoxo,D;xru;eil: '1epe3,D;aHHYro TOqKY (xo, yo) H HMeroru;eil: ,D;aHHblil: yrJIOBoil: K09¢¢HU;HeHT y~ KaCaTeJIbHoil: t (,D;aHHOe HarrpaBJIeHHe B ,D;aHHOil: TO'lKe(pHC. 8)), T. e. y' (xo) = y~ = tg a.

y = y(x)

x

Puc. 8

83

nOHIII>KeHllle nOp"AKa AIII~~epeHu.lllallbHbIX ypaBHeHllliii

B HeKOTOpblX '1acTHblXCJIY'IMXY)l;aeTCH rrOHH3HTb rrOpH)l;OK )l;H<p<pepeHIJ;HaJIbHOro yprumeHHH BTOpOro rrOpH)l;Ka.

3a'laCTYIOOHO B HTOre rrpHBO)l;HTCH K )l;H<p<pepeHIJ;HaJIbHOMY ypaBHeHHIO rrepBOrO rrOpH)l;Ka O)l;HOrO H3 H3Y'IeHHbIXpaHee THrrOB.

PacCMOTPHM HaH60Jlee THrrH'IHbleCJIY'IaH.

YpaBHeHMH BMACI '1/" = f{x)

HHTerpHpOBaHHeM 06eHx '1aCTeil:ypaBHeHHH y" = f(x) OHO rrpHBO)l;HTCH K YPaBHeHHIO rrepBoro rrOpH)l;Ka

y' = Jf(x) dx = F(x) + c1 •

IIoBTopHO HHTerpHpYH rrOJlY'IeHHOepaBeHCTBO, HaxO)l;HM o6IIIee pemeHHe HCXO)l;HOrO ypaBHeHHH:

AM~~epeH4ManbHble ypaBHeHMH F{x, y', yll) = 0, HBHO He cOAep)l(a~e

MCKOMOM ~YHK4MM Y

TaKHe ypaBHeHHH )l;OrrYCKaIOT rrOHH:lKeHHe rrOpH)l;Ka rrO)l;CTaHoBKoil: y' = p, y" =

= p'. ,I4>yrHMH CJIOBaMH, )l;aHHOe ypaBHeHHe paBHOCHJlbHO CHCTeMe )l;H<p<pepeHIJ;H-

aJIbHbIX YPaBHeHHiI: rrepBoro rrOpH)l;Ka

{y' =p,

F(x,p,p') = o.

AM~cJlepeH4ManbHble ypaBHeHMH F{y, y', yll) = 0, HBHO He cOAep)l(a~e

He3aBMcMMoM nepeMeHHoM x

YpaBHeHHH TaKOrO BH)l;a )l;OrrYCKaIOT rrOHH:lKeHHe rrOpH)l;Ka rrO)l;CTaHoBKoil: y' = = p = p(y) (<pOpMaJIbHOe OTCYTCTBHe aprYMeHTa x rr03BOJlHeT C'IHTaTbHeH3BeCTHYIO <PYHKIJ;HIO p <PYHKIJ;Heil: aprYMeHTa y), OTKY)l;a: y" = (P(y))' = p'(y) . y'(x) =

=p' 'p.

TaKHM o6pa30M, ypaBHeHHe F(y, y', y") = 0 PaBHOCHJlbHO CHCTeMe

{ y' =p,

F(y,p,p' p') = o.

ECJlH <PYHKIJ;HH F HBJlHeTCH O)l;HOPO)l;HOil: <PYHKIJ;Heil: CTerreHH k OTHOCHTeJlbHO rrepeMeHHbIX y, y' H y", T. e. F(x, ty, ty', ty") = t k • F(x, y, y', y"), TO )l;H<p<pepeHIJ;H-

aJIbHOe ypaBHeHHe F(x, y, y', y") = 0 )l;OrrYCKaeT rrOHH:lKeHHe rrOpH)l;Ka rrO)l;CTaHOB-

KOil:

y = eJp(x)dx,

r)l;e p(x) - HOBaH HeH3BeCTHaH <PYHKIJ;HH.

84

LIIHTerpMpOBaHMe AM(j>(j>epeHu.ManbHbIX ypaBHeHMiii nopHAKa BblWe BTOpOrO

IIpHeMhI, orIHcaHHble B rrpe,Ll;hI~m;eM rrYHKTe, MO)KHO paCrrpOCTpaHHTb Ha YPaBHeHHH 60JIee BbICOKHX rrOpH,LI;KOB.

06m;ee peIIIeHHe rrpocTeihrrero ,LI;H<p<pepeHII;HaJIbHOro ypaBHeHHH n-ro rrOpH,LI;Ka y(n) HaxO,Ll;HTCH n-KpaTHhIM HHTerpHpOBaHHeM <PYHKII;HH f(x) H CO,Ll;ep)KHT n rrpOH3BOJIbHhIX rrOCTOHHHhIX.

,I1;H<p<pepeHII;HaJIbHOe ypaBHeHHe BH,LI;a F(x, y(k), y(k+l), ... , y(n») = 0, He cO,Ll;ep-

)Kam;ee B HBHOM BH,LI;e HCKOMOfi <PYHKII;HH y, ,LI;OrrYCKaeT rrOHH)KeHHe rrOpH,LI;Ka rro,Ll;- CTaHOBKofi y(k) =p. ,I1;pyrHMH CJIOBaMH, ,LI;aHHOe ypaBHeHHe PaBHOCHJIbHO CHCTeMe

m;ee HBHO aprYMeHT x, ,LI;OrrYCKaeT rrOHH)KeHHe rrOpH,LI;Ka Ha e,Ll;HHHII;y rrO,Ll;CTaHOBKOfi

IIpH 3TOM (rro rrpaBHJIY ,LI;H<p<pepeHII;HpOBaHHH CJIO)KHofi <PYHKII;HH):

HT.,LI;.

2.6.1.HatiTH 06rn;ee pemeHHe )l;H<p<pepeHll;HaJIbHoro ypaBHeHHfl

y" = __1_ + x - sinx.

1 +x2

Q IIHTerpHpYfl, IIOJIyqHM

y' = 1(1 : x2 + x - sin x) dx = arctg x + x; + cos x + C1 •

TIoBTopHoe HHTerpHpOBaHHe (1arctg x dx Ha,n:O 6paTh IIO qacTflM: arctg x =

=u, du = ~,v = x, dx = dv) IIPHBO)l;HT K OTBeTY:

l+x

y =1(arctg x + ~2 + cos X + Cl) dx =

= xarctgx + ~In(l + x 2) + x; + sin x + C1x + C2. •

Ha11.mu o6w,ue peUteHWI OaHH'b/,X ou!Jj!JjepeH'4ua//,'bH'b/,X ypa6HeHu11.:

2.6.2.y" = sin4x + 2x - 3.

85

2.6.6.HaihH qacTHoe pemeHHe ,l1.aHHOrO ,l1.H<p<pepeHIJ;H8JIbHOrO ypaBHe-: HHH, y,l1.0BJIeTBOpHIOm;ee 3a,l1.aHHbIM Haq8JIbHbIM YCJIOBHHM:

y" = (e2X + sin3x)x, y(O) = 1, y'(O) = l.

a CHaq8JIa HaXO,l1.HM o6m;ee pemeHHe. lfHTerpHpYH no qacTHM, HaXO,l1.HM

y' = !(e 2X + sin 3x)x dx =

IIoBTopHoe HHTerpHpOBaHHe no qacTHM (npo,l1.eJIaitTe HY)l{HbIe BbIqHCJIeHHH) npHBO,l1.HT K o6m;eMY pemeHHIO:

B paBeHCTBax (6.4) H (6.5) nO,l1.CTaBHM x = 0, y' = 1, y = 1, oTKY,l1.a nOJIyqaeM CHCTeMY ypaBHeHHit OTHOCHTeJIbHO HeH3BeCTHbIX KOHCTaHT G1 H G2 :

Hait,l1.eHHbIe 3HaqeHHH nOCTOHHHbIX nO,l1.CTaBJIHeM B o6m;ee pemeHHe (6.5). IIoJIyqaeM HCKOMoe qacTHoe pemeHHe

Haii,mu "I,aCmH'bte peUteHWI OaHH'btX oUfp!fiepeH'I.{Ua.lI)bH'btX ypa6HeHuii" yOo6.1/,e- m60p.fl1O'I4ue 3aOaHH'btM Ha"l,a.l/,'bH'btM YC.I/,06W1M:

2.6.7.y" = (x 2 + 7x + 9)eX, y(O) = 1, y' (0) = 4.

2.6.8.y" - 2(2x2 + 2x - 5) cos2x - 4(2x + 1) sin2x, y = 0, y'(O) = o.

2.6.9.y" = l- 32 + 2cosx - xsinx, y(l) = 1, y'(l) = O.

x

2.6.10. y" = (4x 3 + lOx2 + 2x + 2)e2x + 6 sin 3x + 9xcos3x, y(O) = 1, y(O)' = l.

2.6.11.PemHTb ,l1.H<p<pepeHIJ;H8JIbHOe ypaBHeHHe y" - y' ctgx = 2x sin x.

HaitTH TaK)l{e qaCTHOe pemeHHe, eCJIH y = 1, y' = 0 npH x = i'

a ,IJ,aHHoe ,l1.H<p<pepeHIJ;H8JIbHOe ypaBHeHHe BTOpOro nOpH,l1.Ka He CO,l1.ep)l{HT HBHO HCKOMYIO <PYHKIJ;HIO y, T. e. HMeeT BH,l1. F(x, y', y") = O. IIoJIo)l{HM y' = p,

TOr,l1.a y" = p'. IIoJIyqaeM ,l1.H<p<pepeHIJ;H8JIbHOe ypaBHeHHe nepBoro nOpH,l1.Ka p' - p ctg x = 2x sin x - JIHHeitHoe OTHOCHTeJIbHO HeH3BecTHoit <PyHKIJ;HH p = p(x). 06m;ee pemeHHe ~noro ypaBHeHHH Hait,l1.eM nO,l1.CTaHOBKoit p = U· v, p' = u'v + uv'. IIoJIyqaeM:

86

113 nepBoro ypaBHeHH~ Haxo,n:HM In Ivl = In I sin xl, T. e. v = sin x. TIo,n:cTaBJI~~ BO BTopoe ypaBHeHHe, nOJIyqHM u' = 2x, oTKy,n:a U = x 2+Cl . CJIe,n:oBaTeJIbHO,

2.6.14.x 3 y"+x2y'-1=0. 2.6.15. y"+y'tgx-sin2x=0.

2.6.20.(1 + x 2)y" + 2xy' - x 3 = O. ,

2.6.21.HaihH 06rn;ee perneHHe ,n:H<p<pepeHI.J;HaJIbHoro ypaBHeHH~ y" tg Y =

=2(y')2. HaliTH TaK)I{e qacTHoe perneHHe, y,n:oBJIeTBOp~lOrn;ee HaqaJIbHbIM yCJIOBH~M y(l) = ~, y'(I) = -2.

Q ,Il;aHHoe ,n:H<p<pepeHI.J;HaJIbHOe ypaBHeHHe He co,n:ep)l{HT B ~BHOM BH,n:e aprYMeHTa x, T. e. HMeeT BH,n: F(y, y', y") = O. TIpHMeM B KaqeCTBe He3aBHCHMoli

HeHll.e y' = C l sin2 y, KOTopoe TaK)l{e ~BJI~eTC~ ypaBHeHHeM C pa3,n:eJI~IOrn;H-

dy

Mll.C~ nepeMeHHbIMH: - . - 2 - = C l dx, HJIH -ctgy = Clx + C2. TIOJIyqHJIH sm y

06rn;ee perneHHe Hcxo,n:Horo ypaBHeHH~ B He~BHOM BH,n:e. TIo,n:cTaBHM B Hero

87

KOHCTaHT B o6m;ee perneHHe, KOTopoe 3arrHrneM B BH.n;e y = arcctg( -Cl X-C2),

ypaBHeHHjlM rrepBoro rrOpjl,D;Ka H3BeCTHoro HaM THIIa. IIepeIIHrneM ero B BH,D;e

H y') ,D;Hq,q,epeHn:HaJIbHOe ypaBHeHHe IIepBoro rrOpjl,D;Ka, KOTopoe MO)l{HO pernHTb IIO,D;CTaHoBKoit y Ero o6m;ee perneHHe HMeeT BH,D; (Hait,D;HTe ero caMOCTOjlTeJIbHO)

+Cl).

Terrepb OCTaeTCjI pernHTb ,D;Hq,q,epeHI.I;HaJIbHOe ypaBHeHHe IIepBoro rropjl.n;Ka

He pa3perneHHoe OTHOCHTeJIbHO IIPOH3BO,D;HOit y'. Ho B 06m;eM BH,D;e pernHTb ero ,D;OCTaTOqHO XJIOrrOTHO. O,D;HaKO, TaK KaK HaM HY)l{HO HaitTH qacTHoe perneHHe HCXO,D;HOrO ypaBHeHHjI, TO BOCrrOJIb3yeMcjI HaqaJIbHbIMH YCJIOBHjlMH ,D;JIjI orrpe,D;eJIeHHjI IIOCTOjlHHOit Cl, IIOJIaraji B IIOCJIe,D;HeM paBeHCTBe y = 1 H y' = 1. IIpHxo,D;HM K paBeHcTBY

1=J2(~+Cl)'

H3 KOToporo Cl = O. TeM caMbIM, HaM ,D;OCTaTOqHO pernHTb ypaBHeHHe y = y',

OTKY,D;a y = Cex . IIoJIarM 3,D;eCb x = 0, y = 1, HaxO,D;HM C = 1. TaKHM o6pa30M y.. = eX - HCKOMoe qacTHoe perneHHe. •

Hatimu o6'l4ue pemeHUJI cAeay1O'l4UX aug)(pepeHv,uaA'bHUX ypaaHeHuti, a ma.M, eae Y'K:a3aHU Ha"taA'bHUe YCAOaUJI, Hatimu coomaemcmaY1O'l4ee "tacmHoe pemeHue:

2.6.25.1 + (y')2 - 2yy" = O. 2.6.26. 2yy" - 3(y')2 = 4y2.

2.6.27.y" = y'(1 + (y')2).

2.6.28.y" = y'lny', y(O) = 0, y'(O) = 1.

88

2.6.29.y" + y'V(y')2 -1 = 0, yen") = 0, y'(7r) =-1.

2.6.30.3y'y" = 2y, yeO) = y'(O) = 1.

2.6.31.y" = 2y3, yeO) = 0, y' (0) = 1.

2.6.32.npOHHTerpHpoBaTb AH<p<pepeHIJ;HaJIbHOe ypaBHeHHe BTOp01'O IIO-

pjl,II.Ka xy' (yy" - (y')2) = y(y')2 + x4 y3.

a nepenHIIIeM ypaBHeHHe B BHAe

xy' (yy" - (y')2) _ y(y')2 _ X4y3 = 0,

rrOCJIe 'Iero0603Ha'IHM'Iepe3F(x, y, y', y") JIeBYIO 'IacTbnOJIY'IeHHOrOpaBeHCTBa. EcJIH 3aMeHHTb y, y', y" Ha ty, ty', ty", COOTBeTCTBeHHO, TO IIpHXOAHM K paBeHCTBY

TaKOM CJIY'IaeCOOTBeTCTBYIOrn;ee ypaBHeHHe AOIIycKaeT IIOHillKeHHe IIOpjlAKa rroAcTaHoBKoti Y = eJ p dx, rAe P = HeH3BeCTHaji <PYHKIJ;HjI OT x.

OTCIOAa y' = peJ p y" = (p' +p2)eJ p nOCJIe COOTBeTCTBylOrn;HX 3aMeH H dx.

dx, °

COKparn;eHHjI Ha e 3 Jp dx f IIpHXOAHM K AH<p<pepeHIJ;HaJIbHOMY ypaBHeHHIO rrepBoro nOpjlAKa OTHOCHTeJIbHO p

HJIH

(ilpH 3TOM MbI TepjleM peIIIeHHe y == 0, KOTopoe IIOTOM HaAO Ao6aBHTb K OTBeTY). nOJIY'IHJIHypaBHeHHe BepHYJIJIH, KOTopoe MO)l{HO peIIIHTb (IIpeAJIaraeM CAeJIaTb 3TO caMOCTOjlTeJIbHo) IIOACTaHoBKoti p = uv. Ero o6rn;ee peIIIeHHe

HMeeT BHA

P=XVX2 +C1 •

nOCKOJIbKY y = eJp dx, TO HaxOAHM CHa'IaJIa

Jpdx = JXVX2 + C1 dx = lV(x2 + CJ)3 + C2,

a 3aTeM H o6rn;ee peIIIeHHe HCXOAH01'OAH<p<pepeHIJ;HaJIbHOrO ypaBHeHHjI (3a BbI'IeTOM'IaCTHOropeIIIeHHjI y = O)HMeeT BHA

C3 > 0.

Y'IHTbIBM,'ITOIIpH C3 = °KaK pa3 nOJIY'IaeTCjIIIOTepjlHHOe 'IaCTHOe peIIIeHHe y = 0, npHXOAHM K OKOH'IaTeJIbHOMYOTBeTY: y = C3 . etV(X2 +C2 )3,

Pew,um'b iJu!p!pepeHv,Ua.J!'bH'bI.e ypa6'lteH'l.I.R.:

2.6.33.x 2yy" = (y - Xy')2. 2.6.34. 2yy" - 3(y')2 = 4y2.

2.6.35. HatiTH 'IaCTHOepeIIIeHHe AH<p<pepeHIJ;HaJIbHo1'OypaBHeHHjI TpeTbe-

1'0 nOPMKa y'" = 16 cos3 2x + eX - 1, YAOBJIeTBOpjllOrn;ee Ha'IaJIbHbIM YCJIOBHjlM yeO) = -1, y'(O) = -!, y"(O) = 3.

89