Сборник задач по высшей математике 2 том

2.7.113.	1f, arctg2x, arcctg2x, x E (-00,+00).
2.7.114.	sin2 x, cOS2 X,				cos2x, sin2x, x E (-00,+00).
2.7.115.	sin 30:, sino:, sin30:, 1,0: E (-00,+00).
2.7.116.	coso:, cos3 0:, cos 30:, 5,0: E (-00,+00).
2.7.117.	In x, In x 2, In2 x, In3 x,						x E (0, +00).
2.7.118.	eX	,	eX sin2 x	,	eX cos2 x	"	e-2x x E (-00 +00)
							, .

IIor;;a3amb, "tmo He cyw,ecmeyem .IIUHeiJ.H020 OaHOpOaH020 aug)(pepeHv,ua.ll'O- H020 ypaeHeH'USI emop020 nOPJl&a c nOCmOJlHHUMU r;;03!Jj!Jjuv,ueHmaMu, a.llJl r;;omop020 aaHHaJI cucmeMa !JjYHr;;v,uiJ. Jle.llJlemCJI !JjYHaaMeHma.llbHoiJ.:

2.7.119.	sin x, sin 2x.			2.7.120.	cosx, cos2x.
2.7.121.	1			2.7.122.	2	1
	x, x"				x,	x'
2.7.123.	X .			2.7.124.	eX, cosx.
	e ,Slnx.
2.7.125.	x, sinx.			2.7.126.	x, cosx.
2.7.127.	x, x	2	•	2.7.128.	eX, e- x, cosx.
			, Slnx.
2.7.129.	x, x2, cosx.			2.7.130.	x, e	X .
						,smx.

Cocmaeumb .IIUHeiJ.Hoe OaHOpOaHoe aU!Jj!JjepeHv,ua.llbHOe ypaeHeHue c nocmoJlHHUMU r;;03!Jj!Jjuv,ueHmaMu, !JjYHaaMeHma.llbHaJI cucmeMa pemeHuiJ. r;;omopo20 uMeem eua:

2.7.131. e2x , e-3X , eX		2.7.132.	1, eX, e3x .
2.7.133.	cos2x, sin2x, e-x.	2.7.134.	eX, xex, sinx, cosx.
2.7.135.	eX, xex , x2ex, e3x	2.7.136.	eX, xex, sin3x, cos3x.

Cocmaeumb .IIUHeiJ.Hoe aU!Jj!JjepeHv,ua.llbHOe ypaeHeHue c nepeMeHHuMu r;;03!Jj- !Jjuv,ueHmaMu no aaHHoMY 06w,eMY pemeHU70:

2.7.137.

2.7.139.

2.7.141.

2.7.142.

2.7.143.

2.7.144.

C2	2.7.138.
Y =C1 x+ 2'
x
y = C1 X 2 + C2 X 4 + C3 •	2.7.140.
Y = C1 + C2 (x + 1)5 + (	C3 )2

x+1

y = (C1 coslnx + C2 sinlnx)x + xlnx.

y = x(CI + C2 lnx + In2 x).

y = C1x + C2X 2 + 1 + (x 2 + 2x) lnx.

Pemumb ypaeHeH'USI:

2.7.145.	y" - 4y' + 3y = O.	2.7.146.	y" + 4y' + 29y = O.
2.7.147.	9y" + 6y' = O.	2.7.148.	4y" + 12y' + 9y = O.
2.7.149.	5y" + y = O.	2.7.150.	5y" + y' = O.
2.7.151.	y'" - 2y" - 3y' = O.	2.7.152.	y'" + 4y" + 13y' = O.
2.7.153.	y'" + 2y" + y' = O.	2.7.154.	y'" + 2y" - y' - 2y = O.
2.7.155.	yIV - 16y = O.	2.7.156.	yIV + y = O.

110

2.7.157. 2ylll + 9y" + 17y' + 14y = O.

2.7.158. ylV + y" = O.

Pewum'b ypamte'H,'IJ.H" a maM, eiJe ecm'b Ha"ta.lt'bH'bte YC.lt06'IJ.H., Hai1.mu coom6emcm6Y70w,ee "tacmHoe peweHue:

2.7.159.	y" + y' = (x + ~) eX -				2x - 2.
2.7.160.	y" + y' = (1- 4x)e-2x .				2.7.161.		y" + y' = 3e- 2x sinx.
2.7.162.	y" + y' = (2x + 3) sinx + cosx.
2.7.163.	y" -	2y' + y = sin x + e- x.
2.7.164.	ylll + y" = 12x2.
2.7.165.	ylll -	5y" + 8y' -	4y = e2x .
2.7.166.	y" -	3y' + 2y = (x2 + x)e3x .
2.7.167.	y" -	2y' + 3y = e- x cosx.
2.7.168.	y" + y' = cos2 X + eX + x 2 •
2.7.169.	y" + 4y = xsin2 x.
2.7.170.	y" + 4y = x cos x.
2.7.171.	y" -	2y' + lOy = ~ cos 3x + 2 sin 3x.
2.7.172.	y" -	3y' + 2y = sinxsin2x.
2.7.173.	y" -	4y' + 5y = (4x + 22) sin3x -				(28x + 84) cos3x.
2.7.174.	y" -	4y' + 5y = 2x2ex , y(O) = 2,				y'(O) = 3.
2.7.175.	y" -	6y' + 9y = x	2	- X	+ 3, y(O) = ~,		1
							y'(O) = 27.
2.7.176.	y" + 4y = 4(sin 2x + cos 2x), y(7r) = 7r, y'(7r) = 27r.
2.7.177.	y" -	2y' + 2y = 4ex cos x, y(7r) = 7re7r,					y' (7r) = e7r •
2.7.178.	y" -	2y' + lOy = lOx2 + 18x + 6, y(O) = 1, y'(O) = 3,2.
2.7.179.					3
	4y" + 16y' + 15y = 4e- 2x , y(O) = 3, y'(O) = -5,5.
2.7.180.	y" -	2y' = ex(x2 + x -			3), y(O) = 0, y'(O) = 2.
2.7.181.	y" + y = ctgx.				2.7.182.		y" + 2y' + y =	e~x.
2.7.183.	y" + y = ~.				2.7.184.		y" - 2y = 4x2ex2 .
		smx
					2.7.186.		y" +y =	1

cos 2xv'cos 2x

KOHTponbHble Bonpocbl III 60nee CnO)l(Hbie 3aACIHIIISI

2.7.187. npHBeCTfI npHMep <PYHKIl;Hil: Yl(X) H Y2(X), KOTopbIe JIHHeil:Ho 3aBHCHMbI Ha O,D:HOM OTpe3Ke H JIHHeil:Ho He3aBHCHMbI Ha ,D:pyroM.

2.7.188. ,n:oKa3aTb, qTO eCJIH ,D:Ba qaCTHblX pemeHHfI JIHHeil:Horo O,D:HOPO,D:HOro ,D:H<p<pepeHIJ:HaJIbHOrO ypaBHeHHfI BToporo nOpfl,D:Ka HMeIOT 9KCTpeMyMbI B O,D:HOil: H TOil: :lKe TOqKe, TO OHH JIHHeil:Ho 3aBHCHMbI.

2.7.189. KaKHM YCJIOBHflM ,D:OJ1:lKHbI y,D:OBJIeTBOpflTb K09<P<PHIJ:HeHTbI p H q ypaBHeHHfI y" + py' + qy = 0, qT06bI Bce ero qaCTHbIe perneHHfI 6bIJ1H OrpaHHqeHHbIMH.

111

2.7.192.

2.7.194.

2.7.195.

2.7.196.

2.7.197.

2.7.190. llOCTPOIITb .IJ:Be .IJ:II<p<pepeHn:IIPyeMble JIIIHeil:Ho He3aBIICIIMhIe <PYHKn:1I1I Ha OTpe3Ke [a, b], .IJ:JIjI KOTOPbIX IIX onpe.IJ:eJIIITeJIb BPOHCKoro paBeH Hymo TO)K.IJ:eCTBeHHO.

2.7.191. Ha OTpe3Ke [a, b] nOCTpOIITb Tpll JIIIHeil:Ho He3aBllCIIMble <PYHK- n:1I1I, .IJ:JIjI KOTOPbIX onpe.IJ:eJIIITeJIb BpoHcKoro paBeH HYJIIO TO)K-

.IJ:eCTBeHHO.

,lfolCa3amb .//,u'ltefJ:HY70 3a6UCUMocmb rjjy'ltIC'qui1 'Ita ux o6'//'acmu onpeiJMe'ltW&:

sin4 x, cos4x, cos2x, 1. 2.7.193. cos4 x, cos4x, cos2x, 1.

lnx, Inx2 , Inx3, In2 x, In3 x. sinx, sin (x - ~), sin (x + ~).

3HM <PYH.IJ:aMeHTaJIbHYIO cllcTeMY perneHllil: eX, cos x, sin x JIIIHeil:Horo O.IJ:HOPO.IJ:HOrO ypaBHeHlljI, Hail:TII ero qacTHoe perneHlle, Y.IJ:OBJIeTBOpjlIOIIJ:ee HaqaJIbHbIM YCJIOBlljlM y(O) = 3, y'(O) = 4 II y"(O) = -1.

<l>YHKn:1I1I eX, e2x , e3x o6pa3YIOT <PYH.IJ:aMeHTaJIbHYIO CIICTeMY perneHllil: JIIIHeil:Horo O.IJ:HOpO.IJ:HOrO .IJ:II<p<pepeHn:llaJIbHOrO ypaBHeHlljI. Hail:TII qaCTHOe perneHlle, Y.IJ:OBJIeTBOpjlIOIIJ:ee HaqaJIbHbIM YCJIOBII- jiM y(O) = 6, y'(O) = 14, y"(O) = 36.

2.7.198.	llpoBepIIB, qTO Yl = eX			II Y2	= x	o6pa3YIOT <PYH.IJ:aMeHTaJIbHYIO
	cllcTeMY perneHllil: ypaBHeHlljI				y" -	--L..1 y' +			~1Y = 0, Hail:TII
						x-			x-
	o6IIJ:ee perneHlle ypaBHeHlljI (x - l)y" - xy' +								y = (x - 1)2.
2.7.199.	llpoBepIIB, qTO Yl = cos X II Y2				= X cos x o6pa3YIOT <PYH.IJ:aMeHTaJIb-
	HYIO cllcTeMY perneHllil: ypaBHeHlljI y" +2 tgx·y' +(2tg2 x+1)y = 0,
	Hail:TII o6IIJ:ee perneHlle ypaBHeHlljI ctgx·y"+2y' +(2 tgx+ctgx)y =
2.7.200.	= cos2 x.							4y" + y = O.
	Hail:TII o6IIJ:ee perneHlle ypaBHeHlljI 4yIV +
Hai1mu o6w,ue pew,ewu.a ypa6'1te'ltui1:
2.7.201.	yV + 8y'" +		16y' = O.	2.7.202.			yV -	6yIV + gy'" = O.
2.7.203.	ylV -	8y" +	16y = O.	2.7.204.			y" + 4y' + 4y = e-2x lnx.
2.7.205.	yVI -	2yV + 3yIV - 4y'" + 3y" - 2y' + y = O.

Hai1mu "tacm'ltbte pew,e'ltUR, yiJo6.//,em6op.a70w,ue 3aiJa'lt'ltbtM 'lta"ta.//,bHbtM yc'//'o-

6UR,,":

2.7.206. y'" - y' = -2x, y(O) = 0, y'(O) = 1, y"(O) = 2.

2.7.207. yIV - Y = 8ex , y(O) = -1, y'(O) = 0, y"(O) = 1, y"'(O) = O.

2.7.208. Hail:TII IIHTerpaJIbHYIO KPIlBYIO .IJ:II<p<pepeHn:llaJIbHOrO ypaBHeHlljI y" - y = 0, KacaIOIIIJ'IOCjIB TOqKe 0(0,0) npjlMoil: y = x.

2.7.209. Hail:TII IIHTerpaJIbHYIO KPIlBYIO ypaBHeHlljI y" - 4y' + 3y = 0, KacaIOIIIJ'IOCjIB TOqKe Mo(O, 2) npjlMoil: y = x + 2.

112

.M.emoiJ 'UC7CJlIO",e'HWJ.

Pewum'b	ypamte'lt'I.J.R. 9i1J1,epa:
2.7.210.	x 2y" + xy' - Y = O.	2.7.211. x 2y'" - 3xy" + 3y' = O.
2.7.212.	(x + 2)2y" + 3(x + 2)y' -	3y = O.
2.7.213.	x 2y'" = 2y'.

§ 8. ~HTErp~POBAH~E C~CTEM

A~Q)Q)EPEHu.~AnbHbIX YPABHEH~~

~C'Ucme.M.o11. iJUljiifiepe'H'Il,'UaJl'b'H'bI,X ypa6'He'H'U11. Ha3hIBaeTCH cOBoKynHocTb ,Il;H<p<pe-

peHD;HaJIbHhIX YPaBHeHHiI:, KaJK,Il;Oe H3 KOTOPhIX CO,Il;eplKHT He3aBHCHMYIO nepeMeH-

HYIO, HCKOMhIe <PYHKD;HH H HX nepeMeHHhIe.

~

HopManbHaJi CMCTeMa AMclJclJepeH4ManbHbix ypaBHeHMM

~ CHCTeMa ,Il;H<p<pepeHD;HaJIbHhIX ypaBHeHHiI: nepBoro nOpH,Il;Ka, pa3pemeHHhIX OTHOCHTeJIbHO npOH3BO,Il;Hoil:, T. e. CHCTeMa BH,Il;a

{Y~ = /1(x, Yl, Y2, ... , Yn),

Y~.~.~~~~:~l:~2.'.'.'.'.'.~~~'.

Yn = !n(X,Yl,Y2,···,Yn),

r,!l;e x - He3aBHCHMaH nepeMeHHaH, a Yl(X), Y2(X), ... , Yn(X) -

D;HH OT x, Ha3hIBaeTCH 'Hop.M.aJl'b'Ho11. c'Ucme.M.o11..

(8.1)

HeH3BeCTHhIe <PYHK-

~

PemHTb 3TY CHCTeMY 03Ha'laeTHail:TH <PYHKD;HH Yl(X), Y2(X), .. . , Yn(X), y,Il;G- BJIeTBOpHI01D;He CHCTeMe (8.1) H ,Il;aHHhIM Ha'laJIbHhIMYCJIOBHHM:

HOPMaJIbHYIO CHCTeMY MOlKHO npHBeCTH K O,Il;HOMY ypaBHeHHIO nOpH,Il;Ka n (H.JIH MeHbme) OTHOCHTeJIbHO O,Il;HOil: HeH3BecTHoil: <PYHKD;HH, CKaJKeM Yl, npH nOM01D;H CJIe,IJ;yI01D;ero a.rrropHTMa, Ha3hIBaeMoro

)J;H<P<PepeHD;HPyeM nepBoe ypaBHeHHe CHCTeMhI no nepeMeHHoil: x:

ITPOH3BO,Il;HhIe yL y~, ... , y~ B npaBoil: 'IacTH3TOro paBeHCTBa 3aMeHHM HX BbIp3lKeHHHMH H3 CHCTeMhI (8.1). ITOJIY'IHMypaBHeHHe

y~ = H(x, Yl, Y2, ... , Yn).

OTO paBeHCTBO ,Il;H<p<pepeHD;HPyeM no nepeMeHH~iI: x:

y~' = (H)~ + (F2)~1 . y~ + (H)~2 . y~ + ... + (F2)~n . y~.

ITPOH3BO,Il;HhIe y~, y~, ... , y~ B npaBoil: 'IaCTH3TOro paBeHCTBa 3aMeHHM HX BhIpalKeHHHMH, 3a,Il;aHHhIMH CHcTeMoil: (8.1). IIoJIY'IHMe1D;e O,Il;HO ypaBHeHHe

y~' = F3(X, Yl, Y2, ... , Yn).

113

(8.1).

2ho ypaBHeHHe ,Il;H<p<pepeHU;HPyeM no nepeMeHHoil: x H TaK ,n:8JIee ,Il;O Tex nop, noxa He npH,Il;eM K ypaBHeHHlO

yln) = Fn{X,Yl,Y2,···,Yn).

IIOJIyqeHHbIe TaKHM 06pa30M ,n:H<p<pepeHU;H8JIbHble ypaBHeHHH 06'be,n:HHHM B

O,Il;Hy CHCTeMY, K KOTOPOil: npHcoe,n:HHHM nepBOe ypaBHeHHe CHCTeMbI (8.1):
y~	= /1{x,Yl,Y2,···,Yn)
Y?	= F2{X, Yl, Y2, . .. , Yn),
Y?'	= F3{X, Yl, Y2, ... , Yn),	(8.2)

yln) = Fn{x, Yl, Y2, ... , Yn).

IIepBble n - 1 ypaBHeHHiI: CHCTeMbI (8.2) pa3pemHM OTHOCHTeJIbHO nepeMeHHbIx Y2, Y3,···, Yn, BbIpaJKaH HX qepe3 nepeMeHHble x H Yl, a TaKlKe npoH3Bo,n:HbIe yL y?, ... , yln - l ). IIoJIyqeHHble BblpaJKeHHH nO,Il;CTaBHM B nOCJIe,n:Hee ypaBHeHHe CHCTeMbI (8.2). B HTore npH,n:eM K ,Il;H<p<pepeHU;H8JIbHOMY ypaBHeHHlO nopH,n:Ka n OTHOCHTeJIbHO O,Il;HOil: HeH3BecTHoil: <PYHKU;HH Yl.

Ih o61lJ;ero pemeHHH 9Toro ypaBHeHHH MOlKHO nOJIyqHTb o61lJ;ee pemeHHe CHCTeMbI (8.1) HJIH Tpe6yeMoe qacTHoe pemeHHe. 3aMeTHM, 'ITOnopH,n:OK nOCJIe,n:Hero ypaBHeHHH MOlKeT 6b1Tb MeHbme, qeM n, eCJIH npH ero nOJIyqeHHH 6bIJIH HCnOJIb30BaHbI He Bce ypaBHeHHH CHCTeMbI

nOIo1CK Io1HTerplo1pyeMblx KOM6Io1Hau.lo1iii

MHTerpHpOBaHHe CHCTeMbI (8.1) CYllJ;eCTBeHHO 06JIerqaeTCH, eCJIH 9Ta CHCTeMa ,Il;OnYCKaeT HHTerpHpyeMble KOM6HHau;HH ,Il;H<p<pepeHU;HaJIbHblX YPaBHeHHiI:. IIO,Il; HHTerpHpyeMoil: KOM6HHaU;Heil: nO,Il;pa3YMeBaeTCH ,Il;H<p<pepeHU;H8JIbHOe ypaBHeHHe, nOJIyqaeMOe H3 ypaBHeHHiI: CHCTeMbI (8.1) C nOMOIu;blO Onpe,Il;eJIeHHblX npe06pa30BaHHiI:, HO YlKe JIerKO HHTerpHpYlOllJ;eeCH. IIpHMepoM HHTerpHpyeMoil: KOM6HHaU;Heil: HBJIHeTCH ypaBHeHHe BH,Il;a

B03MOlKHO, 'ITO3aMeHoil: nepeMeHHbIX y,n:acTCH nOJIyqHTb ,n:H<p<pepeHU;H8JIbHOe ypaBHeHHe H3BeCTHOrO THna, pemeHHe KOToporo He npe,Il;CTaBJIHeT Tpy,n:a.

CIo1CTeMbl IlIo1HeiiiHblX AIo1~~epeHu.lo1allbHbIX ypaBHeHlo1iii C nOCTORHHblMIo1 K03~~Io1u.lo1eHTaMIo1

,ll;JIH pemeHHH HOPMaJIbHOil: CHCTeMbI JIHHeil:HbIX O,Il;HOPO,Il;HblX ,Il;H<p<pepeH,lJ;HaJIb-

HblX YPaBHeHHiI: BH,Il;a

Y~ = allYl + al2Y2 + ...	+ alnYn,
{Y~.~.~~~~~.~.~~~~~.~.......	~~.2~.~~'.	(8.3)
Yn = anlYl + an2Y2 + ...	+ annYn

114

YP;06HO BOCrrOJlb30BaTbCX MeTOp;aMH JIHHeitHoit anre6pbl, a KOHKpeTHee, MeTop;OM co6CTBeHHbIX BeKTopOB.

)J;06aBHM, 'ITO06IIIee pemeHHe OP;HOPOP;HOit JIHHeitHoit CHCTeMbI rrpep;CTaBJIXeT co6oit JIHHeitHYIO KOM6HHaU;HIO CPYHp;aMeHTaJlbHoit CHCTeMbI pemeHHit, a 06IIIee pemeHHe Heop;HopoP;Hoit CHCTeMbI PaBHO CYMMe 06IIIero pemeHHX COOTBeTCTBYIOIIIero pemeHHX OP;HOPOp;HOit CHCTeMbI H OP;HOro '1aCTHOropemeHHX HeOp;HOpOp;Hoit CHCTeMbI.

2.8.1. PemHTb CHCTeMY .IJ:H<p<pepeHIJ;HaJIbHbIX ypaBHeHHil::

X' =	- 2x - 2y -	4z,
{ y' = -2x + y -		2z,
z' = 5x + 2y + 7z.
a B .IJ:aHHoil: CHCTeMe x, y, z -	HeH3BeCTHbIe <PYHKIJ;HH, a He3aBHCHMruJ ne-

peMeHHruJ t - HX aprYMeHT.

,n:H<p<pepeHIJ;HpyeM nepBoe ypaBHeHHe CHCTeMbI no t:

x" = -2x' - 2y' - 4z'.

BMeCTO y' H z' nO.IJ:CTaBHM HX BbIpaJKeHHjJ H3 BToporo H TpeTbero ypaBHeHHil: CHCTeMbI. IIoJIyqaeM

x" = -2x' - 2(-2x + y - 2z) - 4(5x + 2y + 7z),

OTKY.IJ:a

x" = -2x' - l6x - lOy - 24z.

IIoJIyqeHHoe ypaBHeHHe .IJ:H<p<pepeHIJ;HpyeM no t,	a BMeCTO y' H z' onjJTb
rrO.IJ:CTaBHM BbIpaJKeHHjJ H3 Tex :lKe ypaBHeHHil: CHCTeMbI
XIII = _2X" - l6x' - lOy' - 24z' =
= _2X" - l6x' - 1O( -2x + y -	2z) - 24(5x + 2y + 7z),
XIII = _2X" - l6x' - lOOx - 5Sx - l4Sz.
COCTaBHM HOBYIO CHCTeMY:
X' = - 2x - 2y - 4z,
{ x" = -2x' -16x - lOy - 24z,	(S.4)
XIII = -2x" - l6x' - lOOx - 5Sy -	l4Sz.

CHCTeMa COCTOHT H3 nepBom ypaBHeHHjJ HCXO.IJ:HOil: CHCTeMbI H .IJ:Byx ypaBHeHHil:, nOJIyqeHHbIX nOCJIe.IJ:OBaTeJIbHbIM .IJ:H<p<pepeHIJ;HpOBaHHeM.

113 9TOil: CHCTeMbI HCKJIIOqHM HeH3BeCTHbIe y H z. ,n:ml 9Toro rrporu;e BCero HCnOJIb30BaTb nepBbIe .IJ:Ba ypaBHeHHjJ CHCTeMbI (S.4), H3 KOTOPbIX, nOCJIe

rrpe06pa30BaHHil: (paccMaTpHBruJ -6x' + x" H -5x' + x"), HaXO.IJ:HM
{ 2y = x" - 4x' + 4x,	(8.5)
4z = -x" + 3x' - 6x

115

H 9TH BbIpaJKeHHii 1I0,n:CTaBHM B TpeTbe YPaBHeHHe CHCTeMbI:

XIII = -2x" - 16x' - lOOx - 29(x" - 4x' + 4x) - 37(-X" + 3x' - 6x).

IIoCJIe IIpHBe,n:eHHii 1I0,n:06HbIX CJIaraeMbIX 1I0JIY'IaeMO,n:HO ypaBHeHHe TpeTbero 1I0pil,n:Ka (O,n:HOpO,n:HOe C 1I0CTOilHHbIMH K09<P<PHD:HeHTaMH) OTHOCHTeJIbHO HeH3BecTHoil: <PYHKIJ;HH X = X(t):

XIII - 6x" + llx' - 6x = O.

KOPHilMH ero XapaKTepHCTH'IeCKOrOypaBHeHHii ka - 6k2+ 11k - 6 = 0 ilBJIiIIOTCii 'IHCJIak1 = 1, k2 = 2, ka = 3. CJIe,n:oBaTeJIbHO, 06lIJ;ee pe1I1eHHe 1I0CJIe,n:- Hero ypaBHeHHii HMeeT BH,n:

xoo = C1et + C2e2t + Caeat .

Tellepb HMO 1I0JIY'lHTb3Ha'leHHeMil Yoo H zoo. 9TO JIerKO c,n:eJIaTb, HMeil B BH.n:y CHCTeMY (8.5), co,n:ep:lKalIJ;yIO 2y H 4z, BbIpaJKeHHbIe 'Iepe3x, x' H x".

II09ToMY CHa'laJIaHaxO,n:HM

x~o = C1 et + 2C2 e2t + 3Caeat , x~o = C1et + 4C2e2t + 9Caeat .

OCTaeTcii c,n:eJIaTb COOTBeTCTBYIOlIJ;He 1I0,n:CTaHOBKH:

y = ~(x" -

4x' + 4x) = ~(C1et + 4C2e2t + 9Caeat -

4C1et - 8C2e2t -

-12Caeat + 4C1et + 4C2e2t + 4Caeat ),

oTKy,n:a

lC

t

lC

at

Yoo = '2

1 e +

'2

ae .

AHaJIOrH'IHO,

_I (

"

3'

6

Xoo

)

-

-

C1

e

t

-

C2e2t

-

3 C

at

.

0 KOH-

Zoo - 4

-xoo -

Xoo -

'2 ae

'IaTeJIbHO,

xoo = C1et + C2e2t + Caeat ,

•

{

_

lC

t

+

lc

at

,

Yoo -

'2 1 e

'2

ae

Zoo = -C1et -

C2e2t - ~Caeat.

2.8.2.

Pe1I1HTb CHCTeMY {

X' + Y' -

Y = et ,

"

+ y

+ 2y = cos t

IIpH ,n:aHHbIX Ha'laJIbHbIX

2x

3

4

YCJIOBHilX to = 0, Xo = -17' Yo =

17'

CHa'laJIaIIPHBO,n:HM CHCTeMY K

HOPMaJIbHOMY BH.n:y, T. e. K BH.n:y, pa3pe-

1I1eHHOMY OTHOCHTeJIbHO IIPOH3BO,n:HbIX

X' = -3y + cost -

et ,

{

y' = 4y - cost + 2et .

,1l;aJIee ,n:eil:cTByeM 110 cxeMe, IIpHMeHeHHoil: IIpH pe1I1eHHH IIpe,n:bI.n:ylIJ;ero IIpHMepa.

116

IIepBoe ypaBHeHHe .n;H<p<pepeHIJ,HpyeM no t, nOCJIe 'IeraBMecTO y' no.n;CTaBHM BblpaJKeHHe H3 BToporo ypaBHeHHfl HOBOti CHCTeMbI

x" = -3y' - sint - et = -3(4y - cost + 2et ) - sint - et , T.e. x" ,;" -12y + 3cost - tet - sin t.

113 9Toro ypaBHeHHfl H nepBoro ypaBHeHHfl HCXO.n;HOti CHCTeMbI COCTaBHM

X' = -3y + cost - et ,		(8.6)
{	+ 3cost - tet -
x" = -12y		sint,

H3 KOTOPOit HCKJIIO'IHMY (nepBoe ypaBHeHHe, YMHO)KeHHOe Ha (-4) npH6aBHM

KO BTOPOMY):

x"-4x' =-cost-3et -sint.

IIoJIY'IeHHoeHeo.n;Hopo.n;Hoe ypaBHeHHe BTopora nop».n;Ka C nOCTOflHHbIMH K09<P<PHIIHeHTaMH pernaeTCfl CTaH.n;apTHbIM cnoc06oM no.n;60pa 'IacTHoro perneHHfl. A HMeHHO (B COKpaIIIeHHOM H3JIO)KeHHH):

o

x" = Aet + B cos t + C sin t

et

:

-3A =-3

A= 1,

_

3

-4

x~ = Aet -

B sin t + C cos t

=>

cost:

-4C-B=-1 =>

B -

-17'

1

x~ = Aet -

B cos t - C sin t

sin t :

4B-C= -1

C =

5

17'

OTCIO.n;a x" = et

- 13 cos t + 15

sin t.

OKOH'IaTeJIbHO,

7

7

-

-

C

1

+

C

4t

t

3

t

+

5· t

.

XOH -

XOO + x" -

2e + e

- 17 cos

17 sm

,IWyryIO <PYHKIIHIO YOH MO)KHO HaitTH .n;ByMfl cnoc06aMH.

a) 113 BTopora ypaBHeHHfl CHCTeMbI (8.6) Haxo.n;HM

y = -l2 (x" - 3 cos t + 7et + sin t).

IIo.n;cTaBJIflfl CIO.n;a Hati.n;eHHoe BblpaJKeHHfl ,ll;JIfl X~H' HaxO.n;HM YOH'

6) 113 nepBom ypaBHeHHfl HOPMaJIbHOti CHCTeMbI HMeeM

Y = l(-x' + cost - et ).

OTCIO.n;a, y'IHTbIBM,'ITO

x' = (C

+C e4t + et - ~ cos t + ..Q.. sin t) I = 4C e4t +et + ~sin t + ..Q.. cos t

1

2

17

17

2

17

17'

nOJIY'IHM

Y =! (-4C2 e4t -

et -

~sint -

..Q.. cost + cost - et )

'

OH

3

17

17

4C

4t

-

1 .

4

2 t

.

T.e. YOH = -3

2e

17 smt

+ 17cos t -

3e

117

TaKHM 06pa30M, o6m;ee peIIIeHHe CHCTeMbI HMeeT BH.n:

-

C

1 +

C

2 e

4t

+e

t

3

t

+

5· t

{

X OH -

- 17 cos

17 sm

_

4C

4t

2

t

4

1 .

-

YOH -

-3

2 e

3e

+ 17 cost -

17 smt.

IIo.n:cTaBJuUI Ha'IaJIbHbleYCJIOBHjJ

x

= - {7' Y = 0 H t = 0, onpe.n:eJIHM

KOHCTaHTbI Cl H c2 :

{

Cl + C2 + 1- 137 = - {7'

{Cl = -~,

4

2

+

4

4:::}

1

-3C2 - 3

17 =

17

C2 = -2·

MTaK, 'IaCTHbIepeIIIeHHjJ, y.n:OBJIeTBopjJlOm;He Ha'IaJIbHbIMyCJIOBHjJM, HMelOT

BH.n:		{ x = _1 - 1e 4t			+ e t -	.l. cos t + ..Q.. sin t					•
			2	2		17			17'
		y = ~e4t		_ ~et	+ ..i. cost -			..1. sint
			3	3	17			17·
3aMeqaHHe. ~aJIee 6y.n:eM 3aMeHjJTb X OH Ha x, YOH Ha y H T. .n:.
PeUL1J.m'b aa'H'H,'bte CUCmeMbt autfitfiepe'H'qua'!/''b'Hbtx ypa6'He'Hui1:
	X'	= y + z,
2.8.3.	{ y' = 3x +z,
	z' = 3x +y.
2.8.4.	X'	= 3x -	2y,	x(O) = 1, y(O) = 2.
	{	= 2x -	y,
	y'
	X' + 5x + y = et ,								X' = -x + y + z,
2.8.5.					2.8.6.			{	y' = X -	Y + z,
	{
	y' - X - 3y = e2t •								Zl = X +y - z.
	{ 4XI - y' = sint - 3x,
2.8.7.					2.8.8.			{	X' = 2x+y,
	x' = cost - y.								y' = 3x + 4y.
	X' = x -		2y -	z,				{XI = y,
2.8.9.	{ y' = -x + y + z,				2.8.10.				I _	t	-t
	zI	= x -	z.						Y -x+e +e		•
2.8.11.	PeIIIHTb cHcTeMY ypaBHeHHil: {					X'	= y,
						yI	= x.

Q IIo'IJIeHHoeCJIO)KeHHe 9THX paBeHCTB npHBo.n:HT K HHTerpHpyeMoil: KOM-

	I	I	d(x+y)
6HHallHH: x	+	y = x	+ y, T.e. x + y	=	dt.
OTclO.n:a HaxO.n:HM x + y = Clet .
AHaJIOrH'IHYIOKOM6HHallHIO nOJIY'IaeMBbI'IHTaHHeMypaBHeHHil: HCXO.n:-
HOil: CHCTeMbI
x' - y' = -(x -			y), OTKy.n:a d(x - y)		= -dt, T.e. x - y = C2 e-t .
			x-y

118

OCTaeTCjJ IIO'IJIeHHOCJ10:lKHTb H BbI'IeCTbIIOJIY'IeHHblepaBeHCTBa:

•

X' = y - z,

2.8.12. PeIIIHTb cHcTeMY { Y' = z - x,.

z' = x - y.

o CJIO:lKHB IIO'IJIeHHOBce TpH ypaBHeHHjJ, IIOJIy'IHMHHTerpaJIbHOe ,BbIPaJKe-

Hl1e d(x + Y + z) = 0, T. e. x + Y + z = C1·

YMHO:lKHM IIepBOe ypaBHeHl1e Ha x, BTopoe Ha y, TpeTbe Ha z H IIOJIYqeHHble pe3YJIbTaTbI CJIO:lKHM IIO'IJIeHHO.IIoJIY'IHM,n:pyryIO HHTerpHpyeMyIO KOM6HHaIJ;HIO

X~~ + Y~~ + Z~: = 0, T. e. d(x 2 + y2 + Z2) = 0, oTKy,n:a x 2 + y2 + Z2 ;, C2.

3TI1 ,n:Ba COOTHOIIIeHl1jJ Y:lKe MO:lKHO HCIIOJIb30BaTb .D:JIjJ Toro, 'IT06bI113 I1CXO,n:- HOil: CHCTeMbI IIOJIY'IHTbO,n:HO ,n:11<p<pepeHIJ;l1aJIbHOe ypaBHeHHe OTHOCHTeJIbHO O,n:HOil: He113BecTHoil: <PYHKIJ;HH. Ho MbI IIolIpo6yeM I1CIIOJIb30BaTb TOJIbKO IIepBOe COOTHOIIIeHHe, H3 KOTOpOrO HMeeM Z = C1 - X - y. IIo,n:cTaBHM 9TO BblpaJKeHHe .D:JIjJ Z B IIepBble .D:,Ba ypaBHeHHjJ:

X' = 2y - C1 + X,

{

y'=-2x-y+C1.

,Il;H<p<pepeHIJ,HpYjJ IIepBOe ypaBHeHHe IIO t, IIO,n:CTaBHM 3aTeM BblpaJKeHHe ,n:JIjJ y' 113 BTOpOro ypaBHeHl1jJ: X" = 2y' + X' = -4x - 2y + 2C1 + X'.

A TelIepb H3 IIOJIY'IeHHoil:CHCTeMbI

{X' =X+2y-C1'

x" = x' - 4x - 2y + 2C1

I1CKJIIO'IaeMy - IIOJIy'IHMx" + 3x = C1 , oTKy,n:a

X = C2 cos v'3t + C3 sin v'3t + lCl •

113 ypaBHeHl1jJ x' = 2y - C1 + X HaxO,n:HM y = ~(x' -		X + Cd, T.e.
y = ~ (v'3C3 -	C2 ) cos v'3t - ~ (v'3c2 + C3 )	sin v'3t + lCl •
HaKOHeIJ;, Z = C1 - X -	y, T. e.

119