Сборник задач по высшей математике 2 том

G2e-

x ::I ~. TaKHM 06pa30M, <PYHKIJ;HH tgx H ctgx JIHHeiiHO He3aBHCHMbI B o6J1q.

CTH HX Onpe,ll;eJIeHHfl (X ::I n· ~, n E Z).

•

Ycma?t06Um'b, 7Ca7CUe U3 c.I/,eaY70UJ,UX nap rjjy'H,7C'IJ,U'i1. .l/,U'H,e'i1.'H,O 'H,e3a6UCUM'b&, Q 7Ca7Cue - .l/,UHe'i1.'H,O 3a6UCUMbt:

2.7.2.	arcsin x H arccos x.	2.7.3.	sin x, sin 2x.
2.7.4.	eX , ex2	2.7.5.	1, x.
2.7.6.	sin x, sin2 x.	2.7.7.	sinx cos x, sin2x.
2.7.8.	1 - cos2x, sin2 x.	2.7.9.	1 + cos 2x, cos2 x.

2.7.10. ,Il;aHbI <PYHKIJ;HH YI = eX, Y2 = e-2x . CocTaBHTb O,ll;HOpO,ll;HOe ,ll;H<p- <pepeHIJ;HaJIbHOe YPaBHeHHe BToporo nOpfl,ll;Ka C nOCTOflHHbIMH Ko- 9<P<PHlIHeHTaMH, o6lIJ;ee peIIIeHHe KOToporo HMeeT BH,ll;

a 3aMeTHM CHaqaJIa, qTO ,ll;aHHble <PYHKIJ;HH JIHHeiiHO He3aBHCHMbI Ha BceD

npflMoii, TaK KaK Yyl =	e:	= e3x =!- const. IIycTb Y = Glex + G2e-2x -
2	e-	X

o6lIJ;ee peIIIeHHe HeKoToporo JIO,Il;Y BTOpOro nOpfl,ll;Ka. Tor,ll;a

{Y' = Glex - 2G2e-2x , y" = GI eX + 4G2e-2x .

Pa3peIIIHM 9TY CHCTeMY OTHOCHTeJIbHO nOCTOflHHbIX GI H G2 . BblqHTM nepBoe ypaBHeHHe H3 BToporo, nOJIyqaeM 6G2e-2x = y" - y'. OTCIO,ll;a

G2 = ! (y" - y')e2x .

6

Tenepb nepBoe ypaBHeHHe, ,ll;OMHO:>KeHHOe Ha 2, npH6aBHM KO BTOPOMY:

2y' + y" = 3Gl ex .

OTCIO,ll;a GI = ~(2y' + y")e- X. IIOJIyqeHHble Bblpa:>KeHHfl ,ll;JIfl GI H G2 no,ll;- CTaBHM B Bblpa:>KeHHe ,ll;JIfl y:

y = ~(2Y' + y") + ~(yll - y'),

T. e. y" +y' - 2y :::: O. B HTore MbI nOJIyqHJIH ,ll;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe, KOTOPOMY Y,ll;OBJIeTBOpflIOT <PYHKIJ;HH YI = eX, Y2 = e-2x .

IIoCKOJIbKY peIIIeHHfl YI = eX H Y2 = e- 2x 9Toro ypaBHeHHfl JIHHeiiHO He3aBHCHMbI, TO B CHJIY TeopeMbI 2.5 <PYHKIJ;Hfl Yoo = Gl eX + 2x - ,ll;eii-

CTBHTeJIbHO ero o6lIJ;ee peIIIeHHe. OTCIO,ll;a CJIe,IJ;yeT, qTO 9TO H eCTb HCKOMoe

,ll;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe. •

100

Cocma6Um'b Jl,UHetlHoe OaHOpOaHOe aUtPtPepeHtlUaJl,'bHOe ypa6HeHUe 6mOpOZO nopSl.a'K:a C nOCmOSl.HH'btMU 'K:03tPtPUtlUeHmaMU, aM 'K:omopozo aaHH'bte tPYH'K:- 'lfUU cOCma6M'IOm tPYHaaMeHmaJl,'bHY'IO cucmeMY peweHutl, npea6apUmM'bHO npo6epU6, "tmo aaHH'bte tPYH'K:tlUU Jl,UHetlHo He3a6UCUM'bt:

2.7.11.	sin x, cosx.	2.7.12.	e- X, eX.
2.7.13.	1, x.	2.7.14. xe x , eX.
2.7.15.	e2x , eX	2.7.16.	e2x , xe2x •
2.7.17.	sin 2x, cos 2x.
2.7.18.	HaitTH o6mee peIlIeHHe ypaBHeHHa 2y" - 3y' + Y = o.
a CocTaBHM xapaKTepHCTHqeCKOe ypaBHeHHe: 2k2 - 3k + 1 = O. ITo ero
KopHaM	kl = 1 H k2	= ~ COCTaBHM o6mee peIlIeHHe ,ll;aHHoro O,ll;HOpO,ll;HOrO

ypaBHeHHa, COl'JIaCHOTeopeMe 2.8:

•

Hatlmu o6w,ue peWeHUSI. ypa6HeHutl:

2.7.19.Y" - 5y' + 6y = O. 2.7.20. 2y" + 5y' - 7y = O.

2.7.21.y" + 4y' - 3y = O. 2.7.22. 3y" + y' - 2y = O.

2.7.23.	y" + 25y' = O.		2.7.24. 4y" - 9y' = O.
2.7.25.	HaitTH 06mee peIlIeHHe ypaBHeHHa 4y" + 4y' + y = o.
a XapaKTepHCTHqeCKOe ypaBHeHHe 4k2 +4k +1 = 0 HMeeT ,ll;Ba O,ll;HHaKOBbIX
KopHa kl	= k2 = -~. B TaKOM CJIyqae (CM. TeopeMY 2.8)
	1	1	1
	Yoo = C1 e- 2x	+ C2 xe- P ,	HJIH Yoo = (C1 + C2 x)e- 2x .	•

Hatlmu o6w,ue peWeHUSI. ypa6HeHutl:

2.7.26.y" - 6y' + 9y = O. 2.7.27. y" - 4y' + 4y = O.

2.7.28.	4y" - 12y' + 9y = O.				2.7.29.	9y" + 12y' + 4y = O.
2.7.30.	HaitTH o6mee peIlIeHHe ypaBHeHHa 2y" + y' + 3y = O.
a XapaKTepHCTHqeCKOe ypaBHeHHe 2k2 + k + 3 =							0 HMeeT KOMllJIeKCHbIe
(KoMllJIeKCHo-CollpIDKeHHbIe) KOPHH:
	k		-	-1±V23i		.
		1,2 -			4
B aTOM CJIyqae o6mee peIlIeHHe ypaBHeHHa HMeeT BH,ll;									•
	Yoo = (C		-4-		+ C·2	-4-x	e
	1 cos		V23	x	sm	V23)		_lx
								4 .
Hatlmu o6w,ue peWeHUSI. ypa6HeHutl:
2.7.31.	y" + 4y = O.				2.7.32.	4y" + 9y = O.
2.7.33.	y" + y' + y = O.				2.7.34.	y" -	y' + 6y = O.

101

2.7.35.2y" - 3y' + 5y = O. 2.7.36. 5y" - 3y' + 2y = O.

2.7.37. HaihH 'IaCTHOeperneHHe ypaBHeHHH 3y" +7y' +4y = 0, y,Il;OBJIeTBO-
pHlOmee 3a,n;aHHbIM Ha'laJIbHbIMYCJIOBHHM y(O) = 1, y'(O) = -l
Q XapaKTepHCTH'IeCKOeypaBHeHHe 3k2+7k +4 = 0 HMeeT KOpHH k1 = -1 H
k2 = -l CJIe,Il;OBaTeJIbHO, o6mee perneHHe HMeeT BH,Il; Yoo = G1e- x +G2e-~x.
4	-!
HaxO,Il;HM y~o = -Gle- x - aG2e	3 x . IIo,Il;cTaBJIHH B IIOCJIe,Il;HHX ,Il;ByX paseH-

CTBax x = 0, y = 1, y' = - ~, IIOJIY'laeMcHcTeMY ypaBHeHHiI: OTHOCHTeJIbHO

G1 H G2 :		= 1,	{G1 = 2,
{ G1 + G	2
-G1 -	~G2 = -~		~ G2 = -1.

Hail:,Il;eHHbIe KOHCTaHTbI IIO,Il;CTaBJIHe~ B BblpruKeHHe ,Il;JIH o6mero perneHHH.

IIoJIY'laeMHCKOMoe 'IaCTHOeperneHHe

•

Ha11.mu "I,aCmH'bte peweHUSI aaHH'btX augnpepeHtjuaJI:bH'btX ypa6HeHu11., yao6.11em60pJl'lO'l.4ue 3aaaHH'btM Ha"l,a.ll'bH'btM YC.II06US1M:

2.7.38.y" - 4y' + 3y = 0, y(O) = 6, y'(O) = 10.

2.7.39.y" + 4y' = 0, y(O) = 7, y'(O) =8.

2.7.40.y" - 6y' + 9y = 0, y(O) = 0, y'(O) = 2.

2.7.41.4y" + 4y' + y = 0, y(O) = 2, y'(O) = O.

2.7.42.y" - 4y' + 3y = 0, y(O) = 6, y'(O) = 10.

2.7.43.Hail:TH o6mee perneHHe ypaBHeHHH y" -7y' = 5xex , IIo,Il;6HPM 'IacT-

Hoe perneHHe MeTO,Il;OM HeOIIpe,Il;eJIeHHbIX K09<P<PHII;HeHTOB.

Q XapaKTepHCTH'IeCKOeypaBHeHHe k2-7k = 0 HMeeT ,Il;Ba ,Il;eil:cTBHTeJIbHbIX

KOpHH k1	= 0 H k2 = 7, II09TOMY o6mee perneHHe O,Il;HOPO,Il;HOro ypaBHeHHH
y" - 7y'	= 0 HMeeT BH,Il; Yoo = G1 + G2e7x . IIpaBM 'IacTbHeO,Il;HOpO,Il;HOrO

ypaBHeHHH HMeeT BH,Il; f(x) = P1(x)ekx , r,Il;e P1(x) = 5x - MHoro'lJIeHIIepBOil: CTeIIeHH, a k = 1 - He HBJIHeTCH KopHeM xapaKTepHCTH'IeCKOrOypaBHeHHH. 3Ha'lHT,'IaCTHOeperneHHe HmeM B TaKOM )Ke BH,Il;e: y... = (Ax + B)eX

(Ax + B = Q1 (x) - MHoro'lJIeHIIepBoil: CTeIIeHH C HeH3BeCTHbIMH K09<p<pHII;HeHTaMH). ,n:JIH OIIpe,Il;eJIeHHH K09<P<PHII;HeHToB A H B HaxO,Il;HM

y~ = Aex + (Ax + B)eX = (A + Ax + B)eX , y~ = (2A + Ax + B)eX ,

IIOCJIe 'IeroIIO,Il;CTaBJIHeM BblpruKeHHH ,1l;JIH Y... , y~ H y~ B HCXO,Il;HOe ,Il;H<p<pepeHII;HaJIbHOe ypasHeHHe:

(2A + Ax + B)eX - 7(A + Ax + B)eX = 5xex .

IIocJIe COKpameHHH o6eHx 'IaCTeil:Ha eX H IIpHpaBHHBaHHH K09<P<PHII;HeHToB IIpH cooTBeTcTBYlOmHX CTeIIeHHX x B JIeBoil: H IIpaBoil: 'IaCTHIIOJIY'leHHoro

102

paBeHCTBa npHXO,n;HM K CHCTeMe ypaBHeHHit OTHOCHTeJIbHO HeH3BeCTHblX A

II B:	x:	A - 7A = 5,	{-6A = 5,
	x°:	2A + B - 7A - 7B = 0, T.e.	-5A - 6B = O.
OTclO,n;a A	= -~, B = ;~, a y... = (-~x + ;~) eX. Tenepb B CHJIY Teope-
MbI 2.6 06rn;ee pemeHHe Hcxo,n;Horo ypaBHeHHjI HMeeT BH,n;
		YOH = CI + C2e7x + (-~x + ;~) eX.		•

Hai1mu o6w,ue peweHUSI ypa6HeHui1, HaxoiJSI. ux ",aCmH'bte peweHUSI MemoiJoM HeonpeiJe.lleHH'btX '/l:03tjjgJU'queHmo6:

2.7.44.y" - 3y' + 2y = lOe-x. 2.7.45. y" - 6y' + 9y = 2X2 - X + 3.

2.7.46.y" - 3y' + 2y = 2x3 - 30. 2.7.47. y" - 2y' + 2y = 2x.

2.7.48.y" + 4y' - 5y = 1. 2.7.49. 2y" - y' - y = 4xe2x .

2.7.50. HaitTH 06rn;ee pemeHHe ypaBHeHHjI y" + 6y' + 9y = (x - 2)e- 3X . o XapaKTepHCTHqeCKOe ypaBHeHHe k 2 + 6k + 9 = 0 HMeeT KopeHb k = -3

KpaTHOCTH 2, oTKy,n;a Yoo = (CI + C2 x )e-3X . IIpaBM qaCTb HCXO,n;HOro ypaB-

HeHHjI	HMeeT BH,n; PI (x )eax , r,n;e PI (x) = x - 2- MHOrOqJIeH nepBoit CTe-
rreHH,	a 0: = -3 - KopeHb KpaTHOCTH 2 xapaKTepHCTHqeCKOrO ypaBHe-

HHjI. IIo:3ToMY qacTHoe pemeHHe Heo,n;Hopo,n;Horo ypaBHeHHjI Hrn;eM B BH,n;e y... = X2QI (x )e-3x , T. e. y... = (Ax + B}x2e-3x . ,n:aJIbHeitmHe BblqHCJIeHHjI OcP0PMHM CJIe.n;yIOrn;HM 06pa30M. PaCrrOJIQ)KHM y... , y~, y~, B CTOJI6HK, CJIeBa OT HHX 3aIIHmeM COOTBeTCTBYIOrn;He KO:3cPcPHII;HeHTbI Hcxo,n;Horo ,n;HcPcPepeHII;H- aJIbHOrO ypaBHeHHjI, nOCJIe qero COCTaBHM CHCTeMY ypaBHeHHit OTHOCHTeJIbHO

A H B, rrpHpaBHHBM KO:3cPcPHII;HeHTbI npH o,n;HHaKOBbIX CTerreHjlX nepeMeHHoit x JIeBoit H npaBoit qaCTeit rrOJIyqeHHOrO paBeHCTBa (npH :3TOM e-3x MO:>KHO cOKpaTHTb)

9	y = (Ax 3 + Bx2)e-3X
6	y~ = (3Ax2 + 2Bx - 3Ax3 - 3Bx2)e-3x
1	y~ = (9A x 3 - 18Ax2 + 9Bx2 + 6Ax - 12Bx + 2B)e-3X .

B rrpHBe,n;eHHbIX BblpaLKeHHjlX o,n;HHaKOBbIMH JIHHHjlMH rro,n;qepKHyTbI no,n;o6Hble qJIeHbI.

x 3 :	9A - 18A + 9A = 0,			0=0,
x 2 :	9B + 18A -	18B - 18A + 9B = 0,	==>	0=0,
	12B + 6A -	12B = 1,		1
Xl:				A= 6'
xo : 2B = -2.				B =-1.

3aMeTHM, 'ITOnOJIyqHJIH CHCTeMY H3 qeTblpex ypaBHeHHit C ,n;ByMjI HeH3BeCTHbIMH, npH :3TOM ,n;Ba ypaBHeHHjI TPHBHaJIbHbI. 8TO npH3HaK rrpaBHJIbHOCTH COCTaBJIeHHjI CHCTeMbI.

TaKHM 06pa30M, y... = (ix - 1) x2e-3x , oTKy,n;a 06rn;ee pemeHHe

YOH =

(C

I +

C)X

e

-3x

+

(1

1)

x

2

e

-3x

.

•

2

6x -

103

Ha11.mu o6w,ue peWeH'IJ.R ypa6HeHu11.:

2.7.51.	y" + 3y' - 4y	= (x + 1)ex .
2.7.52.	y" - 2y' + y = (x + 1)ex.
2.7.53.	y" + 2y' + y = (x + 3)e- x.
2.7.54.	2y" + 3y' + y = (1 -		2x)e- x.
2.7.55.	y" + 3y' + 2y	= (1 -	4x)e- 2x .
2.7.56.	y" + 4y' + 4y	= (1 -	4x)e- 2x .

2.7.57. PewHTb ypaBHeHHe y" + 3y' + 2y = (2x + 3) sinx + cosx.
a	PewaJI xapaJ<:TepHCTHqeCKOe ypaBHeHHe k 2+3k+2 = 0, HaxO,n;HM kl = -1,
k2	= -2, oTKy,n;a Yoo = Cle-x + C2e- 2x . qacTHoe peweHHe HeO,n;HOpo,n;HOro

ypaBHeHHH HlueM B BH,n;e

yq = (Ax + B) sin x + (Cx + D) cosx.

TaK KaK HeH3BeCTHble MHOrOqJIeHbI PN(x) = Ax + B H QN(X) = Cx + D

,n;OJDKHbI HMeTb CTeneHb N = max(m, n), r,n;e m = 1 -	CTeneHb MHOrOqJIeHa
Pl (x) = 2x + 3, n = 0 - CTeneHb MHoroqJIeHa Qo (x)	= 1 (CM. cJIyqaii 2 a

Ha c. 97 npH 'Y= 0; MbI CneUHaJIbHO nOMeHHJIH MeCTaMH P(x) H Q(x), 'lTo6bI nOKa3aTb, 'ITO3TO He BJIHHeT Ha CTpyKTypy yq). )l;aJIee

2	yq = (Ax + B) sin x + (Cx + D) cos x,
3	y~ = Asinx + (Ax + B) cos x + Ccosx - (Cx + D) sin x,
1	y~ = 2A cos x - (Ax + B) sin x - 2Csinx - (Cx + D) cosx.

Tenepb npHpaBHHBaeM K03<P<PHUHeHTbI npH no,n;o6HbIX qJIeHax B JIeBoii H npaBoii qaCTHX COOTBeTCTBYIOIUero ypaBHeHHH:

xsinx:	2A - 3C - A = 2,	{A - 3C = 2,
sin x:	2B + 3A - 3D - B - 2C = 3,	3A + B - 2C - 3D = 3,
x cos x:	2C + 3A - C = 0,	:::}
		3A + C = 0,
cosx:	2D+3B+3C+2A-D = 1,	2A+3B+3C+D = 1.

113 nepBoro H TpeTbero ypaBHeHHii HaxO,n;HM: A = ~, C = -i. 113 BTOpO-

ro H qeTBepToro, C yqeTOM IIOJIyqeHHbIX K03<P<PHUHeHTOB, HMeeM: B = ~~,

3
D = - 25'TaKHM o6pa30M,
1 21).	(3	3 )	cos x +	C	-x	+	C	-2x	.	•
YOH = ( gX + 25 sm x -	gX + 25			1e			2e

Pewum'b ypa6HeH'IJ.R:

2.7.58.y" - 7y' + 6y = sinx. 2.7.59. 2y" + 5y' = 29cosx.

2.7.60.y" - 4y = e2x sin 2x. 2.7.61. y" - 2y' - 8y = -8 cos 2x.

2.7.62.y" + 4y' + 4y = (2x + 3) sin x + cosx.

2.7.63.y" - 2y = 2x(cosx - sinx)ex .

2.7.64.PewHTb ypaaHeHHe y" + 16y = 3x sin 4x + cos 4x.

104

o KOPHH XapaKTepHCTH'IeCKOI'OypaBHeHHH k2+16 = °MHHMble: k1,2 = ±4i. C ,IJ;pyroti CTOPOHbI, npaBaH 'IaCTb HeO,ll;HOpO,ll;HOI'O ypaBHeHHH HMeeT BH,ll;

f(x)	= eOX(Po(x)cos4x + Q1(x)sin4x). 3Ha'lHT,a							± f3i = ±4i -	KOpHH
xapaKTepHCTH'IeCKOrOypaBHeHHH, n09TOMY (CM. cJIY'lati26, c. 97).
			y... = x[(Ax + B) cos4x + (Cx + D) sin4x].
I1MeeM:			(AX2 + Bx) cos4x + (Cx2 + Dx) sin 4x,
16	y... =
° y~ =			(2Ax + B) cos 4x - 4(Ax2 + Bx) sin 4x + (2Cx + D) sin 4x +
			+ 4(Cx2 + Dx) cos4x,
1	y~ = 2Acos4x - 8(2Ax + B) sin4x -16(Ax2 + Bx) cos4x +
			+ 2C sin4x + 8(2Cx + D) cos4x - 16(Cx2 + Dx) sin4x.
OTCIO,IJ;a
		x 2 cos4x:		16A -	16A = 0,			C=o,
		xcos4x:		16B -	16B + 16C = 0,
			cos 4x: 2A + 8D = 1,				{ 2A+8D = 1,
		x 2 sin4x:		16C -	16C = 0,		==>	-16A = 3,
		xsin4x:		16D -16A -16D = 3,				-8B+ 2C = 0,
			sin4x:	-8B + 2C = 0.
			3	= C	= 0, D =	11
OTKY,ll;a A = - 16'B						64·
TaKHM o6pa30M,
			YOH = C 1 cos4x + C 2 sin4x -l6x2 cos4x + ~! sin4x.						•
2.7.65.		y" + 25y = cos5x.				2.7.66.	y" + y = 2x cos x + sin x.
2.7.67.		y"+y=xsinx.				2.7.68.	y" +9y = gsin3x-xcos3x.
2.7.69.		y" - 2y' + 5y = eX cos2x.				2.7.70.	5y" - 6y' + 5y = e~x sin !x.
2.7.71.		PemHTb ypaBHeHHe y" - 2y' + 5y = xex cos 2x + x 2 - X + 2.							5
a 06IIIee pemeHHe cooTBeTcTBYIOIIIero O,ll;HOpO,ll;HOrO ypaBHeHHH
					y" - 2y' + 5y =		°

IfMeeT BH,IJ; (npOBepbTe!) Yoo = (C1cos 2x + C2 sin 2x)ex . ,n:aHHOe ,ll;H<p<pepeH1IlfaJIbHoe ypaBHeHHe npe,ll;CTaBHM B BH,ll;e cOBoKynHocTH ,IJ;ByX ypaBHeHHti:

	[	y" -		2y' + 5y = xex cos2x,
		y" -		2y' + 5y = x 2 - X + 2.
	lJacTHoe pemeHHe nepBoro ypaBHeHHH HaxO,ll;HM B BH,ll;e
	y...1 = x[(Ax + B) cos 2x + (Cx + D) sin 2x]eX,
	TaK KaK a ± f3i = 1 ± 2i -		KopeHb xapaKTepHCTH'IeCKOrOypaBHeHHH
	k2 -		2k + 5 = 0, k = 1 ± 2i.

105

COOTBeTCTBYIOrn;He BblqHCJIeHHH, KOTOPbIe npe,II,JIaraeM BbInOJIHHTb CaMOCT().' HTeJIbHO, ,n;alOT

Y'd= x [l6 cos 2x + ~x sin 2X] eX.

qacTHoe pemeHHe BTOpOro ypaBHeHHH Haxo,n;HM B BH,n;e

Y..2 = AX2 + Ex + C.

AHaJIOrHqHO Haxo,n;HM Y..,2	= ~X2 -	215x +	3
			1:5.
ypaBHeHHH HMeeT BH,n; YOH	= Yoo + Y..l + Y..2, T. e.
YOH = (Cl cos 2x + C2 sin 2x)ex+
1	2	1 . 2	x	)	X
+ x ( 16 cos x + B"x sm					e

06rn;ee pemeHHe Hcxo,n;Horo

1 2	1 38 .
+ gX	- 25 x + 125·

Pew!um'b ypa8He'H!IJ.SI (oaHH'ble ypa8HeH'IJ.R" COZ.II.aCHO npa8'blM "taC'TJ'IJIM, npeo-

.ll.aZaeM '4e.ll.eC006pa3H'blM o6pa30M npeoCma8Um'b 8 8uoe c0801CynHocmu 6o.ll.ee npocm'blx):

2.7.72. Y" - 2y' + Y = x 2 - X + 3 + x cos x.

2.7.73.Y" + 5y' + 6y = (x - 2)e- 3X + x 2 + 2x - 3.

2.7.74.	Y" + 6y' + lOy = (x + 6) cos 3x - (18x + 6) sin 3x + 2xe- 3x cos x.
2.7.75.	Y" + 9y = e-3x (x - 2) + 14 + 63x2.
2.7.76.	y" - 2y' + y = sin x + ~ex - ~e-x.
2.7.77.	2x
	PemHTb ypaBHeHHe y" _ y' = ----;=e=:;=
	VI -	e2x

Q 06rn;ee pemeHHe O,n;Hopo,n;Horo ypaBHeHHH y" - y' = 0 HMeeT BH,n; Yoo =

	= Cl	+ C2 ex . IIpaBM qacTb f(x)	=	e2x	Heo,n;Hopo,n;Horo ypaBHeHHH He
				VI -
					e2x

n03BOJIHeT HaiiTH qaCTHOe pemeHHe y.. MeTo,n;OM no,n;60pa (HJIH Heonpe,n;eJIeHHbIX K09<P<PHII,HeHToB), n09TOMY BOCnOJIb3yeMcH MeTo,n;OM BapHaII,HH npOH3BOJIbHbIX nOCTOHHHbIX:

r,n;e Yl = 1, Y2 = eX -	<pYH,n;aMeHTaJIbHM CHCTeMa pemeHHti O,n;HOPO,n;H()'
ro ypaBHeHHH, a Cl (x)	H C2 (x) - pemeHHH CHCTeMbI ,n;H<p<pepeHII,HaJIbHbIX
YPaBHeHHti

{ CfYl + QY2 = 0,			q.l+ Q .ex =o,
	T. e.		{ C'	0	+	C'		X _	e2x
Cfyf + C~y~ = f(x),			l·			2· e		-	---r:===::::::;;;=.
									Vl- e2x
113 BTOpOro ypaBHeHHH HaxO,n;HM					eX
		C~	= -11- e2x				, oTKy,n;a nOCJIe HHTerpH-
pOBaHHH C2 (x) =! eX	dx	=arcsinex				(KoHcTaHTY HHTerpHpOBaHHH
VI - e2x

106

,	x _	e2x		,	C{	=
== -C2 e	-	- Jl- e2x	, T. e., HHTerpHpya,
		C1(x) = -f e2x dx = Vl-e 2x .
nO.J1araeM paBHOii HymO).				11:3 nepBOrO ypaBHeHHa CHCTeMbI nO.J1YQHJ\:I
				Jl- e2x
TeM caMbIM, qaCTHOe peIIIeHHe HMeeT BH,n;
		y... = C1(x)·1 + C2(x)e X = Vl- e2x + eX arcsin eX,
a o6rn;ee peIIIeHHe						•
		You = C1		+ C2x + Vl- e2x + eX arcsin eX.

Pemumb ypa6HeH'WI, ucnOJl.b3YSl MemOa 6apuaquu nOCmO,RHHUX:

X

2.7.78. Y"_y= _e_ eX -1'

2.7.80.y" - 6y' + 9y = ~3x.

2.7.82. y" + y + ctg2 X = O.

2.7.79.y" - 2y' + y = ~X .

2.7.81.y" + 4y = _1_.

cos2 x

2.7.83. y" _ y' = e2x ..,j,=1'-_-e-,2,-x.

2.7.84.y" - 2y' + y = ~.

x2 + 1

2.7.85.HaiiTH qacTHoe peIIIeHHe ypaBHeHHa y" - 2y' - Y = 6xex , y,n;OB.J1e-

TBoparorn;ee 3a,n;aHHbIM HaqaJIbHbIM YC.J10BHaM: y(O) = 2, y' (0) = -5. o 3anHIIIeM ypaBHeHHe 6e3 IIpaBoii qacTH y" - 2y' - Y = O. Ero xapaKTepHCTHqeCKOe ypaBHeHHe HMeeT BH,n; k 2 - 2k - 1 = 0, oTKy,n;a k1,2 = 1 ± 0, T. e. Yoo = C1e(1-v'2)x +C2e(1+v'2)x. qacTHoe peIIIeHHe Heo,n;Hopo,n;Horo ypaBHeHHa

Haii,n;eM MeTo,n;OM IIo,n;6opa:
-1	y... = (Ax + B)eX
-2	-A-2A+A= 6,
	y~ = (Ax + B + A)eX	+ 2A	+ B = O.
1	-B - 2B - 2A
	y~ = (Ax + 2A + B)eX
C.J1e,n;oBaTeJIbHO, A = -3, B = 0 H y... = -3xex. OTcro,n;a
	You = C1e(1-v'2)x + C2e(Hv'2)x - 3xex.

IloJIyqeHHoe BbIpruKeHHe IIpo,n;H<p<pepeHIIHpyeM, a 3aTeM B You H y~u IIo,n;CTaBHM x = 0, y = 1, y' = 1 H3 HaqaJIbHbIX YCJIOBHii H H3 nOJIyqarorn;eiica CHCTeMbI oIIpe,n;eJIHM 3HaqeHHa KOHCTaHT C1 H C2 • llMeeM:

y~u = (1 - ..,/2)Cl e(1-v'2)x + (1 + ..,/2)C2e(1+v'2)x - 3ex - 3xex,

a cooTBeTcTByrorn;M CHCTeMa, 0 KOTOpoii rOBOpHJIH BbIIIIe, HMeeT BH,n;:

{ Cl + C2 = 2 (3aMeHHJIH x = 0,	You = 2),
(1 - 0)C1 + (1 + 0)C2 - 3 = -5	(3aMeHHJIH x = 0, y~u = -5).

IlepBoe ypaBHeHHe, ,n;OMHO:>KeHHOe Ha -(1 + 0), IIpH6aBHM KO BTOpoMy:

-2..,/2Cl = -4 - 2..,/2, Cl = 20 + 4 = 1 + ..,/2. 20

107

,I1;aJIee C 2 = 1 - C 1 = 1 - J2. IIcKoMoe qaCTHOe perneHHe HMeeT BH,ll;
y = (1 + J2)e(l-V2)X + (1 - J2)e(1+V2)x - 3xex .	•

Haf1.mu "I,aCmH'bte pemeHUJ! ypa6HeHuiJ., yiJo6J1,em60p.R'IOw,Ue 3aiJaHH'btM Ha- "I,aJl,'bH'btM yc.n,06U.RM:

2.7.86.y" + y = 4xex , y(O) = -2, y'(O) = O.

2.7.87.y" + y = 4sinx, y(O) = 1, y'(O) = 2.

2.7.88. y" - 2y' - 3y = e	4x	6	= 3:.
		, y(O) = 25 , y'(O)

2.7.89.y" + 2y' - 3y = 48x2ex , y(O) = 1, y'(O) = -~.

2.7.90.y" + 4y' + 4y = 32xe2x , y(O) = -1, y'(O) = 1.

2.7.91. y" - y = 2ex - x 2, y(O) = 2, y'(O) = 1.

2.7.92.y" + 3y' + 2y = 2sin3x + 6cos3x, y(O) = y'(O) = O.

2.7.93.y" + 9y = 6cos3x, y(O) = 1, y'(O) = 3.

2.7.94.y" - y' = 11eX' y(O) = 1, y'(O) = 2.

2.7.95.4y" + y = ctg~, y(7r) = 3, y'(7r) = ~.

2.7.96. HaihH o6m;ee perneHHe ypaBHeHHjI 9itJIepa x 2y" + 2xy' + y = O.

Q IIoJIo)KHM X = et , OTKY,ll;a t = In x. Tor,ll;a HeH3BeCTHM <PYHKn;HjI y = y(x)

CTaHOBHTCjI CJIO)KHOit <PYHKn;Heit aprYMeHTa t: y = y(et ). CJIe,ll;OBaTeJIbHO,

IIcxo,ll;Hoe ,ll;H<p<pepeHn;HaJIbHoe ypaBHeHHe npHHHMaeT BH,ll;

e2t (cP y _ dY ) e- 2t + 2et . dYe- t + y = 0
dt2 dt		dt	'
T.e.	dy + y = 0	HJIH y" + y' + y = 0
cPy +			,
dt2	dt'

r,ll;e y = y(t). 06m;ee perneHHe nOJIyqeHHOrO JIHHeitHoro ypaBHeHHjI C nOCTOjlHHbIMH Km~<p<pHn;HeHTaMH HaxO,ll;HM CTaH,ll;apTHbIM 06pa30M:

Yoo = e	_1 (C .;3	C·.;3 )	.
	2 1 cos Tt +	2 sm Tt

OCTaeTCjI BepHYTbCjI K nepeMeHHoit x, HCnOJIb3Yjl 3aMeHY t = In x. IIoJIyqaeM

1(.;3 . .;3)

Yoo = Vx ClcosTInx+C2smTInx .	•

108

pew:um'b ypa6HeHUS!:

2.7.97.x2y" + 2xy' - 6y = O. 2.7.98. x 2y" + 3xy' + y = O.

2.7.99.xy" +y' = O.

2.7.100. PelIIHTb o606meHHoe ,n;H<p<pepeHU;HarrbHoe ypaBHeHHe 8iiJIepa

			(3x - 2)2y" - 2(3x -				2)y' + 6y = O.
o nOJIO}l{HM			3x - 2 = et , oTKy,n;a t		=		In(3x - 2). Tor,n;a	x
dx = let		dt	= 3e-t . ,Il;arree
dt	3' dx
			y~ = y: . t~ = 3e-t . y:,
	y~2 = (3e-tYD~ . t~ = 3( _e- t . y: + e-t . y:~)3e-t = ge-2t(y~~ -								YD.
IIo,n;CTaBmleM nOJIY'leHHbleBblpa}!{eHHH B Hcxo,n;Hoe ypaBHeHHe:
	9(y:~ -	yD - 6y: + 6y = 0,		T. e.		9y" - 15y' + 6y = 0,		y = y(t).
KOPHH COOTBeTCTBYlOmero xapaKTepHCTH'IeCKOrOypaBHeHHH 9k2 -									15k + 6 =
= 0, T. e. 3k2 -			5k + 2 = 0 paBHbI kl = 1, k2 = ~, n09TOMY
			Yoo(t)	= Cle		t	~t
							+ C2 e 3 •

IIoCJIe 3aMeHbI t = In(3x - 2) H npeo6pa30BaHHii nOJIy'lHMOKOH'IaTeJIbHO

•

Peumm'b ypa6HeHUSI:
2.7.101.	(2x + 1)2y" - 2(2x + 1)y' + 4y = O.
2.7.102.	x2y" + xy' + y = O.	2.7.103. x 2y" - xy' - 3y = O.
2.7.104.	x2y" + 4xy' + 2y = O.

,40'ICa3am'b .n,uHetJ.HY'lO He3a6UCUMOCm'b oaHHux !PYH'IC-qUtJ. Ha ux o6.n,acmu onpeoe.n,eHUSI, HatJ.mu onpeoe.n,ume.n,'b BpOHC'lCO~O:

2.7.105.	x, eX, e- x.	2.7.106.	x, xex, xe- x.
2.7.107.	arctgx, arctg2x, arctg3x.	2.7.108.	1, X, x 2, ... , xn.
2.7.109.	ek1X , ek2X , ek3X (kl , k2, k3 -	pa3JIH'IHble,n;eiicTBHTeJIbHbIe '1HCJIa).
2.7.110:	eX, e2x , e3X , ... , enx .
2.7.111.	cosx, cos2x, ... , cosnx, x E [0,271"].

,40'ICa3am'b .n,uHetJ.HY'lO 3a6UCUMOCm'b OaHHux !PYH'IC-qUtJ. U HatJ.mu onpeoe.n,u-

me.n,'b BpOHC'lCO~O:

2.7.112. 1, arcsin x, arccosx, x E [-1,1].

109