Сборник задач по высшей математике 2 том

MeTO~ JIarpaJDKa

Haii,Il;eM CHa'IaJIa06rn;ee peIIIeHlle COOTBeTCTBYIOrn;erO O,Il;HOpO,Il;HOrO ypaB-

dy

HeHII.H y' + tg x . Y = 0, T. e. dx = - tg x . y. Pa3,Il;eJI.H.H nepeMeHHble, HMeeM

a: =	- tgxdx, In Iyl = In Icos xl + In ICI,	C ¥- 0,
T. e. y = C cos x.	06rn;ee peIIIeHHe 3a,n,aHHoro ypaBHeHII.H	lIrn;eM B BII,Il;e y =

= C (x) cos x (6YKBY C 3aMeHIIJIII Hell3BeCTHoii <PYHKIIHeii C (x)). TIO,Il;CTaBJI.H.H y II y' = C ' (x) cos x - C (x) sin x B ,Il;aHHOe ypaBHeHlle, nOJIy'IIIM

C'(x) cos x - C(x) sin x + tgxC(x) cos x = CO~X'

T.e.

C'(x) cos x = CO~X

(BTOPoe II TpeTbe CJlaraeMble B3allMHO YHH'ITO>KIIJIIICb).OTCIO,Il;a

dC(x)	1	dC(x) =~, C(x) = tgx + C.
dx	= -- ,
	cos2 X	cos2 X

CJIe,Il;OBaTeJIbHO, 06rn;ee peIIIeHHe 3a,n,aHHoro ypaBHeHH.H eCTb

y = (tg x + C) cos x,

T. e. y = C cos x + sin x, KaK II B nepBOM CJIy'Iae.

6) ,I1;aHHoe ypaBHeHHe He .HBJI.HeTC.H JIHHeiiHbIM OTHOCIITeJIbHO y II y', HO

.HBJI.HeTC.H TaKOBbIM OTHOCHTeJIbHO x H x'. Y'IHTbIBa.H,'ITOy'

= ~, npHBe,Il;eM

ypaBHeHHe K BH,IJ;y (3.2):

x

yl=.l =

_ y _

I

X+y2

x' -

1 x = y.

T.e.

x =

--y-,

HJIII

x'

x + y2'

y

PeIIIa.H MeTO,Il;OM

BepHYJIJIH,

nOJIaraeM

x

= uv,

r,Il;e u

= u(y),

v = v(y)

-

<PYHKIIIIII OT y. Tor,Il;a x' = u'v + uv' II

u'v + uv' -

t

uv = y,

HJIH

u'v + u(v' -

tv) = y.

(3.4)

tv = 0:

PeIIIaeM ypaBHeHlle c pa3,Il;eJI.HIOrn;IIMIIC.H nepeMeHHbIMII v' -

dv

v

dv

dy

In Ivl = In ICyl,

C ¥- o.

dy

= Y'

T.e.

v

Y'

BbI6HPa.H O,Il;HO 113 B03MO>KHbIX peIIIeHllii (caMoe npOCToe),

HMeeM: v =

y.

TIO,Il;CTaBJI.H.H v = y B ypaBHeHHe (3.4), nOJIy'IIIMu'y = y,

T. e. y'

= 1,

II,

3Ha'IIIT,u

= y + C. CJIe,Il;OBaTeJIbHO,

x

= uv = (y + C)y

= y2 + Cy, T. e.

x = y2 + Cy -

06rn;ee peIIIeHHe

3a,n,aHHoro ypaBHeHH.Hj y

= 0 -

oc060e

peIIIeHHe.

70

u) YpaBHeHHe npHBO,l1.HTCfI K BH,rry (3.2), T. e. 9TO ypaBHeHHe BepHYJIJm: y' - ~ y = xvy. CHoBa nOJIaraeM y = uv. IIoJIY'IaeMypaBHeHHe

u'v + uv' - ~ uv = xv'Uv

HJIH u'v + u(v' - ~ v) = x.,fUV. PernaeM nepBoe ypaBHeHHe v' - ~ v = 0, pa3,l1.eJIfIfI nepeMeHHbIe: d: = ~ dx, T. e. In Ivl = 4In Ixl + C. BbI6HPM npocTeitrnee perneHHe (npH C = 0), HaXO,l1.HM v = x4. PernaeM BTopoe ypaBHeHHe

C pa3,l1.eJIfllOm;HMHCfI nepeMeHHbIMH: u' x4				= x.,fU . x2, T. e.		~ = ~, OTKy-
,l1.a 2.,fU	= In Ixl + In ICI, C :f.	o. TaKHM		06pa30M, u = ~ In2lxcl, C :f. 0,
H, CJIe,l1.0BaTeJIbHO, y = uv =		~ x4 In2 lxCI,			r,l1.e C :f. 0, -	o6m;ee perneHHe
3a,l1.aHHoro ypaBHeHHfI, y = 0 -			oco6oe perneHHe.			•
PeVJ,umb	ypa6'He'H'U.R:
2.3.2.	y' - 2xy = eX2 .		2.3.3.		xy' + y -	3x2 = O.
2.3.4.	y2 dx + (x + 2) dy = O.		2.3.5.		(x + l)y' - 2y = y2(x + 1)5.

2.3.6. HaitTH KPHBYIO, npOXO,l1.f1~1O '1epe3TO'lKYP(I,O) H TaKYIO, 'ITO OTpe30K, OTceKaeMblit KaCaTeJIbHoit Ha OCH Op,l1.HHaT, paBeH a6cIIHCce TO'lKHKacaHHfI.

x.

Puc. 7

Q IIycTb AC -			KacaTeJIbHM K HCKOMOit KPHBOit B TO'lKeM(x,y) (pHC. 7).
COrJIaCHO YCJIOBHIO OB =				x = ~A. Hait,l1.eM 0p,l1.HHaTY TO'lKHA, nOJIO)J(HB
X = 0 B ypaaHeHHH KacaTeJIbHoit Y - y =								y'(X - x), r,l1.e Y =	~A. lfMe-
eM: Y - y =		-y'x, T.e. Y		= y -	y'x. TaKHM 06pa30M, nOJIY'lHJIHJIHHeitHoe
ypaBHeHHe x		= y - y'x,		HJIM y'	-	! y	=	-1. IIoJIo)J(HB y =	uv, pernHM
ero MeTO,l1.0M BepHYJIJIH: u'v + uv' -						~v	=	-1, T.e. uv' + v (u' -	¥) = -1.
HaXO,l1.HM y,:		~~ -	¥ = 0, ~u =		~, u		=	x. HaXO,l1.HM v, nO,l1.CTaBJIfIfI u:
xv' =	-1, HJIH v'		= -!, OTKY,l1.a v			= -Inlxl +InICI, T.e. v =			Inl~l, r,l1.e
C :f. O.	lfTaK, y = x Inl ~I, r,l1.e C :f.					0 -	ypaBHeHHe ceMeitcTBa HHTerpaJIb-

HbIX KpHBbIX. BbI,l1.eJIHM Cpe,l1.H HHX O,l1.Hy KPHBYIO, npOXO)J.fl~1O '1epe3TO'lKY

71

<PYHKIJ;H~ Y = ClYI + C2Y2

P(l,O): °=		1 . In ICI, a 3Ha'UiT,C = ±l. CJIe)J.OBaTeJIbHO, y =	x In I!I'T. e.
y = -x In Ixl		- ypaBHeHHe HCKOMOit KPHBOit.	•
2.3.7.	HaitTH KpHByID, npoxo)J.~IIU'ID 'Iepe3TO'IKY0(0,0), 3HM, 'ITOyrJIO-
	BOit K09<P<PHIJ;HeHT B JIID60it ee TO'IKepaBeH cyMMe Koop)J.HHaT 9TOit
	TO'IKH.

Pew,umb

au!Jj!Jjepe'Hiqua.llb'Hibl.e

ypa6'H.e'H.'I.LR:

2.3.8.

y' + 2y = 3ex .

2.3.9.

(1 + x 2)y' + 2xy = 3x2.

2.3.10.

2(x + y4)y' -

Y = 0.

2.3.11.

y2 dx + (xy -

1) dy = 0.

2

y' + 2xy = 2xy3.

2.3.12.

xy' + y = Y2

In x.

2.3.13.

2.3.14.

y' + Y cos x = sin 2x.

2.3.15.

x :~ + y = 4x3 •

2.3.16.

y'eX 2 - (xe X

2

- y2 )y = 0.

2.3.17.

x 3y2y' + X2y3 = l.

2.3.18.

y'x3 siny -

xy' + 2y = 0.

2.3.19.

y' -

y = (x + l) eX.

2.3.20.

y'+ _X_y =2.

2.3.21.

' - -y- =t

~

1- x 2

Y

sinx

g 2'

2.3.22.

xy' - Y -

x 3 = 0, y(2) = 4.

2.3.23.

y

, .

ycosx =

1

,y

(11")

1

smx -

4"

= J2.

I

2.3.24.2y2 dx + (x + eY) dy = 0, y(e) = l.

2.3.25.y' - ly = _y2, y(l) =-l.

2.3.26.x cos2 x y' + 2y cos2 X = 2xVY.

2.3.27.y dx + (4 In y - 2x - y) dy = 0.

2.3.28.(y' +y)(x2 + 1) = e- x , y(O) = l.

2.3.29.s' - ssint = 2sin2t, s(O) = l.

2.3.30.cpr' + r - e'P = 0, r(a) = 2a.

2.3.31.dx+ (xy -y3)dy = 0, y(-l) = 0.

2.3.32.y' + 2: = 3x2 W, y(l) = l.

2.3.33.IIycTb YI H Y2 - )J.Ba Pa3JIH'IHbIXperneHH~ ypaBHeHH~ y' +p(x)y = =g(x). IIpH KaKOM COOTHorneHHH Me)l{)J.y nOCTO~HHbIMH CI H C2

6Y)J.eT perneHHeM )J.aHHoro ypaBHeHH~? 2.3.34. MaTepHaJIbHM TO'IKaMaccoit m norpy)l{aeTc~ C HYJIeBoit Ha'IaJIb-

HOit CKOPOCTbID B )l{H)J.KOCTb. Ha Hee )J.eitcTByeT CHJIa T~eCTH H CHJIa conpOTHBJIeHH~ )l{H)J.KOCTH, npOnOpIJ;HOHaJIbHM CKOPOCTH no- rpy)l{eHH~ (K09<P<PHIJ;HeHT npOnOpIJ;HOHaJIbHOCTH k). HaitTH 3aBH-

CHMOCTb CKOPOCTH )J.BH:>KeHH~ TO'IKHOT BpeMeHH.

2.3.35. HaitTH KpHByID, npOXO~IIU'ID 'Iepe3TO'IKY A(1,2), KaCaTeJIbHM K KOTOPOit B npOH3BOJIbHoit ee TO'IKeOTceKaeT Ha OCH op)J.HHaT OTpe30K, paBHblit KBa)J.paTY op)J.HHaTbI TOQKH KacaHH~.

72

2.3.36. CllJIa TOKa			I B 3JIeKTpllqeCKoit	n;enll C conpOTllBJIeHlleM R, KD-
	3<pqmn;lleHToM llH)JyKTllBHOCTll L 11 3JIeKTpo,n:BIDKyrn;eit CllJIOit E
	y,n:OBJIeTBOp~eT ,n:ll<p<pepeHn;llaJIbHoMY ypaBHeHlllO
			L d1 +R1= E.
			dt
	HaitTll 3aBllCllMOCTb CllJIbI TOKa 1= 1(t) OT BpeMeHll, eCJIll:
	a) E 113MeH~eTC~ no 3aKOHY E = kt 11 1(0) = 0 (L, R, k -				nOCTO-
	~HHbIe), k -		K03<P<Plln;lleHT nponOpn;IlOHaJIbHOCTllj
	6) E M3MeH~eTC~ no 3aKOHY E			= Asinwt 11 1(0) = 0 (L,	R, A,
	w - nocTO~HHbIe).
KOHTponbHble BonpOCbl III 60nee CnO)l(Hbie 3aAfl~1II
2.3.37.	HaitTll 06rn;ee peUIeHlle ypaBHeHll~ y' +y~'(x) -~(x)~'(x) = 0, r,n:e
	~(x) -	3MaHHM <pYHKn;Il~.
2.3.38.	PeumTb ypaBHeHll~:			x
	a) xy' -	xeY + 2 = OJ		6) y(x) = Jy(t) dt + x + 1.
Pewumb	awfitfiepeH'U,ua.n,b'H,'bl.e ypasHeH'U.R.:			o

2.3.39.y' - 2xy = 1 - 2x2 , y(O) = 2.

2.3.40.	2yVx
	yx' + 2x = -- , y(O) = 1r.
	cos2 y
2.3.41.	y' cos y + sin y = x.
2.3.42.	dx + (2x + sin2y -	2cos2 y) dy = 0, y(-I) = O.
2.3.43.	(64 y3 - x)y' - 2y = O.
2.3.44.	x - I	2	, y(O) = 2.
	y' + xy = - 2 - eXy
2.3.45.	HaitTll Kp"Bble, y KOTOPbIX nJIOrn;Mb TpeyrOJIbHllKa, 06pa30BaHHD-
	ro OCblO a6cn;Hcc, KaCaTeJIbHoit 11 pa,n:llycoM-BeKTopoM TOqKll Kaca-
	Hll~, nOCTO~HHa 11 paBHa 4.
2.3.46.	KPllBM Y = f(x) npoxo,n:llT qepe3 TOqKY 0(0,0). HaitTH ee ypaB-
	HeHlle, 3Ha~, 'ITOcepe,n:llHa OTpe3Ka ee HOPMaJIH OT JI1060it TOqKll
	KPllBOit ,n:o OCll a6Cn;IlCC JIHlKllT Ha napa60JIe y2 = x.
	Y"a3aHue. Cepe,n:llHa C OTpe3Ka HOPMaJIll llMeeT Koop,n:llHaTbI

( x + ~ yy', ~) .

2.3.47. HaitTll TaKlle <pYHKn;1l1l p(x) 11 g(x), qT06bI peIIIeHll~Mll ypaBHeHll~

y' + p(x)y = g(x) ~BJI~JIllCb <pYHKn;1l1l y = IllY = x 3 + 1.

2.3.48. MO)KHO JIll peIIIaTb ypaBHeHlle y' = y C nOMOrn;blO no,n:CTaHOBKll

y = uv?

2.3.49. MO)KeT JIll peIIIeHlle ypaBHeHll~ y' = y (y =j:. 0) llMeTb TOqKll Mll-

HllMYMa?

2.3.50.,II,.rr~ KaKOit KPllBOit KaCaTeJIbHM B K8.)K,n:Oit ee TOqKe nepneH,n:llKY- JI~pHa PMllyC-BeICTOPY TOqKll KacaIm~?

73

<PYHKIIHil: P(x, y), Q(x, y)

§ 4. YPABHEHLJlH B nOJlHblX ALJI<IJ<lJEPEHLI,LJlAJlAX

P(x, y) dx + Q(x, y) dy = 0

(4.1)

Ha3bIBaeTCH ypa6'He'HUeM 6 nOJl'H'btx aU¢¢epe'H'Il,UaJlaX, eCJIH ero JIeBaH 'IaCTbeCTb

rrOJIHbIil: ,lJ;H<p<pepeHIIHaJI HeKoTopoil: <PYHKIIHH U(x, y), T. e.

dU(x, y) = ~: dx + ~~ dy = P(x, y) dx + Q(x, y) dy.

(4.2)

YpaBHeHHe (4.1) C Y'IeTOM(4.2) MO)KHO.3arrHcaTb B BH,lJ;e dU(x, y) = 0, rro9ToMY ero o6IIIHil: HHTerpaJI HMeeT BH,lJ;

U(x,y) = C.

AnH TOro, 'ITo6bIypaBHeHHe (4.1) 6bIJIO ypaBHeHHeM B rrOJIHbIX ,lJ;H<p<pepeHIIHaJIaX, Heo6xo,lJ;HMO H ,lJ;OCTaTO'lHO,'ITo6bIBbIIIOJIHHJIOCb YCJIOBHe

(4.3)

cI>YHKIIHH U(x, y) MO)KeT 6bITb Hail:,lJ;eHa H3 CHCTeMbI YPaBHeHHil:

au	= P(x,y),	au		= Q(x,y)
ax		ay
JIH60 rro <popMYJIe
	x			y
U(x,y) = jP(x,Y)dX+				jQ(XO,Y)dY,	(4.4)
			YO
r,lJ;e (XO, yo) - HeKOTopaH	<pHKCHpOBaHHaH		TO'lKa H3 06JIacTH		HerrpepbIBHOCTH

H HX 'IaCTHbIXrrpOH3BO,lJ;HbIX.

3aMe"l,a'HUe. ECJIH YCJIOBHe (4.3) He BbIIIOJIHHeTCH ,!I;JIH ypaBHeHHH (4.1), TO B

pH,lJ;e CJIY'IaeBero MO)KHO CBeCTH K ypaBHeHHIO B rrOJIHbIX ,lJ;H<p<pepeHIIHaJIaX yMHO- )KeHHeM Ha HeKOTOpyIO <PYHKIIHIO t(x, y) = t, Ha3bIBaeMYIO «u'Hme~pupY'/OW,UM M'HO- ::HCUmeJleM». HHTerpHPYIOIIIHil: MHO)KHTeJIb JIerKO HaxO,lJ;HTCH B ,lJ;BYX CJIY'IaHx:eCJIH

t = t(x) HJIH t = t(y); B rrepBoM cJIY'Iae
		t(x) = eJ	8P	8Q
			..:8-"Y--,,-_8::...;X,- dx
				Q
ap		aQ
ay-a;;
rrpH'IeMBblp~eHHe	Q	,lJ;OJI)KHO 3aBHCeTb TOJIbKO OT x; BO BTOPOM cJIY'Iae
			8Q	8P
		Jei -8ij			dy ,
		t(y)=e		P

rrpH'IeMrrO,lJ;bIHTerpaJIbHOe Bbrp~eHHe ,lJ;OJI)KHO 3aBHCeTb TOJIbKO OT y.

74

2.4.1.PewHTb ypaBHeHHe eX+y+siny+y'(eY+x+x cos y) = 0, y(ln 2) = 0.

Q 3aIIHweM ypaBHeHHe B ,l1.Hq,q,epeHIl;Ha.rrbHoit q,opMe

(eX +	y +	sin y) dx + (eY + x +	x cos y) dy = 0.
3,L(eCb P(x, y) = eX +	y +	siny, Q(x, y) = eY +	x + x cosy. npoBepHM BbIllOJ1-
HeHHe YCJ10Bm:l (4.3):
aP		aQ	aP	aQ
ay =l+cosy, ax =l+cosy,			T.e. ay -	ax'

11, 3HaqHT, YCJ10BHe (4.3) BbIllOJ1HjleTCjI. CJ1e,L(OBaTeJ1bHO, ,L(aHHOe ,L(Hq,q,epeHIl;11a.rrbHoe ypaBHeHHe eCTb ypaBHeHHe B nOJ1HbIX ,l1.Hq,q,epeHIl;Ha.rrax. Hait,L(eM q,YHKIl;HID U, HCn0J1b3Yjl paBeHCTBa

aU = eX + y + sin y	H aU = eY + x + x cos y.
ax	ay
HHTerpHpYjl nepBoe paBeHcTBo no x	(CqHTaeM y nOCTOjlHHbIM), HaXO,L(HM
U(x, y) = !(eX+ y + siny) dx = eX + yx + xsiny + cp(y),

r,L(e cp(y) - npOH3B0J1bHaji ,L(Hq,q,epeHIl;HpyeM~ (no y) q,YHKIl;HjI. Hait,l1.eM cp(y). npO,l1.Hq,q,epeHIl;HpOBaB nOJ1YQeHHOe paBeHcTBo no y H YQHTbIBaji BTopoe

paBeHcTBo (~~ = eY + x + x cos y), nOJ1YQaeM

~~ = x + x cos Y + cp' (y) = eY + x + x cos y,

oTKY,L(a cp'(y) = eY, T. e. cp(y) = eY + Cl. CJ1e,l1.0BaTeJ1bHO,

U(x,y) = eX + xy + xsiny + eY + Cl.

06rn:HM HHTerpa.rroM jlBJ1j1eTCjI COOTHoweHHe eX + xy + x sin y + eY + Cl = C2

I1J1H eX +		xy + x sin y + eY = C, r,L(e C = C2 -			Ci. Hait,L(eM QacTHbIit HHTe-
rpa.rr ypaBHeHHjI, ,l1.J1j1 Qero nO,l1.CTaBHM HaQa.rrbHoe YCJ10BHe y = 0,							x = In 2
B 06rn:Hit HHTerpa.rr: 2 + °+			°+ 1 =	C, OTKY,L(a C =		3. TaKHM	06pa30M,
eX +	xy +	x sin y + eY = 3 -	HCKOMbIit qaCTHblit HHTerpa.rr.					•
2.4.2.		PewHTb ypaBHeHHe ~ dx + (3y2 + Inx) dy = 0.
I\.			_ Y	_	2	aP _ 1 aQ _ 1
'oJ	B ,l1.aHHOM cJ1YQae P(x, y) - x' Q(x, y) - 3y				+lnx, a	ay - x'	ax -	x'
T. e.	aP	aQ
	ay = ax. CJ1e,l1.0BaTeJ1bHO, 9TO ypaBHeHHe B nOJ1HbIX ,L(Hq,q,epeHIl;Ha.rrax

H, 3HaQHT, HMeeT BH,L( dU(x,y) = 0, r,l1.e ~~ =~, ~~ = 3y2 +lnx. OTCID,l1.a

U(x,y) = !~dx, T.e. U(x,y)=ylnx+cp(y).

(ct>YRKIl;HID U (x, y)	MO:lKHO HaXO,l1.HTb H H3 BTOpOro paBeHCTBa, HHTerpHpyjl
ero no y: U(x, 'II) =	! (3'112 + lox) dy + cp(x).) Tor,l1.a ~~ = (y Inx + cp(y))~ =

== Inx + cp'(y). OTCID,l1.a 3y2 + Inx = Inx + cp'(y), cp'(y) = 3y2 H, CTa.rrO 6bITb,

75

cp(y) = y3+C1 • CJIe.n;OBaTeJIbHO, U(X,y) = yInx+y3+C1, ayInx+y3 = C- 06ID;Hii HHTerpaJI Hcxo.n;Horo ypaBHeHHfI.

3a.Me"taHue. Haii.n;eM <PYHKIJ;HlO U(x,y), HCnOJIb3Yfl <POPMYJIY (4.4). ITo- JIO:>KHM Xo = 1, Yo = 0, Tor.n;a TOQKa (1,0) npHHa.n;JIe:>KHT 06JIacTH HenpepbIB~ HOCTH D = {(x, y): x> o}. lIMeeM:

		x	Y
		U(x, y) = / ~ dx + /(3y2 + In 1) dy,
		1	0
oTKy.n;a U(x, y)		= y In x + y3. CJIe.n;oBaTeJIbHO,		y In x + y3 = C - 06ID;HA
HHTerpaJI ypaBHeHHfI.				•
PeUJ,umb	ypa6HeH'I.LR:
2.4.3.	(2x -	y) dx - x dy = O.	2.4.4.	e-Ydx + (2 - xe-Y) dy = O.

2.4.5.HaiiTH HHTerpHpYlOID;Hii MHO:>KHTeJIb H penlHTb ypaBHeHHe

	(eY + sin x) dx + cos x dy = O.
r..	ap _ Y aQ _ .	ap	=j:.	aQ
'-I3.n;ecb	ay - e , ax - - sm x, T. e.	ay		ax' H, 3HaQHT, ypaBHeHHe He

f1BJIfleTCfI ypaBHeHHeM B nOJIHbIX .n;H<p<pepeHIJ;HaJIax. TaK KaK OTHorneHHe

aQ	of	- sin x - eY
ax -	ay
P	=	eY + sin x =-1

He 3aBHCHT OT x, TO HHTerpHpYlOII1,ll:A MHO:>KHTeJIb MO:>KeT 6bITb HaA.n;eH no
<popMYJIe	Q~ -p;
	t(y) = e J - p- dy
(CM. 3aMeQaHHe Ha c. 74):
	t(y) = eJ(-I)dy = e-Y.

YMHO:>KM Hcxo.n;Hoe ypaBHeHHe Ha t = e-Y , nOJIYQaeM ypaBHeHHe B nOJIHbIX

.n;H<p<pepeHIJ;HaJIax:

(1 + e-Ysin x) dx + e-Y cosxdy = 0

(TaK KaK p~		= -e-Y sin x = e- Y ( - sinx)	= Q~). PernaeM em (6e3 nOflCHe-
HHii):	au = 1 + e-Y sinx, au = e- Y cos x;
a)
	ax	ay
6)	U(x, y) = /(1 + e-Ysin x) dx = x -		e-Y cos x +	cp(y);
B)	~~ =	e-Ycosx + cp'(y), oTKy.n;a	e-Ycosx +	cp'(y)	= e-Ycosx, T.e.
cp'(y) = 0, cp(y) = CI ;
r) U(x,y)		= x - e-Ycosx + C I . TaKHM 06Pa30M X -			e-Ycosx = C -
06ID;Hii HHTerpaJI ypaBHeHHfI.					•

2.4.6.HaATH HHTerpHpYlOlIJ;HA MHO:>KHTeJIb H pernHTb ypaBHeHHe

(x 2 - sin2 y) dx + x sin 2y dy = O.

76

UJl,U t

PeW/1.1mb	ypa6He'H/I.LR:
2.4.7.	(3x - 5x	2y2) dx + (3y2 - 10 x 3 y) dy = O.
		3

2.4.8.(x cos 2y - 3) dx - x 2sin 2y dy = O.

2.4.9.(2x + yeXY ) dx + (1 + xexy ) dy = 0, y(O) = l.

2.4.10.	(	x	+ Y)dX + (x +	Y	)dY = 0, y(v'2) = v'2.
		Jx2	+ y2	Jx 2 + y2
2.4.11.	(x2 + 2xy + 1) dx + (x 2 + y2 - 1) dy = O.

2.4.12.sin(x + y) dx + x cos(x + y) (dx + dy) = O.

2.4.13. (3x 2 + 3x2 1ny) dx - (2y - x:) dy = O.

2.4.14.3x2y + sinx = (cosy - x 3 )y'.

2.4.15.(3x 2 + y2 + y) dx + (2xy + x + eY ) dy = 0, y(O) = 0.

2.4.16.

(x 2 + 2xy) dx + (x 2 -

y2) dy = 0,

y(l) = -l.

2.4.17.

(x-y)dx+(x+y)dy =0.

x2 +y2

2.4.18.

( 2x -

1 -

:2)dx -

(2y -l)dy = O.

2.4.19.

( 2x + e ~) dx + (1 - ~) e ~ dy = O.

2.4.20.

1 .

x

Y

Y

1) d

x+

(

1

Y

x·

Xl) d

0

( -sm---cos-+

- cos --- sm - + -

y=.

y

Y

x2

X

X

X

y2

Y y2

KOHTponbHble Bonpocbl .., 60nee cnO_Hble 3aA3~'"

Pewumb ypa6HeHUR.:

2.4.21.(sin y + y sin x + l) dx + (x cos y - cos x + b) dy = O.

2.4.22.xeY2 dx + (x 2yeY2 + tg2 y) dy = O.

2.4.23.(xchy + shx) dy + (ychx + shy) dx = O.

Pewumb ypa6HeHUR., aonyc'lCa'/oUJ,ue uHmeepupy'lOUJ,u11 MH(h)tCUmeJl,b 6Uaa t = =t(x) = t(y):

2.4.24.y2 dx + xy dy - dy = O.

2.4.25.(1 + 3x2siny) dx - xctgydy = O.

2.4.26.HaitTll YCJIOBllH, IIpll KOTOPbIX ypaBHeHlle

P(x, y) dx + Q(x, y) dy = 0

.n:OIIYCKaeT llHTerpllPYIOIIIllit MHO)J(llTeJIb Bll.n:a t = f(x + y).

2.4.27.HaitTll o6IIIlle llHTerpaJIbI .n:ll<p<pepeHII;HaJIbHbIX ypaBHeHllit:

a) x dx + y dy = OJ

6) x dy + y dx = O.

77

YpaBHeHHe (5.1)

2.4.28. OIIpe,l1.eJUlTb THII ,l1.H<p<pepeHIJ,Ha.JIbHbIX ypaBHeHHti:
a)	(1 -	x2)y' + xy - 3 = 0;			6)	(y + xy2) dx - x dy = 0;
B)	(x + y -		1) dx + (x + eY ) dy;					3
					r) 3y' - 2y = X 2;
)1,)	Y = xy' + y' In y;				e)	2x2 dx -	(x¥ + y2) dy = 0;
)K) Y'(x 2 -			4) = 3;		3) 2x + 3x2y + (x 3 -					3y2)y' = 0;
u) xdx+(x+y)dy=O;					K) y(x-y)dx=x2 dy;
JI)	y2 dx -		(2xy + 3) dy = 0;		M) J17 - 4x -			2x2 dy - dx = 0;
H)	y' -	2xy		.	0) (x3 + y)x' + x - Y = 0;
	-	x2 _ y2'
n) y' -		2JYInx = 0;			p) x(y' - y) = eX;
c) xy' -		3y + x4y2 = 0;			T) xy' = 2(y -			.,;xy);
y) y' = 7x - y ;					<1»	(y2 + 2y + x 2)y' + 2x = 0;
x)	dy = ~ +x.				u.)	y' = ~ -	1;
	dx	x	'			2	y2		3x	2
q) y'tgy = y;					ill)			-
						~ dx +				dy = 0;
						y3		y4
3)	(x + y -		2) dx + (x - y + 4) dy = 0;
10) x dy -			y dx - Jx 2 + y2 dx = 0;
.H)	dx = (siny + 3cosy + 3x) dy.

§ 5. YPABHEHII1S1 JlArPAH>KA 111 KJlEPO

HeKOTopble )J;H<p<pepeHJJ;HaJIbHble ypaBHeHHH nepBoro nopH)J;Ka npHXO)J;HTCH pelllaTb MeTO)J;OM BBe)J;eHHH BcnOMOraTeJIbHOrO napaMeTpa. K qHCJIY TaKHX YPaBHeHHit OTHOCHTCH ypa6'He'HUe JIaepa'H~a

y = xcp(y') + 'I/J(y')	(5.1)
H ypa6'He'HUe KJlepo
y = xy' + 'I/J(y'),	(5.2)

r)J;e cp H 'I/J - H3BeCTHble <PYHKJJ;HH OT y'.

YpaBHeHHe (5.1) HHTerpHpyeTcH CJIe)J;yIOlJJ;HM o6pa30M: 0603HaqaH p = y', 3anHllleM ypaBHeHHe B BH)J;e y = xcp(p) +'I/J(p). )J;H<p<pepeHJJ;HPYH nOJIyqeHHOe ypaBHeHHe no x, HMeeM

p = cp(p) + (xcp' (p) + 'I/J' (P)) ~:,

OTKY)J;a nOJIyqHM (p - cp(p)) ~; = xcp' (P) + 'I/J' (p) - JIHHeitHoe ypaBHeHHe OTHOCH-

TeJIbHO x H ~; • ECJIH ero pellleHHe 6Y)J;eT x = f(p, C), TO o6lJJ;ee pellleHHe ypaBHe-

HHH (5.1) 3anHCblBaeTCH B BH)J;e

X = f(p,C),

{

y = xcp(P) + 'I/J(P) = f(P, C)cp(P) + 'I/J(p).

MO)l{eT HMeTb oco6oe pellleHHe, BH)J;a y = cp(Po)x+'I/J(po), r)J;e po-

KopeHb ypaBHeHHH p = cp(P).

78

Bbl6HpaeM npocTeitrnee perneHHe:

YpaBHeHHe KJIepo liamreTCli qaCTHbIM CJIyqaeM ypaBHeHHli JIarpaH:lKa rrpH <.p(y') = y'. Ero o6IIIee peIIIeHHe HMeeT BH)]. y = Cx+'tj;(C), oco6oe peIIIeHHe rroJIyqaeTCli rrYTeM HCKJIIOqeHHli rrapaMeTpa p H3 YPaBHeHHit y = px + 'tj;(p) H X = -'tj;'(p).

2.5.1.	PenHITb ypaBHeHlIe: y = xy' _ y,2.
a YpaBHeHHe HMeeT BH,ll. (5.2), T. e. 9TO ypaBHeHHe KJIepo. llOJIO)J(HM y' = p.
Tor,ll.a 3a,ll.aHHOe ypaBHeHHe npHHHMaeT BH,ll.
	y = px - p2.	(5.3)

llPO,ll.H<p<pepeHlJ,HpoBaB ero no x, HMeeM: y' = p' x+p- 2pp' , T. e. p' (x- 2p)+p =

=p, HJIH p'(x - 2p) = O. ECJIH p' = 0, TO P = C H, 3HaqHT, 06IIJ,ee perneHHe ,ll.aHHoro ypaBHeHHH eCTb y = Cx - C2 (CM. ypaBHeHHe 5.3). ECJIH x - 2p = 0,

T. e. x = 2p, TO nOJIyqaeM y = 2pp - p2, T. e. y = p2. Oc060e perneHHe 3a,n;aH-

HOro ypaBHeHHH eCTb {		X - 2p	lICKJIlOqM napaMeTp p		(	p = ~	)	,HaXO,ll.HM
		y: p2:
oc060e perneHHe ypaBHeHHH B HBHOM BH,ll.e y =				~2 .				•
Pewumb	ypa6'tte'H!USI:
2.5.2.	Y = xy' + y' -	(y')2.	2.5.3.	y = xy' -		3(y')3.
2.5.4.	PernHTb ypaBHeHHe JIarpaH)J(a: y = x(l + y') + (y')2.

a llOJIO)J(HM y' = p. Tor,ll.a HMeeM y = x(l + p) + p2. ,nH<p<pepeHIJ,HpYH no

x, IIpHXO,ll.HM K ypaBHeHHIO y' = (1 + p) + xp' + 2pp', OTKY,ll.a (x + 2p)p' + 1 =

= 0, T.e. (x + 2P):~ = -1. OTCIO,ll.a x + 2p = -~;, T.e. x' + x = - 2p -

JIHHeitHoe OTHOCHTeJIbHO x H x' ypaBHeHHe. PernHM ero MeTO,ll.OM BepHYJIJIH. llOJIarM x = uv, 1I0JIyqaeM u'v+uv' +uv = -2p, T.e. u'v+u(v' +v) = -2p.

HaXO,ll.HM v, IIpHpaBHHBM CK06KY K HyJIIO H pa3,ll.eJIHH nepeMeHHbIe: v' + v =

= 0, ~; = -v, ~ = -dp, In Ivl = -p+ C.

v= e- p. Tor,ll.a: u'e-P = -2p, T.e. u' = -2pep. OTCIO,ll.a u = -2 JpePdp =

=-2(peP-eP)+C, H, 3HaqHT, u = -2peP+2eP+C. CJIe,ll.OBaTeJIbHO, x = uv =

=e-P(-2peP + 2eP + C) = 2 - 2p + Ce- p. YqHTblBM, qTO Y = x(l + p) +p2,

IIOJIyqHM Y = (2 - 2p + Ce- P)(l + p) + p2. TaKHM 06pa30M, 06IIJ,ee perneHHe ypaBHeHHH HMeeT BH,ll. (B napaMeTpHqeCKoit <POpMe)

X = 2 -	2p+ Ce- P,
{	2p + Ce-P)(l + p) + p2j
y = (2 -

oco60ro perneHHH HeT.				•
Pew,umb	ypa6'H.e'H.USI:
2.5.5.	y = x(y')2 + (y')2.	2.5.6.	xy'	x
			y ----
			- 2	2y"

79