Сборник задач по высшей математике 2 том

o cI>YHK:U;1U1 He'leTHM U II09TOMY pa3JIaraeTC.H B P.H,n: cI>YPhe 110 cuaYCaM (<l>OPMYJIhI (4.6)-(4.7)). B HameM CJIY'Iae f(x) = x, l = 3, CJIe,n:OBaTeJIhHO,

bn = i	3
	Jxsin n;x dx =
	o
[u = X dv = sin n1fX					du = dx		,	v = J sin n1fX dx = -~ cos n1fx]
	,			3'				3		1fn	3
			3		3						3
=-~xcos n1fX I				+ ~ Jcos n1fX dx = -~ COS1fn +..1..:l.... sin n1fX I =
1fn		3	0	1fn	3			1fn	(1fn)2		3 0
					o				sinO) = ~(_1)n+1.
				= _~(_1)n + _6_(sin1fn -
					~			~~		~
	6	00	(-1) n+1 sin n'3x					(Sin 7rX sin 27rX		sin 37rX	)
f(x) = x = 1f		L		n		= *~ -			2 3	+ 3 3	-... .

n=l	•

1.4.25.f(x) = x, (-2,2). 1.4.26. f(x)=x, (-~,~).

1.4.27.f(x) = lxi, (-2,2). 1.4.28. f(x) = lxi, (-~, ~).

1.4.29.f(x) = x2, (-3,3). 1.4.30. f(x)=x2, (-~,~).

Pa3.!UXHCumb a pRO r!Jyp'be OaHH'bIe tjjYH'lCV,UU Ha UHmepaaAe (-1f, 1f):

1.4.31.	f(x)	= {9,	-1f < x < 0, 1.4.32.	f(x) = {a,	-1f < x < 0,
		5,	°< x < 1f.	b,	°< x < 1f.
1.4.33.	f(x)	= ~ - Ixl·
1.4.34.		2
	f(x) = cos ax (a - He :u;e.uoe 'IUCJIo).

~(~ + x), -1f < x< 0,

1.4.35.f(x) = { 2a 1f

1f(2-x), O::;;X<1f.

1.4.36. f(x) = {O,	-1f < X < 0,
x,	0::;; x < 1f.

1.4.37.lIpu IIOMOlIl;U pa3JIO:lKeHU.H U3 3a,n,a'lU 1.4.35 BhI'IUCJIUTe CYMMY P.H-

111		2 + ...
,n:a1+2"+2"+ ... +	(2k -
3 5		1)

50

1.4.38. MCnOJIb3Yil pa3JIQ)KeHHe 3a,n;a,{H 1.4.36, HaA,D,HTe CyMMy pil,D,a

1	1	. 1	+ ... ;
a) 1 + 32	+ 52	+ ... + (2k _ 1)2

1111

6)1 - 2 + 3 - 4 + 5 - ... (pil,D, JIeA6HHu;a).

1.4.39.Pa3JIQ)KHTe B pil,D, <DYPbe no cHHycaM <PYHKu;HIO

X,	0:::;	x < ~,
f(x) = {	11"	x < 11".
0,	2":::;

Pa3J1rotCumb e P.RO tPYPbe Ha UHmepeaJle (-l, l) cJleOy70ut,ue tPYHftOqUU:

1.4.40. f(x) = x.

1.4.41.

f(x) = Ixl·

1.4.42.f(x) = x 2 •

60nee CnO)KHble 3aACIHMfI

1.4.43.	Pa3JIO)KHTb B pil,D, <DYPbe Ha HHTepBaJIe (-11",11") <PYHKu;HIO
	f(x) = eX.
1.4.44.	Pa3JIQ)KHTb B pil,D, <DYPbe no CHHyCaM Ha HHTepBaJIe (0,11")		<PYHK-
	u;HIO
	a) f(x) = x 2 ;
	6) f(x) = cos ax, r,D,e a - u;eJIoe ,{HCJIO.
1.4.45.	Pa3JIO)KHTb B Pil,D, <DYPbe no KocHHycaM Ha HHTepBaJIe (0,11")		<PYHK-
	U;HIO f(x) = sin ax, r,D,e a -	U;eJIoe ,{HCJIO.
1.4.46.	Pa3JIO)KHTb <PYHKu;HIO f(x)	= x 2 B pil,D, <DYPbe no cHHycaM Ha OT-
	pe3Ke [O,~].

1.4.47. Pa3JIO)KHTb B pil,D, <DYPbe Ha OTpe3Ke [0,3] <PYHKu;HIO

X,	0:::;	x:::; 1;
f(x) = { 1,	1 < x < 2;
3-x,	2:::;	x:::; 3.

1.4.48. Pa3JIO)KHTb <PYHKU;HIO f(x) = eX B Pil,D, <DYPbe Ha HHTepBaJIe [-l, l].

UHmeZpa.lt'bHoit 'lCPU60it.

= cp(x),

rllaBa 2. AlII<D<DEPEH4111AllbHbIE YPABHEHlIIfI nEPBOrO nOPfiAKA

§ 1. OCHOBHblE nOH~TVI~. YPABHEHVI~

C PA3AEJ1~IOIJ...I,VlMVlC~ nEPEMEHHblMVI

F(x, y, y') = 0,

(1.1)

CBH3hIBaIOw;ee MelK~ co6oit He3aaHCHMYIO nepeMeHHYIO, HCKOMYIO (HeH3BecTHYIO)

cPYHKn;HIO y(x)	H ee npOH3Bo,n;HYIO y' (x)	Ha3hIBaeTCH iJu!p!pepeH!4Ua.lt'bH'bI"M, ypa6He-
HueM nep60ZO	nop.1liJ'lCa.	$:

ECJIH ypaBHeHHe (1.1) MOlKHO 3anHcaTb B BH,n;e y' = f(x, y), TO rOBOpHT, 'ITO OHO pa3peIlIHMO OTHOCHTeJIbHO npOH3BO,n;HOit. 2ho ypaaHeHHe HHOr,n;a 3anHChIBaIOT B BH,n;e dy = f(x,y)dx rum, 60JIee o6w;o,

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

(,n;HcPcPepeHn;HaJIbHaH cPopMa).

~PeweHueM (RJIH UHmeZpa.ltOM) ,n;HcPcPepeHn;HaJIbHOrO ypaBHeHHH nepBoro no-

pH,n;Ka Ha3hIBaeTCH JII06as cPYHKn;HH y KOTopaH npH no,n;CTaHOBKe B 9TO ypaaHeHHe 06paw;aeT ero B TOlK,n;eCTBO. rpacPHK cPYHKn;HH y = cp(x) B 9TOM CJIy'lae

Ha3hIBaeTCH TIpon;ecc HaxolK,n;eHHH peIlIeHHit ,n;aHHoro ,n;HcP-

cPepeHn;HaJIbHOrO ypaBHeHHH Ha3hIBaeTCH UHmezpup06aHueM 9Toro ypaaHeHHH. $:

~ 3a,n;a'laOThICKaHHH peIlIeHHH ,n;HcPcPepeHn;HaJIbHOrO ypaBHeHHH nepBoro nopH,n;-

Ka (1.1), y,n;OBJIeTBOpHIOw;ero 3a.n;aHHoMY Ha"ta.lt'bHOMY YC.lt06UIO	y(xo) = Yo, Ha3hI-
BaeTCH 3aiJa"teit Kowu.	$:

reOMeTpH'IeCKH9TO paaHOCRJIbHO CJIe~IOw;eMy: Tpe6yeTcH HaJ:iTH HHTerpaJIbHyIO KPHBYIO ypaBHeHHH (1.1), npoxo,n;HID;yIO '1epe3TO'lKYMo(xo, yo).

~06'14UM peweHueM ypaaHeHHH (1.1) Ha3hIBaeTCH TaKaH cPYHKn;HH

y = cp(x, C),

(1.2)

r,n;e C - npOH3BOJIbHaH nOCTOHHHaH, 'ITO:

1) npH JII060M KOHKpeTHOM 3Ha'leHHHC OHa HBJIHeTCH peIlIeHHeM 9Toro ypaB-

HeHHH;

2) ,n;JIH JII06oro ,n;onYCTHMoro Ha'laJIbHOrOYCJIOBHH y(xo) = yo Hait.n;eTcH TaKoe 3Ha'leHHenOCToHHHoit C = CO, 'ITOcp(xo, Co) = Yo. $:

B HeKOTophIX CJIy'lasxo6w;ee peIlIeHHe ,n;HcPcPepeHn;HaJIbHOrO ypaBHeHHH npHXO,n;HTCH 3anHChIBaTb B HeHBHOM BH,n;e: ~(x, y, C) = O. Tor,n;a COOTHOIlIeHHe ~(x, y, C) = 0 Ha3hIBaeTCH o6'14UM UHmeZpa.ltOM 9TOro ypaaHeHHH.

reOMeTpH'IeCKHo6w;ee peIlIeHHe (06W;Hit HHTerpaJI) npe,n;CTaBJIHeT co6oit ceMeitCTBO HHTerpaJIbHhIX KpHBhIX Ha nJIOCKOCTH Oxy.

52

~lfaCmH:b/,M peWeH.UeM ));H<J>4>epeHn;HaJIbHoro ypaJ3HeHHH nepBOrO nOpH));Ka Ha-

3bIBaeTCH cPYHKn;HH

y = tp(X, Co),

nOJIY'IaeMaH H3 o6m;ero pemeHHH (1.2) npH KOHKpeTHOM 3Ha'leHHH nOCToHHHoit

C=Co.

lfaCmH.'b/,M UH.mezpaJtOM ypaBHeHHH (1.1) Ha3bIBaeTCH paBeHCTBO <I>(x, y, Co) =

= 0, nOJIY'IeHHOeH3 o6m;ero HHTerpaJIa npH cPHKcHpoBaHHoM 3Ha'leHHHC. ~

TeopeMa 2.11. nYCTb B AlllcI>cI>epeH~lIIanbHoM ypaBHeHlII1II y' = I(x, y) cl>YHK~III"

I(x,y) III ee 'laCTHa"npo1ll3BOAHa" I;(x,y) HenpepblBHbl B HeKOTopoA o6naCTill D nnOCKOCTIll Oxy. TorAa An" mo6oA TO'lKIllM(xo, yo) ED cYLLleCTByeT III nplllToM eAIIIHCTBeHHoe peweHllle y = y(x) 3Toro ypaBHeHIII". YAoBneTBOp"IOLLlee Ha'lanbHoMy ycnoBlll1O y(xo) = Yo.

B KaJK));Oit TO'lKe(Xo, yo) E D '1HCJIOI(xo, yo) BbIpaJKaeT yrJIOBoit K03cPcPHn;H- eHT KacaTeJIbHoit K KPHBOit Y = y(x). TI03TOMY KaJK));Oit TO'lKe06JIaCTH D ypaBHeHHe y' = I(x, y) CTaBHT B COOTBeTCTBHe HeKOTopoe HanpaBJIeHHe - reOMeTpH'IeCKH ero M02KHO H306pa3HTb '1epTO'lKoit(CTPeJIKOit), npOXO));Hm;eit '1epe33TY TO'lKY.TeM caMbIM ypaBHeHHe y' = I(x, y), (x, y) E D Onpe));eJIHeT noJte H.anpasJteH.uit Ha nJIOCKOCTH.

MH02KeCTBO TO'leK(x, y) E D, B KOTOPbIX y' = k, r));e k - nOCTOHHHaH, HJIH, 'ITO TO 2Ke caMoe, I(x, y) = k (JIHHHH ypOBHH cPYHKn;HH I(x, y)), Ha3bIBaeTCH U307CJtUH.Oit

));HcPcPepeHn;HaJIbHOrO ypaBHeHHH. B TO'lKaxH30KJIHHbI HanpaBJIeHHe nOJIH O));HHaKOBO, T. e. HanpaBJIeHHH KaCaTeJIbHbIX B TO'lKaxH30KJIHHbI (HJIH COOTBeTCTBYIOm;He '1epTO'lKH)napaJIJIeJIbHbI.

TIpH));aBaH k MH3KHe '1HCJIOBbIe 3Ha'leHHH, M02KHO nOCTpOHTb ));OCTaTO'lHYIO rYCTYIO ceTb H30KJIHH, a C HX nOMOIlJ;bIO - npH6JIH2KeHHO HapHCOBaTb BH)); HHTerpaJIbHbIX KPHBbIX, T. e. pemeHHit ));HcPcPepeHn;HaJIbHOrO ypaBHeHHH. 9TOT MeTO));,

MemoiJ U307CJtUH., HJIH rpacPH'IeCKHit(reOMeTpH'IeCKHit)MeTO)); pemeHHH ));HcPcPepeH- n;HaJIbHbIX YPaBHeHHit, oco6eHHO n;eHeH B TOM CJIY'Iae,KOr));a pemeHHe, o6m;ee HJIH '1acTHoe,ypaBHeHHH He BbIpaJKaeTCH B 3JIeMeHTapHbIX cPYHKn;HHX - HHTerpaJI He 6epeTCH.

HeKOTopbIe ));HcPcPepeHn;HaJIbHbIe ypaBHeHHH MorYT HMeTb TaKHe pemeHHH, KOTopbIe He nOJIY'IaIOTCHH3 o6m;ero HH npH KaKHX 3Ha'leHHHXnpOH3BOJIbHoit nOCTOHHHOit. 9TH pemeHHH He HBJIHIOTCH '1acTHbIMHH n03TOMY Ha3bIBaIOTCH OC06'b/,MU.

Oco6bIe pemeHHH MorYT HMeTb TOJIbKO Te ypaBHeHHH, )l;JIH KOTOPbIX HapymaIOTcH YCJIOBHH TeopeMbI cym;ecTBOBaHHH H e));HHCTBeHHocTH pemeHHH.

~YpaBHeHHe BH)l;a

(1.3)

Ha3bIBaeTCH iJug}(pepewu,Ua./t'bH.'b/,M ypaSH.eH.UeM C pa3iJeJISI'lOw,UMUCJI nepeMeH.H.'b/,Mu.

~

1TeopeMa CYIIIeCTBOBaHHH H e)l;HHCTBeHHOCTH peweHHH )l;H<p<pepeHn;HaJIbHoro ypaBHe-

HHH nepBoro nOpH)l;Ka.

53

,Il;.nH 9Toro ,!l;OCTa-

CBO,!l;HTCH K ypaBHeHHIO

(1.4).

(1.3).

YpaBHeHHe (1.3) nYTeM ,!l;eJIeHHH Ha npOH3Be,!l;eHHe Ql(Y)· P2(X) npHBO,!l;HTCH K

ypaBHeHHIO C pa30e.lleH,H/b/,MU nepeMeH,H,'b/,MU

(1.4)

(K09cPcPHIJ;HeHT npH dx 3aBHCHT TOJIbKO OT x, a npH dy - TOJIbKO OT y).

06W;Hit HHTerpaJI nOJIY'IeHHOrOypaBHeHHH HaxO,!l;HTCH nO'IJIeHHbIMHHTerpHpo-

!	H(x)	! Q2(y)
	P2(X) dx +	Ql(Y) dy = C.

3aMeTHM, 'ITOypaBHeHHIO (1.3) MorYT Y,!l;OBJIeTBOpHTb pellIeHHH, nOTepHHHbIe npH ,!l;eJIeHHH Ha Ql(Y) . P2(x), T. e. nOJIY'IaeMbIeH3 ypaBHeHHH Ql (y) . P2(X) = o. ECJIH 9TH pellIeHHH He BXO,!l;HT B Hait,!l;eHHbIit 06W;Hit HHTerpaJI, TO OHH HBJIHIOTCH OC06bIMH pellIeHHHMH ypaBHeHHH

YpaBHeHHe y' = /1 (x) . h(y)

		dy
TO'lHOnOJIOlKHTb y' = dx H pa3,!l;eJIHTb nepeMeHHbIe.
2.1.1.	IIoKa3aTb, 'ITO ,Il;aHHaJI <PYHKIJ;HlI			lIBJIlIeTClI peweHHeM ,Il;aHHOrO
	,Il;H<p<pepeHIJ;HaJIbHOro ypaBHeHHlI.
	a) y = (x + C)eZ, y' -		y = eZ;
	6) y = -	22 , xy2 dx -	dy = 0;
	B) x 2 -	X		2x + y = o.
		xy + y2 = C, (x - 2y)y' -

a a) HaXO,Il;HM npOH3BOAHYIO ,Il;aHHoit <PYHKIJ;HH: y' = e Z +(x+C)ez . Tenepb

nO,Il;CTaBHM 3HaqeHHlI y H y' B 3a,IJ;aHHOe ypaBHeHHe: eZ+(x+C)eZ-(x+C)eZ =

= eZ. IIoJIyqHJIH TOlK,Il;eCTBO eZ=eZ. CJIe,IJ;OBaTeJIbHO, <PYHKIJ;HlI y =(x + C)eZ

lIBJIlIeTClI peweHHeM ypaBHeHHlI y' - y = e Z •

6) CHaqaJIa HaxO,Il;HM dy: dy = (- 22 )'dx = 43 dx. IIo,Il;cTaBHB 3HaqeHHlI
x	Z	X

Y H dy B ,Il;aHHt>e ypaBHeHHe, nOJIyqHM TO)l{,Il;eCTBo: X· ( - :2)2dx- x; dx = 0,

T. e. 0 = O. 3HaqHT, <PYHKIJ;HlI y = - 22 - ,Il;eitcTBHTeJIbHO peweHHe HCXO,Il;HOrO

x

ypaBHeHHlI.

B) Hait,Il;eM npOH3BO.D:HYIO HellBHoit <PYHKu;HH, AJIlI qero npO,Il;H<p<pepeHIJ;H- pyeM 06e qacTH ypaBHeHHlI x2-xy+y2 = C no x: 2x-y-xy' +2yy' = 0, OTKy-

Y -2x
,Il;a y' = -2--' x ¥- 2y. IIo,Il;cTaBHM nOJIyqeHHOe BbIp~eHHe AJIlI y' B ,Il;aHHoe
y-x	-2x	+ y = O. YpaBHeHHe
,Il;H<p<pepeHIJ;HaJIbHoe ypaBHeHHe: (x -	2y) . L - 2 - 2x
	y-x

o6paIIIaeTclI B TO)l{,Il;eCTBO, T. e. <PYHKIJ;HlI x 2 - xy + y2 = C lIBJIlIeTClI HHTe-
rpaJIOM HCXO,Il;HOro ypaBHeHHlI.
2.1.2.	IIoKa3aTb, 'ITO3a,Il;aHHbIe <PyHKIJ;HH lIBJIlIIOTClI peweHHlIMH COOT•-

BeTCTBYIOIIIHX ,Il;H<p<pepeHIIJIaJIbHbIX ypaBHeHHit:

a) y = In C()sx, y' = - tgx;

54

6) x 2 + 2xy = C, (x + y) dx + x dy = 0; B) y = C· sinx, y'tgx - y = 0;

r) y = Ce- 3x , y' +3y = 0;

.n.) y - x = CeY , (X - y + l)y' = 1;

e)y = Ce x3 , dy - 3x2ydx = O.

2.1.3.IIpoBepHTb, lIBmnOTClI JIH peweHHlIMH ,D,aHHbIX ,D,H<p<pepeHII.HaJIbHbIX ypaBHeHHit YKa3aHHbIe <PYHKII.HH:

a)y = 3(x ~ 1)' y' = 3y2;

6)v=f(l-e-~) adv+bv-c=O·

b	' dt	'
B) Y = 3 -	e- x 2 , xy' + 2y = e- x 2	;

r) x 2 + t 2 - 2t = C, x dx + t = 1. dt

2.1.4.PewHTb 3a,D,aqy KOWH:

a) y' = sin5x, y (i) = 1;

6) : = 3, x = 1 npH t =-1.

o a) IIpoHHTerpHpyeM 06e qacTH ypaBHeHHlI:

y = ! sin 5x dx = -i cos5x + C.

Tenepb Hait,D,eM qacTHoe peweHHe ypaBHeHHlI. IIo,D,cTaBHB x = i H Y = 1

B Hait,D,eHHoe peweHHe, nOJIyqHM HCKOMoe 3HaqeHHe C: 1 = -i cos 5; + C, oTKY,D,a C = 1. TaKHM 06pa30M peweHHeM 3a,D,aqH KOWH lIBJIlIeTClI <PYHKII.HlI

y= -i cos5x + 1.

6)IIHTerpHpYll, HaXO,D,HM: x = 3t + C, oTKy,D,a, C yqeTOM HaqaJIbHOrO YCJIOBHlI, HMeeM: 1 = 3 . (-1) + C, C = 4. IICKOMoe qacTHoe peweHHe ecTb

<PYHKIJ.HlI x = 3t + 4.

•

2.1.5.	PewHTb 3a,D,aqy KOWH:	6) y' = e-3X , y(O) = ~.
	a) y' = 2x + 1, y(2) = 5;
		3

2.1.6.CocTaBHTb ,D,H<p<pepeHII.HaJIbHOe ypaBHeHHe no 3a,D,aHHOMY ceMeitCTBY HHTerpaJIbHbIX KpHBbIX:

a) y = Cx3 ;
6) ceMeitcTBo napa6oJI, C BepwHHoit B HaqaJIe KOOp,D,HHaT H OCbIO,
COBna,D,aIOlIJ.eit C OCbIO a6CII.HCC.
o a) IIpo,l1,H<p<pepeHII.HpoBaB no x paBeHCTBO y = Cx3 , nOJIyqHM: y' = 3Cx2.
KpoMe Toro,	OqeBH,D,HO, C = Y3. IIO,D,CTaBJIlIlI 9TO BblpaJKeHHe ,l1,J111 C B
	x
paBeHCTBO y'	= 3CX2, nOJIyqaeM HCKOMoe ,D,H<p<pepeHII.HaJIbHoe ypaBHeHHe:

y' = 3· Y3 . x 2, T. e. xy' = 3y. x

6) 3a,D,aHHoe B YCJIOBHH ceMeitcTBo napa60JI onpe,D,eJIlIeTClI ypaBHeHHeM y2 = Cx. OTCIO,D,a 2y· y' = C. IICKJIIO"tfllB H3 paBeHCTB y2 = Cx H 2y· y' = C napaMeTp C, nOJIyqHM ,D,H<p<pepeHII.HaJIbHoe ypaBHeHHe 2xy' - y = O. •

55

2.1.7.	Iho6pa3HTb ceMeil:cTBo HHTerpaJIbHbIX KpHBbIX ,ll;HCp<pepeHIJ.HaJIb-
	HOrO ypaBHeHHH:
2.1.8.	a) y' = 3;	6) y' = ~ .
	CocTaBHTb ,ll;H<p<pepeHIJ.HaJIbHbIe ypaBHeHHH 3a)];aHHblX ceMeil:cTB
	HHTerpaJIbHbIX KpHBbIX:
	a) y = ~;	6) x 3 = C(x2 _ y2).
2.1.9.	CocTaBHTb ,ll;H<p<pepeHIJ.HaJIbHOe ypaBHeHHe:
	a) IIpOIJ.eCCa H3MeHeHHH TeMIIepaTypbI TeJIa B cpe,ll;e C TeMIIepaTY-
	poil: to, eCJIH CKOPOCTb H3MeHeHHH TeMIIepaTypbl IIPOIIOPIJ.HOHaJIbHa

pa3HOCTH TeMIIepaTYP TeJIa H Cpe,ll;bI;

6) IIpOIJ.eCCa H3MeHeHHH 'IHCJIeHHOCTHHaCeJIeHHH CTpaHbI, C'IHTaH, 'ITOCKOPOCTb IIpHpOCTa HaCeJIeHHH IIPOIIOPIJ.HOHaJIbHa ero 'IHCJIeHHOCTH.

a a) 0603Ha'lHM'Iepe3T(t) TeMIIepaTYPY TeJIa B MOMeHT BpeMeHH t. CKOPOCTb H3MeHeHHH TeMIIepaTypbI TeJIa paBHa ~~. Pa3HOCTb TeMnepaTYP TeJIa H Cpe,ll;bI paBHa T - to. Tor,ll;a ,ll;H<p<pepeHIJ.HaJIbHOe ypaBHeHHe IIpOIJ.eCCa COmaCHO YCJIOBHIO 3a)];a'lH 6Y,ll;eT TaKHM: ~~ = -k(T - to), r,ll;e k > 0 -

K09<P<PHIJ.HeHT IIpOnOpIJ.HOHaJIbHOCTH. ECJIH T - to > 0, TO CKOPOCTb H3MeHe-

HHH TeMnepaTypbI OTpHIJ.aTeJIbHa, T. e. 'ITOTeMIIepaTypa TeJIa nOHIDKaeTCH; eCJIH T - to < 0, TO CKOPOCTb nOJIO)KHTeJIbHa - TeJIO HarpeBaeTCH.

6) 0603Ha'lHM'IHCJIeHHOCTbHaCeJIeHHH CTpaHbI B MOMeHT BpeMeHH t 'Iepe3 N(t). Tor,ll;a ,ll;H<p<pepeHIJ.HaJIbHOe ypaBHeHHe npOIJ.eCca H3MeHeHHH 'IH-

CJIeHHOCTH HaceJIeHHH 6Y,ll;eT TaKHM d:; =kN, r,ll;e k > 0 -			K09<P<PHIJ.HeHT
npOnOpIJ.HOHaJIbHOCTH.			•
2.1.10.		CocTaBHTb ,ll;H<p<pepeHIJ.HaJIbHOe ypaBHeHHe H3MeHeHHH CKOPOCTH
		IIpH 3aMe,ll;JIeHHOM IIPHMOJIHHeil:HoM ,ll;BIDKeHHH TeJIa MaCCbI mo IIO,ll;
		,ll;eil:cTBHeM CHJIbI COIIpOTHBJIeHHH Cpe,ll;bI, IIPOIIOPIJ.HOHaJIbHOil: KBa-
		,ll;paTY CKOPOCTH (k - K09<P<PHIJ.HeHT IIPOIIOPIJ.HOHaJIbHOCTH). IIc-
		IIOJIb30BaTb BTOPOil: 3aKOH HbIOToHa.
2.1.11.		CocTaBHTb ,ll;H<p<pepeHIJ.HaJIbHOe ypaBHeHHe H3MeHeHHH MaCCbI pa-
		,ll;HH B 3aBHCHMOCTH OT BpeMeHH «<pa)];HoaKTHBHblil: pacna)];»), C'IH-
		TaH, 'ITOCKOPOCTb paCIIa)];a pa,ll;HH npHMO IIpOnOpIJ.HOHaJIbHa (Ko-
		9<P<PHIJ.HeHT k > 0) ero KOJIH'IeCTBYB Ka)K,ll;bIil: MOMeHT BpeMeHH.
2.1.12.		,il;aHo ,ll;H<p<pepeHIJ.HaJIbHOO ypaBHeHHe y' = x2 • IIocTpoHTb IIOJIe
		HaIIpaBJIeHHil:. MeTO,ll;OM H30KJIHH IIOCTPOHTb npH6JIIDKeHHO rpa-
		<PHKH HHTerpaJIbHbIX KpHBbIX. CpaBHHTb HX C	TO'lHbIMH HHTe-
a		rpaJIbHblMH KpHBblMH.
	IIMeeM f(x, y) = x2 , f~(x, y) = O. YCJIOBHH TeopeMbI cYIIIecTBOBaHHH H

e,ll;HHCTBeHHOCTH BbIIIOJIHHIOTCH BO Bcex TO'lKaXnJIOCKOCTH Oxy. Qepe3 Ka- )K)l;yIO TO'lKYnpOXO,ll;HT e,ll;HHCTBeHHaH HHTerpaJIbHaH KpHBaH H pa3JIH'IHbIe HHTerpaJIbHble KpHBbIe He nepeceKaIOTCH.

56

TIpM X

y

x

x

x=-2 x=-l	x=l x=2
Puc. 3	Puc. 4

IIPM X = 0 M J1I060M Y E (-00, +00) MMeeM y' = 0, T. e. BO Bcex TO'IKaXOCM

Oy TIOJIe roPM30HTa.JIbHO (pMC. 3). IIPM X = 1 M J1I060M Y E (-00, +00) MMeeM y' = 1 (TIoJIe o6pa3yeT yroJI 45° C OCbIO Ox), = 1 TIOJIe TaIOKe o6pa3yeT C OCbIO Ox yrOJI 45°. IIoJIe CMMMeTpM'IHOOTHOCMTeJIbHO OCM Ox. IIOCTPOMM TeTIepb MHTerpa.JIbHbIe KpMBbIe, KOTOPbIe B KroK,n;Oit TO'IKeKacaIOTCjI «TIOJIjI».

IIOJIY'IeHHbIeKpMBbIe HaTIOMMHaIOT Ky6M'IeCKMeTIapa60JIbI (pMC. 4). TO'IHbIe HHTerpa.JIbHbIe KpMBbIe MMeIOT BM,n; y = ~3 + C. •

j{.!&SI c.I&eiJY'lOt4UX iJurjjrjjepe'H'Il,Ua.!l(b'H'btX ypa6'He'HuiJ. noCmpOUmb nO.l&e 'Hanpa6-

.l&e'HuiJ. U npu6.1&u;)tCe'H'H'bt.M 06pa30M noCmpOUmb 'He?ComOp'bte U'Hme2pa.l&b'H'bte "'pU6'bte

2.1.13.	y' = -x + y.	2.1.14.	y' = x-I.
2.1.15.	PeIIIHTb ypaBHeHMe (x -	xy2)dx + (y -	yx2)dy = O. MMeeT JIM OHO

OC06bIe peIIIeHM.H?

a IIpeo6pa30BblBM, 3aTIMIIIeM ,n;aHHoe ypaBHeHMe B BM,n;e (1.3):

x(1 - y2)dx + y(1 - x 2)dy = O.

2ho ypaBHeHHe C pa3,n;eJIjlIOrn;MMMCjI TIepeMeHHbIMM. Pa3,n;eJIMM o6e 'IaCTM ypaBHeHMjI Ha (1 - y2)(1 - x 2). IIOJIY'IMMypaBHeHMe C pa3,n;eJIeHHbIMM TIe-

57

peMeHHbIMH	~dx+~dy=O.
	I - x I - y
IIHTerpHPYlI o6e qacTH ypaBHeHHlI, HMeeM:
	-~In 11 - x 21_ ~In 11 - y21 = -~In IGI,

(npOH3BOJIbHYIO nOCTOllHHYIO 3,D,eCb Y,D,06HO 3anHcaTb HMeHHO Tax: -~In IC!),

T.e. (1 - x 2)(1 - y2) = C, r,n;e C ¥- OJ aTO B03MOXHO, Tax In IGI MoxeT npHHHMaTb JII06bIe ,D,eiicTBHTeJIbHbIe 3HaqeHHlI. IIoJIyqHJIH 06rn;Hit HHTerpaJI HCXO,D,HOrO ypaBHeHHlI. IIpH ,D,eJIeHHH Ha (1 - y2)(1 - x 2 ) MbI MOrJIH nOTepllTb peweHHlI y = 1, y = -1, x = 1, x = -1, HO OHH co,n;epxaTclI B 06rn;eM HHTe-

rpaJIe, eCJIH nO,D,CTaBHTb ,D,OnOJIHHTeJIbHOe 3HaqeHHe C =	O. TaxHM o6pa30M,
oc06blx peweHHit ,D,aHHoe ypaBHeHHe He HMeeT.	•

Pew,um'b aurjjrjjepeHqua.tt'bH'bte ypa6HeH'U.R.:

2.1.16.(1 + y) dx - (1 - x) dy = o.

2.1.17.~dx + YVI - x 2 dy = O.

2.1.18.xyy' = I - x2. 2.1.19. y'(1+ y) = xy sinx.

2.1.20. eY(I + y') = 1.

2.1.21. y' - xy2 = O.

2.1.22.HaitTH qacTHoe peweHHe ypaBHeHHlI

ydx + ctgxdy = 0,

Ylx=~= -1.

a 2ho ypaBHeHHe HMeeT BH,D, (1.3). Pa3,n;eJIlIlI nepeMeHHbIe H HHTerpHpYll, HaXO,n;HM 06rn;ee peweHHe 3a,n;aHHoro ypaBHeHHlI:

tgxdx + t dy = 0,	!tgxdx + !~ =	In ICII,		C I	¥- 0,		oTKy,n;a
In Iyl-In I cos xl = In ICII, Iyl =			ICI cos xl,			T. e.
y=±Clcosx,	HJIH y=Ccosx
IIO,D,CTaBJIlIlI B Hait,D,eHHoe o6rn;ee peweHHe		y	= -1	H x	=	~ (HcnOJIb3yeM
HaqaJIbHoe YCJIOBHe), HaxO,n;HM nocTollHHYIO C. A HMeHHO:
	-I=Ccos~,	C=-2.
CJIe.n;oBaTeJIbHO, HCKOMoe qacTHoe peweHHe HMeeT BH,D,: y =						-2 cos x.		•

Hatimu "tacmH'bte pew,eH'U.R. aurjjrjjepeHqua.tt'bH'btX ypa6HeHuti:

2.1.23.2..jYdx - dy = 0, y(O) = 1. 2.1.24. y' = 8..jY, y(O) = 4.

2.1.25.y' sin x - yIny = 0, y (~) = 1.

2.1.26.(1 + y2) dx + (1 + x 2) dy = 0, y(I) = 2.

2.1.27.Onpe,D,eJIHTb qHCJIeHHOCTb HaCeJIeHHlI POCCHH qepe3 20 JIeT, CqH-

Tall, 'ITOCKOPOCTb npHpOCTa HaceJIeHHlI npOnOPIJHOHaJIbHa ero HaJIHqHOMY KOJIHqeCTBY, H 3HalI, 'ITOHaceJIeHHe POCCHH B 2000 ro.n.y

58

COCTaBmlJIO 145 MJIH '1MOBeK,a npHpOCT HaceJIeHHB: 3a 2000 ro,n; 6bIJI paBeH 0:%. (BbI'IHCJIHTbnpH 0: = 2%, 0: = -1%.)

a 0603Ha'lHM'1HCJIeHHOCTbHaCeJIeHHB: POCCHH B MOMeHT BpeMeHH t '1epe3

N = N (t). ,I1;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe HCCJIe.n:yeMoro npoIJ;ecca (CKO-

POCTb	«npHpocTa»		'1HCJIeHHOCTH HaceJIeHHB:) HMeeT BH,n; dJ: =				kN, r,n;e
k > 0	-	K09<P<PHIJ;HeHT npOnOpIJ;HOHaJIbHOCTH (CM. 3a,n;a'lY2.1.9). OTcIO,n;a
HaxO,n;HM, 'ITOd;			= k dt, oTKy,n;a In INI -		In ICI = kt, T. e. lIn ~I= kt, T. e.,
Y'lHTbIBM,'ITON > 0, HMeeM N = Cekt -					o6ru;ee penreHHe ypaBHeHHB:. Co-
rJIacHo YCJIOBHIO 3a,n;a'lHN =				145 npH t	= O. HaxO,n;HM '1aCTHoepenreHHe:
145 = Ceo, T.e. C			= 145, N	= 145ekt .	Haii,n;eM 3Ha'leHHeK09<P<PHIJ;HeHTa
k, 3HM,		'ITOB KOHIJ;e 2000 ro,n;a, T. e. npH t = 1, HaceJIeHHe POCCHH paB-
HO N	=	145 + 1~0 . 145 MJIH '1eJIOBeK: 145 + 1~0 . 145 =				145ek .	OTcIO,n;a
ek = 1 + 1~0' T.e. k = In (1 + 1~0). PaBeHcTBo N = 145ekt						Tenepb MOXHO

nepenHcaTb TaK: N = 145 (1 + 1~0)t. TaKHM o6pa30M '1epe320 JIeT '1HCJIeH-

HOCTb HaCeJIeHHB: COCTaBHT:		•
npH 0: = 2%:	N = 145·(1,02)20 ~ 215 (MJIH '1eJIOBeK);
npH 0: = -1%:	N = 145·(0,99)20 ~ 119 (MJIH '1eJIOBeK).

2.1.28.TeJIo ,n;BH:>KeTCB: co CKOPOCTbIO, npOnOpIJ;HOHaJIbHOii npoii,n;eHHoMY nyTH. KaKoii nyTb npoii.n;eT TeJIO 3a 5 ceKyH,n; OT Ha'laJIa,n;BH:>Ke-

HHB:, eCJIH H3BeCTHO, 'ITO3a 1 ceKyH.n:y OHO npoxo,n;HT nyTb 8 M, a 3a 3 ceKYH,n;bI - 40 M?

2.1.29.IhBecTHo, 'ITOTeJIO OXJIa:>K,n;aeTCB: B Te'leHHe15 MHH OT 100° ,n;o 80°. Qepe3 CKOJIbKO MHHyT TeMnepaTypa TeJIa nOHH3HTCB: ,n;o 40°, eCJIH TeMnepaTypa oKpyxaIOru;eii cpe,n;bI COCTaBJIB:eT 10°? (CKOPOCTb OXJIa:>K,n;eHHB: TeJIa npOnOpIJ;HOHaJIbHa pa3HOCTH TeMnepaTYP TeJIa H OKpyxaIOru;eii cpe,n;bI, CM. 3a,n;a'lY2.1.9.)

2.1.30.B 3a,n;aHHOM ceMeiicTBe KpHBbIX HaiiTH JIHHHIO, y,n;oBJIeTBOpB:IOIIJ;yIO Ha'laJIbHoMYYCJIOBHIO: l; 6) y2 - x 2 = C, y(O) = 1.

2.1.31.Y6e,n;HTbCB:, 'ITO3a,n;aHHaB: <PYHKIJ;HB: B:BJIB:eTCB: penreHHeM COOTBeTcTByIOru;ero ,n;H<p<pepeHIJ;HaJIbHOrO ypaBHeHHB::

!e; dx, xy' - y == xe:t;

6) In(4x+8y+5)+8y-4x = C, (x+2y+l)dx-(2x+4y+3)dy = O.

2.1.32.CocTaBHTb ,n;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe ceMeiicTBa KPHBbIX,

.D:JIB: KOTOPbIX oTpe30K JII060ii KacaTMbHoii, 3aKJIIO'Ie~Hblii Me:>K.n:y KOOp,n;HHaTHbIMH OCB:MH, ,n;eJIHTCB: nOnOJIaM B TO'lKeKaCaHHB:. (McnOJIb30BaTb reOMeTpH'IeCKHiiCMbICJI npOH3BO,n;Hoii).

59