TEMA 12. 1ftjHjepeHu1IaJIbHbIe ypaBilellhlsi, B aipaypax

4IH4J4JePeHLWaJThHMM H3MBTC5I

ypaBHeHlle, CB5I3EIBIOW H3BHCHM}O HepeMeHHyIo, HH3BCTH}O JYHKUH}O H eeHOH3BOHbI

ypaBHeHHe, co.Lep)Kawee HpOH3BOLHy1O He3aBHCHMOH HepeMeHHoH

ypaBHeHHe, KOTOO JIFKO

ypaBHeHHe, KOTOO pemaeci.Lw1Jc1epeHuHpoBaHHeM

PeTJIHTh Hc1x1epeHuHaJmHoe ypaene — 3T0 oaae wc1x1epeHuHpoBaHHe BHHH5J

HaxoKeHHe H3BHCHMO11 HepeMeHHoH

HpOH3BO,LHOH c1yHKuHH

4IH4xjepeHuHaJmHoe H3MBTC5J.TIHHeHHMM, ecJrn

HH3BCTH51 y B HBO11 CTHHH

Bce HOH3BOHM HeH3BeCTHOH c1YHKUHH B HepBOH CTHHHOHO J1HHF1HO OTHOCHTJThHO y H ee HOH3BOHMX

pemee 3IIHCMBTC5I B BHLe 5JBHOH YHKUHH

O6MKHOBeHHMM LH434epeHuHaJmHbIM H3MBTC5J KOTOO HOCTO

KOTOO CO.Lep)KHT TOJThKO H3BHCHMIO nepeeyio H HH3BCTH}O YHKUHIOB KOTOOM HH3BCTH5{ 4YHKuH5I 3BHCHT OT BX HMHHMX

B KOTOOM HH3BCTH5{ 4YHKuH5I 3BHCHT OT OLHOH nepeMeHHoH

4HCILO HOCTO5{HHMX B O6WeM HJHHH w1xepeHuHaJmHoro BHHH5I oHpe,LeJI5IeTc5{

HO5IKOM LHc1xePeHuHaJmHOrO BHHH51 cTapmell CTHHMO HeH3BeCTHOH 4YHKUHH

BHOM ilpaBoll qCTH

cTapmell CTHHMO He3aBHCHMOH HepeMeHHoH

4aCTHMM pemee Hc1Jc1JePeHuHaJmHOrO BHHH5I llepBoro HOp5ILKa H3MBTC5Ipemee HH y = x

pemeHHe, oiiyaioweeci 113 o6iuer’o pemeHu HH oHpe,TeJ1eHHoM 3HHHH HOCTO5fflHOFI

C

pemee HH y =

pemee B BHLe CTHOFO BX 4YHKL1HH

4IHc1Jc1JePeHuHaJmHMM HepBoro HO5LTK H3MBTC5IypaBHeHHe, B KOTOOM H3BHCHM5I llepeMeHHa5{ X B HepBOH CTHHHypaBHeHHe, B KOTOOM HH3BCTH51 4YHKL1H5{)’ B HepBOH CTHHH

ypaBHeHHe, KOTOO CO,Tep)KHT HpOH3BO,THyIO HeH3BecTHOH 4YHKL1HH TOJThKO HepBoro HO5IK

HBO1kI CTHH1kI

4IM4J4JePeHLWaJThHOe ypaene H3MBTC5J JTHHeHHMM ypaBHeHHeM HepBOFO nopn4Ka, eCJTHOHO.LIHHeHHO OTHOCHTJThHO X My

OHO.LIMHeMHO OTHOCMTJThHO X My’

— CBO4MTC51 K yBHHMflM C Pa34CJ11OWHMHCn HCCMCHHMMH OHO.TIMHCMHO OTHOCMTCJThHO y My’

eyHKuMSlf(x, y) SIBIISICTCSI O,RHOpO4HOM 4YHKUMCM CBOMX FMCHTOB k-ro nop5rnKa, ecTifi

f(tx,) = tkf(x,y)

y =?

y=kx

Cpe,LH.LHc1xepeHuHaJmHbIx ypaBHeHHH:

a) 2y’—xy2 = e_X; 6) y’+5xy = sin2x; B)yy’—y = e2X;)y’+=tgx

y

.TIHHeI1HMMFI w14x1epeHuHaJmHbIMH BHHH5IMH neporo nop5ILKa 5JBJ]5110TC51 BHHH5IB)

6)

B,F)

a,B)

YpaBHeHHe y’=f(x, y) HThIBTC5I oHopoHMM, ecii

f(x,y)= 0

cj?yHKuwIf(x, y) 5{BJ15{TC5J OLHOpOLHOH 4YHKUHeH CBOHX FMHTOB HJTBOFO nopIwca Bce B IIepBOH CTHHH

cl?yHKUWI f(x, y) 5{BJ15{TC5J OLHOpOLHOH 4YHKUHeH CBOHX FMHTOB HepBoro nop5I,TKa

113 wffj)cjepeHuHaJmHbIx ypaBHeHHH:

a)y’ +y =x; 6) y’—2y = cosx; B) y’+= sin2x r) y’—xy=e

y

He 5{BJ15ITC5J I[HHeHHbIM LH4xepeHuHaJmHMM eporo Hop5ILKa TO.JTbKO ypaee

a)

6)

B)

R)

113 H4J4JepeHuHaJmHMX ypaBHeHHH:

a)(y) —y=x 2 ;6)y

7

+xy=e;B

) xy —y 3 =sinx;r)y +xy=e

5{BJ15{TC5{ I[HHeHHJ,IM HepBoro Hop5{LKa

a)

6)

B)

113 HHMX LH4xepeHLu1aJThHbIx ypaBHeHnFT:

a) y’+3xy=cosx;6)xy’—y=x2y;B) y’—2xy=sin2x;r) 2xy’—y=xy2

51BI151TC51 l3epHyIEIIH ypaee

HOp51OK.L114x1epeHuHaJmHoro BHH115I

11Op51.LKOM HaliBbIcifiell HpoH3Bo,LHoH, BXOL51WeH BHoKa3aTeIleM CTeIIeHH H3B11C11MO11 HMHHOF1HOKa3aTe.TleM CTIIHH HeFI3BeCTHOH cIYHKUHH

HOp51.LKOM padlloJlodceHwI 11pOH3BO,LHOH

PemeHHeM,Lw1x1epeHuHaJmHoro BHHH5J y’ = f(x, y) H3MBTC51

— rno6a5l 4yHKuH5I

4yHKu1151 y = q(x), KOTO51 upil HOCTHOBK B 3T0 O6paWaeT ero B TO)KLeCTBO

— rno6a5l w14x1epeHuHpyeMa5I 4yHKuwI

— rno6a5l HHTerpFlpyeMa5I 4YHKUH5I

B JmHeiiHOM BHHHF1 y’ + p(x)y = q(x) IYHKUHH p(x), q(x) 5IBII5IIOTC5I

TOJThKO BO3PCT10WHMH HF13BCTHMM11 cI2YHKUH5JMH

F13BCTHMM11 cI?YHKUH5JMII He3aBHCHMOH nepeMeHHoH Xowia 113 4)yHKUIW H3BecTHa5I, Lpyra5{ HH3BCTH51

O6ee pemee Lw14ePeHuHaJmHOrO BHHH51 y” = f(x,y,y’) coepnrrOH HOH3BOJThH1O HOCTO51HH1O

emipe HOH3BOJThHM HOCTO51HHM

T11 HOH3BOJThHM HOCTO51HHM Be HO113BOJThHM HOCTO51HHM

113 wlcl)cl)epeHuHaJmHbIx ypaBHeHHl:

a) y y’ + 2x = e2X; 6) y” + — = sin 2x; B) y’ + 3xy2 = cos x; r) y’ — = xeX

I1HH11HMM 51BJ151TC51

CpeH 2w4J4epeHuHaJmHMx ypaBHeHHH:

a) y’+2xy2 =eX; 6)y2y’—2y = sinx; B)y’—cosx; r)y’+3xy = e2X

y

IIHHeHHMMH M4xfepeHuMaJThHMMH BHHFI5IMFI iieporo nop5ILKa 5JBJI5JIOTCSJ BHHFI5I

a,B)

—6,B)

a)

r)

Ho.Lw1x1epeHuMaJmHoro BHHI45I HOHHMTC5I HaxoKeHMe FlHTerpa.rla OT llpaBoI4 qaCTil BHCHH5I

pemee,Lw1x1epeHunaJmHoro BHCHH5IHaxoKeHMe FiRTerpaJia OT (j)YHKUIIII yHaxoKeHMe FiRTerpaJia OT nepeMeHHon X

Cpe,LM wffl4epeHunaJmHbIx ypaBHeHnH:

a)xy’ + 3y = 2x2 6)yy’ — 2x = e3X; B)y’ — = xsin x; r)y’ + 3xy3 = tgx

IH4HF1HMM 51BJ151TC5J ypaee

a)

—6)

B)

r)

O6wee pemee BHHH5I y’ — y = 0 HMT BH

1

x+C

—y=Cx

—y=e x+c

Cx

Ecilil y(O) = 1, TO ‘-IaCTHOC pemee BHCHH5I y’ + y = y = eX_l	0 HMCT BH
y = e_X
—y=e x+1 —y=e 2x
YpaBReRMe l3epHyJU[H HMCCT	BHL

= f(x,y)

y’+p(x)y=q(x)

+ a1y’ + a2y = f(x)y’+p(x)y=q(x)y

YpaBReHile l3epHyJl.T1H 5{BJI5{CTC5I JIHHCHHMM BHCHHCM llpFI

—n=2

n=O

O6ee pemel-me BHHFT5I xy’ — in X =0 nee BII

y=ln2x+C

y=(2inx—1)+C

in2 x +c

y=ln(Cx)

YpaBReHile y’ + p(x)y = q(x)y’1 H3MBTC5J

IH4HF1HMM

I1F1HF1HMM eporo nopiwca fl-FO HOP5LTK

13epnyuin

4IwjcjepeHuHaJmHoe ypaee y’ + p(x)y = q(x) HamiBaeTci

13epnyuin

OHOOHbIM

.LWIHeHHMM eporo nopiwca

C P3CJ15IIOWHMHC5{ HMHHMMH

O6wee pemee BHCHH5{ l3epHyJiTIH y’ + p(x)y = q(x)y’1 coeprn

n HOF13BOJThHMX HOCTO5{HHMX Be HOH3BOJThHM HOCTO5IHHMC

6ecKoHeHoe ‘-IFICJIO HOH3BOJThHMX HOCTO5{HHMX

OH HOH3BOJThHIO HOCTO5IHHIO

HOp5IKOM wff1Jc1JepeHuHaJmHoro BHCHH51 H3MBTC5I capmai cenen HCH3BCCTHOH 4YHKUHH

HOp5I,LOK HaHBMCIIIeH npow3BoLHoH, BXOL5{weH B BHCHHC

capma cenen HC3aBHCHMOH nepeeno x HO5IOK HailMenbifiell npow3BoLHoH, BXOWIWeH B

Haamnoe yciornie Lw134epeHuHaJmHoro BHCHH51 y’ = f(x, y)6yeT 3aaHo, CCYIH BBHHHF1

H3BCTHO OHO 113 peuien H3BCTHO o6ee pemee

H3BCTHO aene 4YHKuHH y npn x =

npaa acm HOCTO5{HH

HaaJmHoe yciioBlle y(x) = B,LwjxjepeHUMaJmHOM BHHH11 y’ = f(x,y) 3a,LaeTc5I ,LII[5I

opeeiiewi o6ero pemeHi

Дата добавления: 2015-10-30