Diferansiyel Denklemler - Doç.dr. Tahsin Engin Ders Notları

8/10/2019 Diferansiyel Denklemler - Do.dr. Tahsin Engin Ders Notlar

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www.ffjafrsebtdiro.gbj

JhfeaosdfroeA@F^IEW_FDAFEMDFJDF^

Ab.Ar.QihsoeFeloe

Fydd>338


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>.9.L^>.>.NÊG JF^QFNFA@FÎEW_FDAFEMDFJDF^FLFEFDNIMO>.7.NÊG JF^QFNFDEFFÂ@FÎEW_FDAFEMDFJDF

>.1DEFF^BDJI_IENÊG JF^QFNFAFEA@FÎEW_FDAFEMDFJDF >.1.9.AfomfedfroefIyrdinodorQoptfNoroegoJfrtfnfafeAfemdfjdfr>.1.>.HbjbkfeQoptfNoroegoJfrtfnfafeAfemdfjdfr>.1.7.QijAo`friesoyfdAfemdfjdfr

>.1.1.NizzfdQopAo`friesoyfdAfemdfjdfr>.6.NÊG JF^QFNFAFEAFEMDFJDF EWWQFJIQM_IMDIOJ>.0.JHFEAWDMV_LVDIJIDIÔ>..DEFF^NIOJWO[DOMPF\^BEWMOIE@BEMW_BEDIÔ7.7.HBJBKFEAFEMDFJDF QFB^W7.1.WINQMIQI_ODOHBJBKFEAFEMDFJDF 7.6.HBJBKFEBDJI_IEAFEMDFJDF QFB^W7.0.HBJBKFEBDJI_IEAFEMDFJDF^=NFD^W[MIQWI_ODI _EQFJ7.. DEFF^ A@FÎEW_FD AFEMDFJ WWQFJDF EE FDJEIW_BE _EQFJ DF[J6.7.[AFF^_EQFJ DF[J6.1.JIQ^W_EQFJ DF[J6.6.JHFEAWDMV_LVDIJIDIÔ


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NDJ6DISDIGFAEJ

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0.1.QF^WDISDIGFAEJ0.6.A@F^IEW_FDAFEMDFJDF EDISDIGFAEJDF[J0.0.MBEPBDW_BEQFB^FJ0.


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9. NDJ

NÊG JF^QFNFAFEA@FÎEW_FDAFEMDFJDF^

9.9.L^

Ao`friesoyfd afemdfjdfr uzue yddirar, aeyiai bu òzomsfd nodojdfr vf jhfeaosdom

aiddireai efjdo nor yfr tutjimtiar. Nodoj iaijdir vf jhfeaosdfr lfefddomdf afoojf

uriyie sostfjdfro oegfdfrdfr vf ao`friesoyfd afemdfjdfr jhfeaosdfrf nor sostfjafmoieihtir afomfedfroe afoojoeo oegfdfjf vf òzomsfd bdiy aihi oyo iediji bdiei lftoror.

Nodoj vf jitfjitom rfegodfroef yefdom jitfjitom rftojo uzue srfaor nodoj vf

jhfeaosdom ìmdtfdfro iriseai nor iedijizdm mbeusu bdjutur. Nodoj vf jhfeaosdom

ìmdtfdfro rfegodfroe tfbrom jitfjitomdf iridiree bm ai oyo bdjia mbeusueaihfjòmordfraor. [ori nu tirz rftoj rfegodfroe prbndfj zjf nfgfrodfroeo

lfdotorfnodjfdfroef yirajg bdijijimtiar. Nu uyludiji bu eovfrsotfaf nu afrstfe

niirszdm briee % 63dfrf miair mirjtr. Nu osf abi nodojdfro vf jhfeaosdomndjdfroooeefjdonormiyptr.Nutirtjivfiedijizdmlfefddomdfjitfjitomrfeojo

lrfe rfegodfrdf abi nodojdfro vf jhfeaosdom rfeojo lrfe rfegodfrf ìrmd ofromdo

afrsdfroe bdujisydi sbeudiejtr. Nydfgf abi nodojdfro vf jhfeaosdom ìmdtfdfrorfegodfroef mfeao aosopdoedfro ofrosoeaf mirditmdir prbndfjdfro zfnodjfdfro ooe

jitfjitomafrsdfroeomfeaodfrovfrjfybdueunfeojsfjodfraor.

Ao`friesoyfd afemdfjdfro gfnorsfd afemdfjdfrafe iyrie fe efjdo zfddom `bemsoybe

trfvdfro ofrjfdfroaor. Ao`friesoyfd afemdfjdfroe oegfdfejfso oyo nor jitfjitom idtyipslfrfmtoror vf abdiysydi rfegodfroe nu afrsf nidijiaie egf nijd vf nijsz

afomfe,srfmdovfsrfmsoz`bemsoybe,iaovfmsjotrfvdfr,ìrmdirvfirtrjdirodfoetflrid

lonotfjfdmbeudirlzafelforjfdfromfsoedomdfefrodor.

9.>.NI[OQFJFDQIEOJDIJIDI

Nor yi ai aihi ìzdi `bemsoybeue trfvdfroeo ofrfe afemdfjdfrf ao`friesoyfd afemdfjaoybruz.Aofrnoroìafydfao`friesoyfdafemdfjnortimj`bemsoybedirodfnuediretrfvdfroiriseaimoodomoyotfjsodfafr.Numivrijodmbdirim90


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faodjfmtfaor.Abdiysydiao`friesoyfdafemdfjoefdaffaodjfsoprbndfjhimmeaiyftfrdonodlosihono bdjiy, prbndfjf aihod bdie afomfedfronfdordfyfnodjfyo,uylue nisotdftorjfdfr vf

virsiyjdiryipinodjfyo,muddiedigimòzomsfdprfesopvfmieuedirnodjfyovfafaommitdonor

ieidozyipinodjfyolfrfmtoror.Iiainizrefmdfrvfrodjotor.

refm99Efwtbeuehirfmftyisis

Efwtbeue omoego mieueueu muddieirim az nor ozlo nbyuegi @ muvvftoeoe ftmoso idteai

hirfmft fafej mtdfdo nor gosjoe mbeujueu s tiejdiyie ao`friesoyfd afemdfjo fdaf

faoeoz.

zjAoeijomafrsdfrojozafehzvfovjftiejdiree

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bdirim vfrodaooeo nodoybruz. Efwtbeue omoego mieueuMuvvft:Mtdf vjf fmdoeaf oìaffaodaooeafe

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: ao`friesoyfdafemdfjofdaffaodor.

refm9>EfwtbeuesbujimieueuNidielti nfdordo sgimdisihopmrfsfd jftidnor gosoj sgimdQ3 bdie sgim su ofrosoef

nrimdybr. Gosjoe nidiel sgimd su sgimdeaie am osf, gosjf s tries`fronidiyigi nodoejfmtfaor. Nuei lrf gosjoe hfrhielo nor t ieeai sgimde Q(t) vfrfe

ao`friesoyfdafemdfjonfdordfyoeoz.

zjWuyue nudueauu min jmfjjfd fmodaf yidtdj aefdoj (fvrfyf s miyn ybm) vf

nueilrffefrkoeoembrueujuprfesonoeouyludiyidj.Gosojsgimsuyinrimdamtietsrfsberigosjoefefrkosoeafmo irt,gosjoe yzfyoeafegosjftiejdi(mbevfmsoybe)lffe s

fefrkosomiairbdigimtr.Nueilrf

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[ijieaodojoeosbesuzmmidajzai(dojotaurujueai,yieot3)

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ao`friesoyfdafemdfjofdaffaodor.Nuafemdfjmrfsfdgosjoesgimdezijiee`bemsoybeu

bdirim oìaf ftjfmtfaor. Ao`friesoyfd afemdfjdfr òzomsfd bdiy, nijsz afomfe(dfr)oe

nfdordo nor iridmtimo affrdfro ooe tiejdiyinodor. refoe nu afemdfj mrfsfd gosjoe

jfrmfzoeafe yzfyoef miair bdie sgimdm afoojoeo tiejdir, nu serdire aeai

lffrsozaor. Iyrgi afemdfj, gosjoe sgim suyi aidarda ieaie otonirfe (t:3) sgimde

vfrorvfabdiysydifdaffaodfgfmzj3tirideailffrdobdigimtr.


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Nor yi ai aihi ìzdi nijd afomfeoe tfm nor afomfef lrf iao trfvdfroeo ofrfeao`friesoyfd afemdfjdfrfIaoAo`friesoyfdAfemdfj (IAA) afeor. Nueue yieeai ofrosoeaf

noryiaiaihiìzdinijdafomfeoe,noryiaiaihibmnijszafomfeflrftrfvdfro

nudueie afemdfjf osfMsjoAo`friesoyfdAfemdfj (MAA) aoyfgfoz. Nu afrstfIAA mbeusu

zfroeaf aurudigimbdupMAAmbeusuaihi bm dosiesstazfydfraffdf idejimtiar. Iao

norao`friesoyfdafemdfjfrefmbdirimGbtx>xfy1yx7y x> +:+

vfrodfnodor.

Nor ao`friesoyfd afemdfjaf fe ymsfm jfrtfnfdo trfvoe jfrtfnfso ao`friesoyfd afemdfjoe

jfrtfnfsoeo vfror. refoe sttfmo afemdfj 7. jfrtfnfafeaor afeor. Nueue yieeai,

jfrtfnfydfsmimirtrdienormivrijbdieafrfgfyfafafoejfmlfrfmor.Norao`friesoyfd

afemdfjafnudueiefeymsfmjfrtfnfdotrfvoessef,nuao`friesoyfdafemdfjoeafrfgfsoafefgfmtor.

Nor ao`friesoyfd afemdfjafmo nijd afomfe vf tj trfvdfro noroego afrfgfafe osf,

ao`friesoyfd afemdfjf doeffr ao`friesoyfd afemdfj afeor. Abdiysydi ofrosoeafy>7 f,ysoe,yy,yy,)y(,y lono tfrojdfr nudueie afemdfjdfr doeffr afodaor. Nueue

yieeaiafemdfj xde,f,xsoe,yx,x7

xsoe> treafeoìafdfrofrfnodor.

Aihilfefdnoroìafydfffrnorao`friesoyfdafemdfj

)x(^y)x(`...y)x(`y)x(`y e)>e(

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9)e( :++++

`brjueai oìaf faodfnodoybrsi afemdfjf doeffraor aoyfgfoz, imso hidaf doeffr bdjiyie nor

ao`friesoyfd afemdfj sz mbeusuaur. Nu afemdfjaf ffr 3)x(^ : osf doeffr ao`friesoyfdafemdfjhbjbkfeaor.Imsoaurujaiafemdfjhbjbkfebdjiyieao`friesoyfdafemdfjiaeidr.

refm97Ao`friesoyfdafemdfjdfroese`dieardjisIiaimoao`friesoyfdafemdfjdfrose`diearez.

zj(9) 3y7y :+ (>.jfrtfnfdoeffrhbjbkfe)

(>) xsoex>y7y +:+ (>.jfrtfnfdoeffrhbjbkfeafod)

(7) 3yy7y :+ (>.jfrtfnfdoeffrafod)

(1) x>fygbsyxsoey :++ (7.jfrtfnfdoeffrafodhbjbkfeafod)

Ao`friesoyfd afemdfjdfr nijd afomfe vf trfvdfroeoe mitsiydiree aurujuei lrf af

se`dieardjimtiar. Ffr nu mitsiydir norfr sinotsf afemdfj sinot mitsiyd ao`friesoyfdafemdfj, ffr nijsz afomfef nid `bemsoybedir osf afomfe mitsiyd ao`friesoyfd

afemdfj iae idr. refoe xsoey>y :+ afemdfjo sinot mitsiyd, xzxzxgbsh > :+ osfafomfemitsiydnorao`friesoyfdafemdfjaor.

9.7.A@FÎEW_FDAFEMDFJDF E[JDF


7/90

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Nor prbndfj ooe ao`friesoyfd afemdfjoe fdaf faodjfso lfefddomdf mbdiyar. Aofr yieaie nuafemdfjoezjeenuduejis osf lfefddomdfzbraur.Nor ao`friesoyfd afemdfjoezj

nizfe nor yi ai normi afì oetflrid idji odfjoeafe onirft bdinodsf af nu tr aurujdir

lfefddomdf ostoseiar. Qj ao`friesoyfd afemdfj topdfroef uyludieinodfe lfefd nor zj

yetfjo ef yizm mo jfvgut afodaor. fotdo se`diri iyrdie ao`friesoyfd afemdfjdfr ooe

nuedirizlzjjftbtdir

lfdotorodjotor.Nizfenorao`friesoyfdafemdfjozjfmnorafeìzditfmeooenfrinfrmuddiedjiseeyie srinutfmeomdfrafmoyftfrdonoruzjiedmazfyo

vf hefr lfrfmtoror. Niz ao`friesoyfd afemdfjdfr siafgf ustigi yipdj nor timj

jieopdisybedirdi zdfnodormfenizdireeieidotomzjdfroojmieszbdinodor.Abdiysydi

norao`friesoyfdafemdfjozjfmnodojafezoyiafnorsieitaidlonoaor.

Gfnorsfd afemdfjdfroe zjeaf lfefddomdf 393x : treafe nor afemdfjosidiyie iyrm affrdfroe (mmdfroe) nuduejis hfaf`dfeor. tf yieaie nor ao`friesoyfd

afemdfjo zfrmfe, nfdordo nor iridmti afemdfjo sidiyie `bemsoybedir irier. _umiraimo

gfnorsfd afemdfjo sidiyie affrdfr > vf 6 tor. Bysi 3yy : ao`friesoyfd afemdfjoeoe nor zjbdauueulstfroeoz.

zjPfrodfezjao`friesoyfdafemdfjosidijidar.

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f>y

fyideirim ao`friesoyfd afemdfjaf yizdrsi 3f>f> x>x> : bdur vf afemdfj

sidier.

refm96Norao`friesoyfdafemdfjoezj>

x9y : oìafsoeoe 3xyy :+ ao`friesoyfd afemdfjoeoe nor zj bdauueulstfroeoz.Pfrodfezjehfrxaffroooelffrdobdupbdjiaeorafdfyoeoz.

zj

Pfrodfe zj ao`friesoyfd afemdfjaf yizdrsi >x9 3xx9>

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fdaf faodor,

abdiysydi vfrodfe zj afemdfjo sidijimtiar. zje tiej irid ooe 3x9 > bdjis lfrfmtoo imtr. Nuriaie (9 )(9 ) 3 9 9x x x vf x + yizdirim 9x9 fdaf

faodor.


8/90

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refm90Norao`friesoyfdafemdfjoelfefdzj

7x>Gxfy x> +: oìafsoeoe >3x8y1y1y :+ ao`friesoyfdafemdfjoeoe,Gsinotoeoehfrhielonoraffroooe,zjbdauueulstfroeoz.

zj

7x>Gxfy x> +: , >Gxf>Gf>)xf>f(Gy x>x>x>x> ++:++: vfx>x>x>x>x> Gxf1Gf1)xf>f(G>Gf>y +:++: nudueupafemdfjafyfroefyizirsim,

:+ y1y1y + )Gxf1Gf1( x>x> )>Gxf>Gf(1 x>x> ++ >3x8)7x>Gxf(1 x> :++fdaffaodor. Abdiysydivfrodfezjlfefdzjar.

ojaotfmrirnorsunieybsueiaidardiemrfsfdgosojrefojozfaefdoj(refm9>).Fdaf

faodfeao`friesoyfdafemdfjoezjeafegosjoesgimdeezijiediafoojo,

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bdirim fdaf faodor. Nu oìafafG nor mfyò sinottor. Mbdiydmdi lstfrodfnodor mo nu zjGeoeida affrafe nijsz bdirim ao`friesoyfd afemdfjo sidir. Abdiysydi Geoe idinodfgfo

sbesuz siyai affrf mirdm sbesuz siyai zfd zj fdaf ftjf ojmie virar. Lfefd

zj ofrosoeaf gosjoe nidiel sgimd (t:3 ieeai) bdieQo nuduejiaeaie nu sbeu

srproz afodaor. Abdiysydi nidiel sgimd nfdortodaooeaf, vfrodfe ao`friesoyfd afemdfjfbdie zj af zfd nor zj mojdoo mizier. Nozoj odlodfeaoojoz zj af,Q fmsfeoeoQo

sgimdeai mfsfe zfd zj bdigimtr. Nuriaie u efjdo sbeu mjimtiar; Nfdordo nor

prbndfjoe tfm nor zj bdsi ai, nu prbndfjo tfjsod fafe ao`friesoyfd afemdfjdfroe

sbesuz siyai zjf sihop bdjidir jjmear. Nueue efafeo, ao`friesoyfd afemdfjoe,nijd afomfedfrdf nijsz afomfedfrafmo afoojdfr iriseaimo nor odomo bdjiseaie

nimi nor fy bdjijisar. Ao`friesoyfd afemdfj, nor `bemsoybe vfyi trfvdfroeoe nfdordo

nijsz afomfe affrdfroef mirdm lfdfe nijsz afomfe affrdfro mbeusueai nodloofrjfz. Wbeu bdirim iye òzomsfd bdiydi odlodo pfm bm ìrmd prbndfj iye ao`friesoyfd

afemdfjdf oìaf faodor. @irmddm osf fdaf faodfe lfefd zjafe nozoj odlodfeaoojoz

prbndfjoe zfd zjef lffnodjfjozo sidiyie zfd irtdire tiejdiejisar. Ffr nu

irtdir nijsz afomfeoe iye affro ooe vfrodjosf nu irtdiri nidiel irtdir,nijszafomfeoenorafeìzdiaffroooenfdordfejosfnuirtdiriserirtdiraoyfgfoz.

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9/90

-? -


lat

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ao`friesoyfdafemdfjoydfnfdordoaor.Gosojodmhzszbdirimnrimdaeaie

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aztP yizdinodor. Nor aofr irt osf gosjoe nidielti h ymsfmdooeaf

nuduejisirtar.Aofrnoroìafydf,

htz :: )3(

Hfromoirtainijszafomfeoe(t)iyeaffroeafvfrodaooooenuprbndfjnornidielaffrprbndfjobdjimtiar.

Norao`friesoyfdafemdfjozjfaf,afemdfjosidiyiey:y(x)`bemsoybeueuenuduejisirzu

faodor. Iegim bu zijie nu jjme bdjiz vf yimdim zj tfmeomdfro zj ooe tfm

idtfreito`midr.Wiysidyetfjdfrnuefafedfbrtiyimj yimdimzjybddirar.Mipid

zjdfroe fdaf faodaoo ieidotom yetfjdfrdf ni

js

z afomfeoe sbesuz affroef mir

d

msbesuz siyai nijd afomfe affro hfsipdijim jjmear. _ieo ieidotom z`bemsoybeu, vfrodfe zj irideai, srfmdo nor `bemsoybeaur. Aofr tiri`tie siysid

yetfjdfr,iegimnijszafomfeoeaihiegfafetiejdiej affrdfroefmirdmlfdfe

nijdafomfeaffrdfroeoyimdimbdirimvfrjfmtfaor.

9.1.A@FÎEW_FDAFEMDFJDF EAB^VAIEEQFLÎD_BDV_DI[JDF

Niz ao`friesoyfd afemdfjdfr doeffraor vf trfvdfro ofrfe tfm nor tfrojf sihop bdup,

nodoejfyfe(irieie)`bemsoybeuenorirpiebdauutfrojdfroofrjfzdfr.Ffroetflridodfjo

yipdinodoybrsi, ao`friesoyfd afemdfj af abruaie oetflriddfjf tfmeooydf zdfnodor

afjfmtor. Nueue yie

eai aofr niz

ao`friesoyfd afemdfj trdfro doeffr bdjiyie tfrojdfrfsihoptorvfnuybddizdjfdfrojjmeafodaor.

: 3fxy x0> abruaieoetflridybduydizdfnodor.

:+ 3fxxy7y x0> abruaie oetflrid ybduydi zdfjfz, em nodoejfyfe`bemsoybeuenorirpieaurujueaiebdauu7xytfrojoabruaieoetflriddfefjfz.

Norao`friesoyfdafemdfjabruaie oetflridybduydizdrmfetfrojtfroj oetflrffaodorvf

noroetflridsinotofmdfeor.Hfroetflrisybeiajeaitrfvdfroejfrtfnfdfronorardrvfnuei mirdm nor nimi oetflrid sinoto fmdfeor. Abdiysydi nor ao`friesoyfd afemdfjoe lfefd

zjeaf,ao`friesoyfdafemdfjafnudueiefeymsfmjfrtfnfdotrfvoejfrtfnfsomiair

mfyòsinotfdaffaodor.

refm98AbruaieoetflrisybeodfzjIiaimoao`friesoyfdafemdfjdfroeabruaie oetflridybduydizdpzdfjfyfgfmdfroeo

nfdortoeozvfzdfnodorbdiedirzez.

(9) 37y6y :+

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(7) 31yy> :


10/90

-93 -


zj(9)

Nu afemdfj, omoego tfrojoeoe nodoejfyfe `bemsoybeu (yieo nijd afomfeo)

ofrjfsoeafeabdiyzdfjfz.

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Nu afemdfj doeffraor vf trfvdo tfm tfrojo nuduejimti vf aofr tfrojdfraf

nodoejfyfe `bemsoybe y nor irpie vfyi ìmtr aurujueai afodaor. Nu efafedf afemdfj

zdfnodor.Afemdfj>.jfrtfnfafebdauueaieirtirai>mfzoetflridid

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Afemdfjafyfroefmbeursivfhfromotiriàxodfirpdrsi

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aihioetflrididjimsurftoydf axGaxx>ay 9> : vfyi >91 GxGx>

9y +: yizdirim

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Nuafemdfjdoeffrafodaorvfabruaieoetflridybduydizdfjfzlonolreybr.

Iegim aommitdo nor mbetrbd odf yy> tfrojoeoe >y eoe trfvoeafe nimi nor fybdjiaimilrdjfmtfaor.Bhidafafemdfjufmodafazfedfefnodor.

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a > : .Afemdfjoehfromotiriàxodfirpdirim 3ax1)y(a > : yizdpoetflrididersi 9

> Gx1y : vfyi 9Gx1y +: lfefdzjfdaffaodor.

refm9?Wfrnfstajfhirfmfto933jymsfmdomtfenrimdienorgosjoe7sieoyfsberimoyfrafeymsfmdooeovfhze

nfdordfyoeoz.

zj[fjoeo z:3 minud fafdoj vf gosjoe nrimda ymsfmdoo af z:933j ididj. Hivi aorfego

ohjid faodaooeaf gosjoe nu sfrnfst ajf hirfmftoeoe lz : ao`friesoyfd afemdfjoydf

tiejd bdauueu nodoybruz. Irt irai omo mfz oetflrid idirim 9Glt)t(Pz +:: vf

>9>

GtGlt>

9)t(z ++: fdaffafroz.Nuzjdfrsfrnfstajfhirfmftoyipiehfrgosoj

ooeiyear. Aommit faodorsf nor ao`friesoyfdafemdfjoelfefd zj,nodoejfyfe`bemsoybe

odf nijsz afomfe iriseai nor odomoaor vf mfsoedomdf nor nijd afomfef iot nor trfv

tfrojoofrjfz.

Lfefd zje>iaftmfyòsinotonuduejimtiarvfnuedirnodjfmsozoegosjoe7ssberimo

mbeuju ooe nor hfsipdiji yipdijiz. Nu srproz nor auruj afodaor em nu mbeuj

gosjoe itda ymsfmdof vf gosjoe odm hzei nid bdirim afofnodor. Bysi lfefd zjnuedirmbeusueaihonornodlovfrjfz.Aihiegfafnfdortodaoolononorao`friesoyfdafemdfj

nijdvfnijszafomfedfrafmoafoojdfririseaimoodomoyotiejdirvfprbndfjfzl

vfrodfe nijsz afomfe affroef mirdm lfdfe nijd afomfe affrdfroeafe

ftmodfejfz.refoe933jyfroef>63jaedsfyao,fdaffafgfojozlfefdzjyoef

iye midigimt. Iegim abid bdirim gosjoe 7 s sberimo mbeuju nu aurujai ìrmd nor yfraf

bdigimt. Lfefd zj afemdfjoeo nozoj prbndfjojozf uylue hidf lftorfnodjfm ooe

ofrosoeafmosinotdfroeprbndfjafvfrodfembuddirilrfnfdordfejfsolfrfmor.Numbuddir


11/90

-99 -


3at

az)3(P

3t

:::

933)3(z :

bdirim vfrodjotor. Noroego mbudujuzu hz afemdfjoeaf yizidj; 9G3l3)3(P +:: ,

.3G9: moego mbudu ai mbeuj afemdfjoeaf yfroef mbyidj;

>9 G3G3l>

9933)3(z ++:: , 933G>: . Abdiysydi irieie zfd zj o`iafso

933lt>

9)t(zz

> +:: bdigimtr. Hz o`iafso osf G9:3 bdauueaie lt)t(P : bdur. t:7

sieoyfsberimoaffrdfrhfsipdiersi,

86.66933)7)(89.?(>

9)s7(z

> :+: j vf 17.>?)7)(89.?()s7(P :: j/s (ii

yed).


12/90

-9> -


Ndj9odfodlodoprbndfjdfr

@ozomsfdprbndfjdfrojbafddfjfafgfnorsfdafemdfjdfrefafeyftfrsozmidr,ao`friesoyfd

afemdfjdfrfefafeohtoyiauyudjutur.

Winot hzdi (P3) ajfmtf bdie nor pirite hirfmftoeo tiejdiyie ao`friesoyfd

afemdfjoeoEfwtbeueomoegomieueueu( ij@

: )muddieirimfdaffaoeoz.Mtdfsoj bdie nor ti yfrafe yumiryi abruP3 afy hzydi itdjimtiar. Efwtbeue

hirfmftmieueueilrfgosjoeyfrafeymsfmdooeovfhzezijiee`bemsoybeubdirimfdaf

fafnodfgfoeozao`friesoyfdafemdfjoyizez.

_iydir lfefddomdf af`brjisybe jomtirdirydi brietd vf ye srfmdo bdirim afelf

mbeujuei abru bdie nor muvvft bdutururdir. refoe x miair lfrodjo nor yiye

bduturigi muvvft@:mx odf vfrodor vfnuriaimyiy mitsiys iae idr. _iy mitsiysmbdie

nydf nor yiy uguei, yiy afelf mbeujueaiymfe (x:3)j mtdfdo nor gosoj nidiejimti vf

mtdf yiye uguei nid bdirim yfrfmojo vf yiy muvvftoeoe ntedfom ftmosoydf sidejyipjiyi tfrm faodjfmtfaor. Efwtbeue hirfmft mieueuei lrf mtdfeoe mbeujueu,

nidiel mbeujuei lrf (x:3), zijiee `bemsoybeu bdirim vfrfe ao`friesoyfd afemdfjo

yizez.Sdtbeyuj, îayuj vf G91 lono Mirnbe ozbtbpdiree, nimi nor fdfjfet vfyi iye

fdfjfetoe nimi nor ozbtbpueu bduturjim zfrf tinoo bdirim nbzueauu nodoejfmtfaor.

Nbzueji hz hfez nbzuejij jomtir odf abru brietd bdirim afojfmtfaor. Nor

riaybimto` jidzfjfeoe hfrhielo nor t ieeaimo jomtireJ(t) idirim, nu mtdfeoe zijiediafoojoeovfrfeao`friesoyfdafemdfjofdaffaoeoz.Ebt;Norafomfeoe(refoeI)niminorafomfedf(refoeN) abru brietdbdjisjitfjitomsfd bdirimI:mN bdirim oìaf faodor.

Nuriaimbrietmitsiysiaeidr.

>.9. Nijd vf nijsz afomfe efaor, nor `bemsoybeai nuedir nornoroeafe eisd iyrt

fafrsoeoz.>.>.Qrfvoelfbjftromiedijeimdiyez.

>.7.Iaovfmsjoao`friesoyfdafemdfjdfririseaiefìrmvirar.>.1.x:6ebmtiseaimotfftoxfmsfeoefpiridfdbdienor`(x)`bemsoybeueuenuebmtiaimo

noroegotrfvombeusueaiefsydfyfnodorsoeoz.>.6. Nor ao`friesoyfd afemdfjoe jfrtfnfso odf afrfgfsoeo nfdortoeoz. 7. jfrtfnfafe , >.

afrfgfafenorao`friesoyfdafemdfjrefovfroeoz.

>.0.Iiaimo`bemsoybediretiejiridmdirenfdordfyoeoz.

(i) >x+ (n) xde)9x( (g) x/9 (a) >x/xgbs (f))9x(x

f x>

>.719 f>x7soex>71> fttx7soex :

(g) )txde(` >>7 : , >7

>7

t

`2,

x

`

:

= (a) xsoef)9x(` x71 : , 2ax

a`1 :

(f)7

x

6

x9

xtietxf`

: , 2

t

`2,

x

`>

6>

6 :

:

= (`) xxsoe` >0: , 2ax

a`0 :

(l) xde>7gbsx` : , 2

t

`8 :

>.8.

Iiaimooetflridodfjdfroeoyipez.

(i) ( ) ++ axx6soefx x7> (n)

+6

>

ax

x

6x7de (g) ( )axxt7t>soex >t> ++


13/90

-97 -


(a) ( ) ++ axxtgbsf7)x(y tx> (f) ( ) +x

7

axx>det)x(y (`)( ) )9x(>x

ax7

(l) >x>ax

(h) ax9x

t>> + (o) xaxde (k) xaxsoex

(m) xaxsfgxtie (d) axxdex 7> (j) + )>x)(>x)(9x(

xax>

(e) xaxdex>.?.

Iiaimomivrijdir,hfrnoroefrefmdfrvfrfrfm,imdiyez.

(i) Iaoao`friesoyfdafemdfj,(n)Msjoao`friesoyfdafemdfj,(g)afrfgf,jfrtfnf(a)

Hbjbkfeao`friesoyfdafemdfj,(f)Winotmitsiydao`friesoyfdafemdfj

(`)

Nidielaffrprbndfjo,(l) Weraffrprbndfjo, (h)Werirtdir

(o) Doeffrao`friesoyfdafemdfj,(k)Doeffrbdjiyieao`friesoyfdafemdfj,(m)Lfefd

zj

(d)zfdzj,(j)Qfmodzj

>.93.

Iiaimo ao`friesoyfd afemdfjdfroe jfrtfnfdfroeo, doeffr bdup bdjiamdire vf

sinot/afomfemitsiydbdaumdirenfdortoeoz.

(i) 7x xxy6yx>yx,3yf>y,3yyx7y,x8y7y :++:+:+:+

(n)

3y6yyxf>yy,xgbtfy>y,3fy,9xsoey>y xxyx> :+:+:++:+

(g) 9xdexzsoexsoezxz :+

7 Pfrodfe `bemsoybedire yieeaimo ao`friesoyfd afemdfjoe nor zj bdauueu

lstfroeoz.

(i) 9x>yvfx6y,3y >9 +::: (n) x>

>x>

9 f7yvffy,3y1y :::

(g) 1>xde

9>

xyvffy,3y1yx>yx :::

(a) xgbshyvffy,fy,3yy 7x

>x

9 ::::

(f) ( )x>gbsx>soef7yvfx>soefy,3y7y>y x>x9 +:::+(`) x>xde7

x>>

x>9 fxf6yvfxfy,fy,3y1y1y

::::+

1.9.Eftrao`friesoyfdafemdfjdfrabruaieoetflridybduydizjffdvfrodoaor.

1.>. eg jfrtfnfafe doeffr vf hbjbkfe nor ao`friesoyfd afemdfj yizez. Nu

afemdfjoe zjeafe mi tief mfy`o sinot fdaf faodor. Nu mfy`o sinotdfro nfdordo nor

prbndfjooenudjimostfrsfeoz,miiaftmbudnfdortjfdosoeoz.

1.7. Iiaimo ao`friesoyfd afemdfjdfroe abruaie oetflrid ybduydi zdpzdfjfyfgfmdfroeooegfdfyoeoz.zdfnodfgfmaurujaibdiedirzez.

(i) 3xfyf,3x7soeyy>,3yy,3y x7x :+:+:+:

(n) 3xyy,3x8yy1,3xf1y 7x1 :::

(g) 3x8yyx,3xgbsfy,3yy,3y6y 1y ::::

1.7. ^:3.> j ipeai mrfsfd fmdf sihop riaybimto` nor jiaaf ofrosoeaf l3:193


14/90

-91 -


3m

l

ar

aQr

ar

a

r

9 3>>

:+

ao`friesoyfdafemdfjoydftiejdier.Nuafemdfjoedoeffrbdupbdjiae,sinotmitsiyd

j vfyi afomfe mitsiyd bdauueu, jfrtfnf vf afrfgfsoeo, hbjbkfe bdup bdjiae

nfdordfyoeoz.Afemdfjzjeafelfdfgfmmiiaftsinotjfvguttur,nuedirnudinodjfm

ooe hielo mbuddir efrorsoeoz. Nu afemdfjo zfrfm mrfsfd gosoj ofrosoeafmo sgimdmaidjyiripe`bemsoybeubdirim(Q(r))fdaffaoeoz.Fdaffttooeoe o`iafyonorsgimdmyirip(Qr)frosoeaflstfroeoz.

1.1. Mided D:3.6 bdie lfeo nor auvir lz eef ididj. Auvire sbd yz (x:3)jmfjjfd fmodaf yidtdj bdup aofr yz (x:D) eo`brj bdirim 73 bG sgimdmtiar.

Auvir ofrosoeaf x3>.33fl)x(l : o`iafsoef lrf s rftodjfmtfaor. Auvir ofrsoeaf

sgimdesiafgfxabrudtusueaiafotooeo(Q:Q(x))vfnuafoojoe 3m

)x(l

ax

Qa>

>

:+

ao`friesoyfd afemdfjo uyiregi bdauu nodoeaooef lrf, m:96 \/jM, l3:9633 \/j7

idirim auviraimo sgimdm aidje, yieo Q:Q(x) `bemsoybeueu vf yidtdj yzfyafmosgimdhfsipdiyez.(Ebt;_idtdj yzfyafaQ/ax:3ider.).


15/90

-96 -


>. NDJ

NÊG JF^QFNFAFEDEFFÂ@FÎEW_FDAFEMDFJDF^

>.9.L^

Sfmbmuyludijiainornymdeafoojhz(noroegotrfv),nunymdemfeaosoefvf

nijsz afomfef nidar. Nu tr prbndfjdfr lfefdaf )y,x(`y : `brjueai oìaffaodordfr.Nunisotlrej,nutrafemdfjdfroezjeeafnisotbdigifmdoeafyiednor iedijiyi efafe bdinodor. Niz ostosei aurujdir aeai nu tr afemdfjdfro zjfaf

mirdidie zbrdumdirdi aihi ymsfm jfrtfnfdo afemdfjdfro zjfaf mirdidie zbrdumdir

iye azfyaf bdinodor. Noroego jfrtfnfafe ao`friesoyfd afemdfjdfro zjfaf ef yizm mo

lfefdnorybmybmtur.Nuefafedfnoroegojfrtfnfafemdfjdfrafmfeaoiridireaiidtse`diriiyrdj vfhfrnorseòoeìrmdyetfjdfrlfdotorodjotor.Nundjafnoroegojfrtfnfafe

afemdfjdfroeeisdse`dieardaieditdigim,iraeaiesostfjitomnoryimdijdihfrzijie

zj jjme bdie noroego jfrtfnfafe doeffr afemdfjdfr vf uyludijidir

odfefgfmtor.Iraeaiedoeffrbdjiyietrdfrooevfrodfenorzjirideaizjevirbdupbdjiatirtdigimtr. Nu seì lorfe afomfedfroef iyrdinodor top, hbjbkfe vf tij ao`friesoyfd

toptfmoafemdfjzjdfrozfroeafaurudigimtr.

>.>.NÊG JF^QFNFA@FÎEW_FDAFEMDFJDF^FLFEFDNIMO

Qiejeaieiedidigizfrfnoroegojfrtfnfafeao`friesoyfdafemdfjdfrafsiafgfnoroego

trfv yfr idr. y nijd, x af nijsz afomfeo lstfrjfm zfrf nydf nor afemdfj

3)y,y,x(` : `brjueai vfrodor. Nu ndjaf siafgf noroego trfvoe abruaie nijd vf

nijsz afomfe goesoeafe yizdinodaoo )y,x(`y : treafe afemdfjdfr zfroeafaurigiz.Nydfgfzj

+: Gax)y,x(`yfmdoeafoìaffaodfnodfgfmtor.Iegimbumfznuyizjtirz,vfrodfeao`friesoyfdafemdfjoe

nor oetflrid afemdfjf aetrdjfsoeafe aihi tf nor sbeu lftorjfz. Pfrodfe`(x, y)`bemsoybeueue siafgfxf nid bdauu nisot aurujdir iegim abruaie oetflrid ybduydi

zjf uylueaur. refoe 6x0y > : ao`friesoyfd afemdfjoeaf`(x,y) siafgf nijsz

afomfef nidar vf abruaie oetflrid ybduydi lfefd zj Gx6x>y 7 +: bdirimmbdiygi fdaf faodor. Nuriai u hususue idte ozjfm lfrfmor. Noroego jfrtfnfafe norao`friesoyfd afemdfjo zfrmfe nijsz afomfe bdirimx yi ai y sfodfnodor. Nu tr nor

afooj nizfe zj zbr bdie ao`friesoyfd afemdfjo, zj aihi mbdiy nor hidf

lftorfnodor.refoe

9y7soex)9y(

f

ax

ay>

y>

+++:

ao`friesoyfdafemdfjodoeffrbdjijiseirijfe,iiaimoafemdfjdoeffraorvfmfsoebdie

norzjybduvirar.

y>y>

>

f

9y7soe

xf

9y

ay

ax +

+

+

: afemdfjoxflrfdoeffraor.


16/90

-90 -


>.7.NÊG JF^QFNFDEFFÂ@FÎEW_FDAFEMDFJDF

Noroegojfrtfnfafedoeffrnorao`friesoyfdafemdfj,

)x(^y)x(Sy :+ (>.9)

`brjueai vfrodor. Hitrditjim lfrfmor mo )y(^x)y(Sx :+ afemdfjo afxf lrf doeffraorvf nuriai ieditdigim yetfjdf zdfnodor. PfrodfeSvf^ `bemsoybedir elrdfe zj

irideaixfnidsrfmdo`bemsoybedirar.Nutrafenorafemdfjoezj,ffrafemdfjoe

sbd yie tfm nor tfrojoe trfvo fmdoeaf oìaf faodfnodorsf, sriaie nor odfjf aefgfmtor.Nueue ooe afemdfjoe sbd tiriè nor tfrojoe trfvo hidoef lftorfnodfgfm nor irpiee

iriejislfrfmor.Afemdfjoehfromoyiee(x)`bemsoybeuodfirpidj.

)x(^)x(y)x(S)x(y)x( :+ (>.>)

] Z y)x(y)x(y)x(

+

:

(>.7)

bdauueaie, (>.>) fotdooeoe sbd tirièe ] Z y)x( eoe idj bdinodjfso ooe)x(S)x()x( : irt sidiejidar. Nudjiyi idtjz irpiee s`raie ìrmd bdauu

aurujai,nuafemdfjooetflrffafrsfm

)x(S)x(

)x(:

vfyi de ( ) ( )a

x S xax

: yizdirim +: 9Gax)x(S)x(de fdaf fafroz.(x)irpieyideznrimdpvf

G9sinotoaommitfid

ejiyirim

: ax)x(Sf)x( (>.1)

fdaffaodor. etflridsinotoeoenuiijiaiaihodfaodjfsolfefdzjzfroeafnorafoomdoo

ybd ijiyigimtr. Iyrgi (>.1) afemdfjoeoe si tiri` hfr mbudai pbzoto` bdigieaie

afemdfj,

: ax)x(Sf)x( (>.6)

bdirim ai oìaf faodfnodor. (>.6) afemdfjoydf tiejdieie `bemsoybei oetflrid irpieaoyfgfoz.(>.9)afemdfjoeoesbdtiriìrtmtfmnortfrojoetrfvofmdoeafoìaffaodaooeafe

] Z )x(^)x(y)x( : (>.0)yizdirim

+: Gax)x(^)x(y)x( vfyinijdafomfeoyideznrimirimao`friesoyfdafemdfjoelfefdzj,

] Z +: Gax)x(^)x()x(y 9

(>.


17/90

-9< -


bdirim fdaf faodjo bdur. Noroego jfrtfnfafe doeffr nor ao`friesoyfd afemdfjoe (>..9)afemdfjoeafvfrodfefmdflftorodjfsolfrfmor.

refm>9

97)>(,?7 :: yxyy doeffrnidielaffrprbndfjoeozez.

zj

y trfvoeoemitsiys9vfS(x):7,^(x):?xbdauulrdjfmtfaor.etflridirpie,

xaxaxxS fffx 77)(

)(

::: fdaffaodfrfm(>.( :y bdauueaie, 0>7 09>797 :++: fGGf .Nydfgfiriajzzfdzj,

970 07 ++: xfy x

uieimiairyipdieodfjdfrafvfrodfenorzjirideaiS(x)vf^(x)o`iafdfroeoesrfmdo

`bemsoybedir bdjis lfrfmtoo vurludiejt. Ffr nu `bemsoybediraie noroeoe vfyi omosoeoe

srfmsozdomebmtidirvirsi,zjndlfsosrfmdodooebdauuidtndlfdfrfiyrdjidar.Nueu

norrefmdflrfgfoz.

refm>>

7)>(,69

9 > ::+

+ yxyx

y nidielaffrprbndfjoeozez.

zj

9

9)(

+:

xxS `bemsoybeu x:9 ebmtiseai srfmsozaor. B hidaf zj 4x49 vfyi

94x4irideaiirijijzlfrfmor.etflridirpiejz,

9)( 9de9

)(+:::: ++

xfffx xxax

axxS, x59 ooe jutdim affr oo pbzoto`tor. Abdiysydi

94x4irideai 9)( +: xx yizdinodor.(>.


18/90

-98 -


] Z

9)9(9>

)17(6

9>

>396

9

9

7

6

1

6

9

96)9(

9

9

7

7171>

++

++

:

+

++

:

++

+:++

+:

x

G

x

xx

Gxx

xG

xx

xGaxxx

xy

PfrodfembuduyfroefyizirimGsinotoeonudidj;

7

)9>(9>

)1>7(>67

7

:+

++

+: G

G, nydfgfirieiezfdzj=

)9(7

)17(6 7

+

++

:xx

xxy bdigimtr.

>.1DEFF^BDJI_IENÊG JF^QFNFAFEA@FÎEW_FDAFEMDFJDF

Doeffr vf noroego jfrtfnfafe ao`friesoyfd afemdfjdfrdf oìaf faodfe nidiel affrprbndfjdfro eospftfe zjdfro mbdiy bdie prbndfjdfraor. Nueue efafeo (>.


19/90

-9? -


>.1.9.AfomfedfroefIyrdinodorQoptfNoroegoJfrtfnfafeAfemdfjdfr

Noroego jfrtfnfafe nor ao`friesoyfd afemdfjoe ),( yxhy : fmdoeaf vfrodaooeo aefdoj.Ffrafemdfjoesi yie,

)(

)(),(

yl

x`yxh : fmdoeaf o`iaf faodfnodoybrsi, nu ao`friesoyfd afemdfj afomfedfroef

iyrdinodortoptfaor.

refm>7Afomfedfroefiyrdinodorafemdfj

9)>(,> > :: yxyy nidielaffrprbndfjoeozez.

zj

Pfrodfeafemdfj xy

ayxy

ax

ay>>

>

> :: fmdoeafafomfedfroefiyrdptfrojtfrojoetflrf

faodorsf=

>>

>

>

9>9

> >

9

999> :

::

xyxyxyay xyxy

vfsbeubdirim,

>7

9

xy

: fdaf faodor. Aommitf faodorsf, vfrodfe nidiel irt abruaie oetflrid odfjo

sriseaihfsinimitdj,nydfgfzfdzjfdaffaodjotor.

refm>1

3)( 1>1 :++ ayyxyax ao`friesoyfdafemdfjoeozez.

zj

Pfrodfeafemdfj 39

1

> :+

+ ayy

x

axfmdoeafyizdpabruaieoetflrffaodfnodor.

:++ 39

1

> ayy

x

axG

yx :+

6)irgtie(

6

irieiezjar.

refm>6Brtblbeid(Aom)_relfdfrNornoroeo ?33 iydi mfsfe abrudiri nornoroeoe brtblbeidar afeor. Nu efafedf abrudiraie

noroeoe fojo j osf nuei brtblbeid bdie aofroeoe fojo 9/j bdigimtr (lfbjftroafenornoroefaomomoabrueuefojdfroirpjee9bdauueuhitrdiyez).Qiejlfeodftfgfm

bdursim,refoenorxyazdfjoeafmo@(x,y):Gfroiodfsoeoehfrnorfroso,iyeazdfjafmo

L(x,y):Mfroiodfsoeoehfrnorfrosoeoaombdirimmfsoybrsi,nufrodfrnornoroeoebrtblbeidar.Pfrodfenorfroiodfsoefbrtblbeid(aom)bdiefroiodfsoeonudjimooeuiajdirozdfejfdoaor;

9 Pfrodfeo`iafeoe(@(x,y):G)xflrftrfvoidepax

aybduturudur.

> Nuomoafemdfjafe(@(x,y)vfnueuetrfvoeovfrfeafemdfj)Gybmfaodor.

7 moegoafemdfjafax

ayyfroef

ay

ax ideirimfdaffaodfgfmao`friesoyfdafemdfj

zdr.


20/90

->3 -


Nu iajdir nor uyludiji odf lrfdoj. refoe xgy: fro iodfsoef aom bdie fro iodfsoeo

nudjim ostfyfdoj. Pfrodfe oìafafe gx

yyx@ ::),( bdauu iedidjimtiar. Abdiysydi

gax

ay: fdaffaodor.Hfr omoafemdfjafgyffotbdauueilrfnufotdomdfroesbdyiedirai

nornordfroeffotbdjidar,aofrnoroìafydf=

x

y

ax

ay: nudueur. Nydfgf odm omo iaj tijijdij bduybruz. ojao af trfvo

ay

ax odf yfr

afotorfdojvfzdfgfmao`friesoyfdafemdfjox

y

ay

ax: bdirimnfdordfjo bdidj.Afemdfj

azfedfeorsf 3:+yayxax ao`friesoyfdafemdfjofdaffaodormonuaiafomfedfroefiyrdjvizoyfttfbdauueaietfrojtfrojoetflrffaodfnodor.

myx

yayxax :+:+ >>3>>

, vfyi >>> ^yx :+ fdaffaodjo bdur.

Nuriai >> ^m: idam. xgy: froiodfsofojogbdievfjfrmfzafelffeabrudirtfjsod

fafr.tfyieaie >>> Myx :+ froiodfsoosfjfrmfzobrokoeebmtisbdievfyirip^bdiejfrmfzgod fjnfr iodfsoaor. Lfrfmtfe af nu omo fro iodfso norfydfro nornordfroeo aom i odf

mfsfrdfr.

Aom yrelfdfr mbeusu, siafgf lfbjftroydf odlodo afod, aofr jhfeaosdom idiedireai ai

efjdoyfrtutir.refoe>nbyutdunorfdfmtromidieeaimomuvvftozlodfro,sinotpbtiesoyfd

ozlodfroeoebrtblbeidyrelfdfroaor.>nbyutdunorstries`froprbndfjoeafsim ozlodfro,

sinotsgimdmozlodfroeoebrtblbeidyrelfdfroaor.Noraofrefjdouyludiji aiimmiedir

jfmieooeafaor. > Nbyutdu nor immie imj prbndfjoeaf imj ozlodfro, sinot pbtiesoyfd

ozlodfroeoebrtblbeidar.

>.1.>.HbjbkfeQoptfNoroegoJfrtfnfafeAfemdfjdfr

Afomfedfroef iyrdinodor`brjilftorodfnodfgfm ao`friesoyfd afemdfjdfrafenoro afhbjbkfe

toptfbdiedirar.Noroegojfrtfnfafenorao`friesoyfdafemdfjffr,

)(v`x

y`y :

:

fmdoeafyizdinodoybrsinuafemdfjfhbjbkfeaorafeor.refoe

yx

yxy

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77 :

hbjbkfe toptf nor ao`friesoyfd afemdfjaor. em afemdfjoe si yie siafgf veoe

`bemsoybeubdirimoìaffaodfnodor.

yx

yxy

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77 : oìafsoeafy:vxaejuyludiersi,

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777 99

)(v

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vxx

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ueu ebt ftjfjoz lfrfmor mo nuriai muddieajz hbjbkfe tfrojo, 9. Ndjafmoiedijeaie ìrmdar. Sfm bm ao`friesoyfd afemdfjoe hbjbkfe bdup bdjia nisot nor


21/90

->9 -


mbetrbddf brtiyi mirdinodor. Mirjim o`iafdfr ooe hbjbkfedom tfsto uyludiejis tivsoyf

faodor. ),( yx`y : ao`friesoyfd afemdfjoeoe hbjbkfe bdup bdjiae iedijim ooe

afemdfjafmo tjxdfr x odf, tjydfr af y odf yfr afotorodaooeaf ffr afemdfjoe si

tiri`eaimo `(x, y) `bemsoybeu e `(x,y) hidoef lfdfnodoybrsi afemdfj eoego afrfgfafehbjbkfeaor afeor. Hbjbkfe nor ao`friesoyfd afemdfjo zfrmfe iiaimo aejdfro

uyludiyigiz;

vxy: vfhfromoyieexflrftrfvobdie vvxy +:

refm>0Hbjbkfeao`friesoyfdafemdfj

xy

xyy

+

: ao`friesoyfdafemdfjoeozez. vxy: vf vvxy +: aejdfroeo

uyludirsim,

9

9

+

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+

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xvxvvx vfyi

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+

+:

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vvx fdaffaodorojaoafemdfjafomfedfroef

iyrdinodor.Azfedfjfyipdrsi=

xax

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+ vfyi :+++:+

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ax

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9 > :++ vftfmrirazfedfjfyipdrsi=

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x

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:

fdaffaodor.(Ebt; xx :> )nudueur.

refm>> :+ ayxaxxyy ao`friesoyfdafemdfjoeozez.

zjAfemdfjoeomoegoafrfgfafehbjbkfebdauuimtr.

vxy: vf vvxy +: aejuyludiersi,

3)()( >>> :++ vvxxxvxxv vf lfrfmdo siafdftorjf yipdirim 3)()( > :++ vvxvvnudueur.

y

x

vG

xGx

vx

ax

v

avv

ax

avx :::+:+:+

9dedede

933

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> afelfefdzj,

( )Gxx

yde

: bdirimfdaffaodor.


22/90

->> -


refm>8

3)9(>)>9( // :++ ay

y

xfaxf yxyx ao`friesoyfdafemdfjoeozez.

zj

Pfrodfe ao`friesoyfd afemdfjaf xdfr x vf ydfr y odf yfr afotorodaooeaf sbeugueafojfaoo lrdr. B hidaf afemdfj s`reg afrfgfafe hbjbkfeaor. Iegim zj ooe

vxy: aejyfroef vyx: aejmbdiydmsidir.Nueilrf=

vyx: yavvayax +: oìafdfroao`friesoyfdafemdfjafyizdirim=

3)9(>))(>9( :+++ ayvfyavvayf vv fdaf faodor. Afomfedfroef iyririm azfedfjfyipirsim,

:+

++

3>

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fv

f

y

ay

v

v

Gfvy v de>dede :++ vfyi Gfvy v :+ )>( .Iegim vy

x:

bdauueaie Gfy

xy yx :+ )>( vfyi Gyfx yx :+ > .

G sinotoeoeeflito`bdijiyigieaiehirfmftdfjutdimaffrmidardinodor.

3> 5:+ Gyfx yx

>.1.7.QijAo`friesoyfdAfemdfjdfr

3),(),( :+ ayyxEaxyxJ ao`friesoyfd afemdfjoeo lz eef ididj. NuriaiJ vfE,xyazdfjoeafmo ^ ndlfsoeaf noroego trfvdfro nudueiex vf yyf nid srfmdo `bemsoybedirbdsue. 3),(),( :+ ayyxEaxyxJ `brjueaiyizdienoroìafeoetijao`friesoyfdbdinodjfsoooe afemdfjoe sbd tiri` ),( yxu fmdoeaf nor `bemsoybeue tij ao`friesoyfdo, si tiriè ai

trfvdfeaooeafs`rbdinodjfsoooenorsinotbdjislfrfmor.Abdiysydiirieielfefdzj

Gyxu :),( fmdoeafbdjidar.zjetijao`friesoyfdoeoididj;

] Z ] Z 3),(),(

),( :

+

: ayy

yxuax

x

yxuGayxua . Pfrodfe ao`friesoyfd afemdfjdf nu

afemdfjiriseainfmdfefenornfezfrdombdauuimtr.Bhidaf 3),(),( :+ ayyxEaxyxJ ao`friesoyfdafemdfjoeoetijao`friesoyfdtopoefuyluebdinodjfsoooe=

axuyxJ :),( vf

ayuyxE :),( bdjis lfrfmtooeo sydfyfnodoroz. Nuei odivf bdirim,

3),(),( :+ ayyxEaxyxJ fmdoeaf vfrodfe nor ao`friesoyfd afemdfjoe tij ao`friesoyfdafemdfjbdinodjfsoooe=

x

yxE

y

yxJ

:

),(),(bdjislfrfmor.


23/90


24/90

->1 -


93),( 7 :+:y

xxyxu

Nizfevfrodfenorao`friesoyfdafemdfjnor oetflridirpieyirajyditijao`friesoyfdhidf

lftorodfnodor. Ffr 3),(),( :+ ayyxEaxyxJ afemdfjo nueaie egfmo yetfjdfrdfzdfjoybrvfiyezijieaiaitijao`friesoyfdafodsf,

),(

),(),(

)(yxE

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yxE

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yxJ

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: bdjimzfrfoetflridirpie : axxlfyx

)(),( o`iafsoeafe

nudueur. Nydfgf afemdfjoe hfr omo tiri` nu irpiedi irpdirim tij ao`friesoyfd hidf

lftorodfnodor.

>.1.1.NizzfdAo`friesoyfdAfemdfjdfr

9( ) ( )

3>>>999

:+++++ aygynxiaxgynxi ao`friesoyfdafemdfjo

Nu afemdfj ffr g9 : g>:3 osf hbjbkfeaor. g9 3, g> 3 osf afemdfj iiaimo

aejdf hbjbkfedftorodor.

99

99

ayaymyy

axaxhxx

:+:

:+:

Nydfgf afemdfj

3)()( 99>9>99999 :+++ ayynxiaxynxi hbjbkfe afemdfjoefaer.

Iegim jn

n

i

i::

9

>

9

> osf yumiraimo aej zjf ltrjfz. Nu aurujai

9> iji : vf 9> jnn : aejydf afemdfj

3Z)(]Z)]( >99999 :+++++ aygynxijaxgynxi bdur.

Nu afemdfjf vynxi :+ 99 aej uyludirsim afemdfj afomfedfroef iyrdinodoraurujilfdor.

refm>99( ) ( ) 3919 :++ ayyxaxyx ao`friesoyfdafemdfjoeozez.

zj

99

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ayaymyy

axaxhxx

:+:

:+:

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3,9391

39 ::

:+: mh

mh

mhbdjidar.Nydfgf afemdfj


25/90

->6 -


( ) ( ) 31 999999 :++ ayyxaxyx fmdoeafhbjbkfebdur.Nuafemdfjozjfmzfrf=

+:

:

999

99

vaxavxay

vxyaejuyludirsimafemdfj,

3)19()91( 99> :+++ avvxaxv hidoeo idr.Afomfedfroef iyrrsim,

391

19>

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ax fdaffafroz.

gv

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vx :

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++

91

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gvvex :+++ )>(tie>991>9de 9>9 fdaf faodor. Niti itiajz afomfedfrafe lfro

aefrsfm,

93

99

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999

999

::

:+:+:

:+:+:

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y

x

yv

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yy x x g

x

+ + + :

fdaffaodor.

>.Nfrebuddo ao`friesoyfdafemdfjo

Nfrebuddo ao`friesoyfd afemdfjo eyx^yxSy )()( :+ , 3,9 ee `brjueai vfrodor zj

ooe

:

:

ayyeau

yu

e

e

)9(

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aejuyludieaeaiafemdfjdoeffrafemdfjhidoeflftorodjo

bdur.Doeffrdftorodfeafemdfjoelfefdfmdo

)()9()()9( x^euxSeax

au:+ hidoeflfdor.

refm>9>

> dexy y y x + : ao`friesoyfdafemdfjoeozez.

zj

Pfrodfeafemdfj >9 dex

y y yx x

+ : bdirimyizdaeainorNfrebuddoafemdfjobdauuimtr.

9 9 > 9eu y y y : : : vf >u y y : aejdfro afemdfjaf yfroef yizdr vf lfrfmdoazfedfjfdfryipdrsi,


26/90

->0 -


9 dexu u

x x : fmdoeaf uyi lrf doeffr nor afemdfj fdaf fafroz.

9 9de( ) de 9( )

axS x ax xx xx f f f fx

: : : : : bdirim oetflrid irpie nudueup lfefd zj

oìafsoeafyizdrsi,

9 9 de de 9( ) ( )

9/

x xu u x ax G x G

x x x x x

: : + : + +

vfyi9

de 9x Gxy

+ + : fdaffaodor.

7.Gdioriutao`friesoyfdafemdfjo

Nu afemdfj

+:

ax

ay

ax

ayxy fmdoeafaor. Nu afemdfjoe lfefd zj ooe

ax

ayy : dfr

yfroef mfyò bdirim sfodfe nor g sinotoeoe mbeudjisydi nudueur. _ieo lfefd zj,

)(ggxy +: bdur.

Nu afemdfjoe tfmod zj osf )(ggxy +: lfefd zj odf nu afemdfjoe hfr omo

tirièe gyf lrf trfvoeo idirim nudueie 3)( :+ gx afemdfjdfroeafe gyo fdojoefftjfmsurftoydf nudueur.

refm>9>>yyyxy +: ao`friesoyfdafemdfjoeozez.

zj

PfrodfeafemdfjGdioriut afemdfjoaor.Lfefdzjooe gax

ay ideirim >ggxgy +:

bdirim fdaf faodor. Qfmod zj ooe hfr omo tiriè gyf lrf trfvoeo ididj;

>

93>9

+::+

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>)9(

1

9)9(

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++++: xxxx

y vfyi >)9(

1

9+: xy nudueur.

1.Dilrielfao`friesoyfdafemdfjo

Nu afemdfj

+

:ax

ay

ax

ayxy fmdoeaf vfrodor vf lfefd zj ooe py : aej

uyludier.

refm>977yyxy +: ao`friesoyfdafemdfjozez.


27/90

->< -


zj

( )y p x : aej yipdaeai 7pxpy +: fdaf fafroz. Hfr omo tiri` xf lrf

trftorsfm,

pppxppy ++:: >7)( vfyixp

pp

:

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> nudueur. Azfedfjf yipdirim

>

7

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p

p

x

ap

ax:+ ao`friesoyfdafemdfjonudueurmo nuaixflrf noroegojfrtfnfafe doeffr

nor ao`friesoyfd afemdfjaor. Nu afemdfjoe zjeafep

gpx +: >

6

7 nuduruz. Aihi

egfaf 7pxpy +: bdauueunudjutum,abdiysydi,

7pxpy +: ,p

gpx +: >

6

7iriediezjfiotpirijftromafemdfjdfraor.

6.ôggitoao`friesoyfdafemdfjo

Nuafemdfj >( ) ( ) ( )y S x y Y x y ^ x : + + bdirimvfrodorvf S(x)s`raieìrmdaffrdfridr.Nuafemdfjoe ffr y9 lono nor zfd zj vfrodjosf, nu aurujai lfefd zj af

nudueinodor.Iiaimoaejdfrnuijidimuddiedr;

9 9 >9 uy y y yu u

: + : ideirimao`friesoyfdafemdfjafyfroefyizidj.

>

9 9>

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u uu

: + + + +

(9)

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>9 9 9( ) ( ) ( )y S x y Y x y ^ x: + + nuduruz. (>)

(>)afemdfjo(9)afemdfjoeafyfroefyizdpazfedfjfyipdrsi,

] Z9> ( ) ( ) ( ) 3u S x y Y x u S x + + + : doeffr afemdfjoef udidr mo nu ai oetflrid irpie odfzjbdienorafemdfjaor.

zfdauruj;

Ffr vfrodfe ôggito afemdfjoeoe y9 vf y> lono omo zfd zj nodoeoybr ai lfefd zj

ostfeoybrsinuaurujailfefdzjbdirimiiaimooìaffdaffaodor.


28/90

->8 -


Lfefdzj;( )> 9( )9

>

S x y y axy yGf

y y

:

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>

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9 13y y y

x x + + : ao`friesoyfdafemdfjoeoenorzfdzj 9

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x: bdauueilrfnu

afemdfjoelfefdzjenudueuz.

zj

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9 > 9 >u uy y y y

u x u u x u

: + : + : : bdur.Afemdfjafyfroefmbyidj;

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> > 9 9 > 9 13

u

x u x x ux u x

+ + + + :

Azfedfjfyipirsim,

69u u

x : nudueur.

6

96 de( ) 6de

6

9( )

axS x ax xx xx f f f fx

: : : : :

16 6 6

6

9( 9)

1 19/

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x

: + : + : +

fdaffaodor.Aofryieaie, 99

y yu

: +

bdauueaie>

xu

xy:

nudueur._umiraimoo`iafafyfroefyizdaeai,

>

x

u xy: :

6

1

x

Gx+ sbeugueiudidr.


29/90

->? -


>.6.NÊG JF^QFNFAFEAFEMDFJDF EWWQFJIQM_IMDIOJ

u iei miair noroego jfrtfnfafe ao`friesoyfd afemdfjdfroe zjef yefdom fotdo

yetfjdfr rfeaom. Nu yetfjdfr uyludijiai mirdidie bu prbndfjo zjfm ooe

yftfrdoaor,ìmitnorao`friesoyfdafemdfjoenuyetfjdfroenoroydfzdfnodfgfoeoelirietoso

ybmtur.Nizfenoryiaiomoyetfjyftfrdobdinodormfe,nizfehonoroeoeofyirijia

lononorauruj brtiyi minodor. Hfr omo aurujai ai prbndfjf sostfjitom nor jietm frfvfsoeaf

yimdijyipjimìyaidbdinodor.

),( yx`y : ao`friesoyfd afemdfjoeoe e lrdfe zj irideai zjee virbdauueu vf nu zje tfm bdauueu aefdoj. Mfeaojozf u sbrudir sbrup gfvip

nudjiyiididj;

9. Afemdfjabruaieoetflridybduydizdfnodoybrju2

Vyludijiai mirdidie pfm bm ao`friesoyfd afemdfj abruaie oetflrf faodfrfm

zdfnodfgfm fmodafaor. )(x`y : fmdoef lftorodfnodfe tj afemdfjdfr nu ybddizdfnodor.

>. Pfrodfeafemdfjdoeffrjoaor2Qj doeffr afemdfjdfroe, mirdidigim bdie oetflriddfr ideinodaoo srfgf zj

yipdinodor.

7. Afemdfjafomfedfroefiyrdinodorjoaor2

Pfrodfeao`friesoyfdafemdfjafxdotfrojdfrnortiri`ti,ydotfrojdfrafaofrtiri`titbpdieinodoybrsi, afemdfj afomfedfroef iyrdybrafjfmtor.Afomfedfroef iyrdie

afemdfjabruaieoetflridybduydizdr.

1. Afemdfjhbjbkfejoaor2Afodsfhbjbkfehidflftorodfnodorjo2Afomfedfroef iyrdijiyie nor afemdfj, ffr hbjbkfesf, xyv /: lono nor

pirijftrfeoetiejdiejisydiaiojiafomfedfroefiyrdinodoraurujilftorodfnodor.6. Afemdfjtijao`friesoyfdjoaor2Afodsf,tijao`friesoyfdhidoeflftorodfnodorjo2

Afemdfjf tij ao`friesoyfddom tfsto yipdr. Ffr tij osf zj yipdr. Ffr tij

afodvfaofr`brjdiriaiuyjuybrsiafemdfjotijao`friesoyfdaurujilftorfgfm

noroetflridirpieirier.ojaonusostfjitomybdunorrefmdfpfmotorfdoj;

refm>91yxxfy +: de ao`friesoyfdafemdfjoeozez.

zjdm mbetrbd fafgfojoz mbeu afemdfjoe jfrtfnfsoaor. Pfrodfe afemdfj noroego

jfrtfnfafeaor, em omo vfyi aihi ìzdi jfrtfnfafe trfv tfrojo nuduejijimtiar.

Iraeaie abruaie oetflrisybe odf zdfnodordooef nimidj. Afemdfjoe si tiri`,nodoejfyfe`bemsoybe,emomovfyiaihiìzdijfrtfnfafetrfvtfrojonuduejijimtiar.

Iraeaie abruaie oetflrisybe odf zdfnodordooef nimidj. Afemdfjoe si tiri`,

nodoejfyfe `bemsoybe yyf nid bdauueaie afomfedfroef iyrijiyz. Afemdfj doeffr

afodaor,si tiri`tielfdfgfmbdie yf tfrojodoeffrafodaor.Afemdfjonoriznisotdftorjfyf

ididj.yxyxx

fxffffy

:: de

fdaffafroz.Afemdfjafomfedfroefiyrdinodoraurujilfdjotor.


30/90

-73 -


Gxffaxxfayf xyxy +:: )9( vfhfromotirièabiddblirotjisideirim=( )Gxfy x +: )9(de

>.0.NÊG JF^QFNFAFEDEFFÂ@FÎEW_FDAFEMDFJDF EV_LVDIJIDIÔ

@ozomtf, noybdbkoaf vf sbsyid nodojdfraf mirdidie bm siyai prbndfjaf nor nymde

afoojhzeenymdemfeaosoydfbrietdbdauulzdfejotor.Aofrnoroìafydf,ffry

odlodfeodfenymdetieeaimoaffroysf,nuaurujai,

myat

ayvfyiy

at

ay:

yizdinodor. Nuriai m briet sinoto bdup afefysfd vfyi lzdfjsfd sbeudiraie nfdordfeor.

Afemdfjoesbdueaimotfroj,ynymdeezijiediafoojhzetfjsodfafr.Nuafemdfj

noroego jfrtfnfafe doeffr nor ao`friesoyfd afemdfj bdup afomfedfroef iyrdjim surftoydf

mbdiygizdr;mtfyy 3:

Nuriai y3, y nymdee t : 3 ieeaimo affroaor. @ozomsfd bdiydirai lfefddomdf afoojsrfmdo nor `bemsoybe zfddooeaf bdjisei rijfe noybdbko vf sbsyid nodojdfraf mfsomdoafoojdfrszmbeusuaur.Norhiyvietreesiysvfyinormbdbeoafmonimtfrosiyseaimo

afoojdfr nuei refmtor. Iegim siy bm ìzdiysi afooj zijiee srfmdo nor `bemsoybeu

bdirim aedfnodor vf nu efafedf yipdigim hfsipdiji hitis lfefddomdf jimud serdir

ofrosoeafmidr.

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siy bdjim zfrf vf t:3ieeaimo siyyE3 idirim, eùsue afooj hz (irt vfyi izidji

hz)=mtfEtE 3)( :

bdigimtr. Nuriai m abuj vf dj hzdir iriseaimo ìrm tiejdiyie eft eùs afooj

hzar.refoem:3.396/yd,9333af96domnoreùsirtafjfmtor.

refm>90îaybimto`nbzuejivfîaybimto`Mirnbe_i QiyoeoSdtbeyuj, îayujvfG91 bdirim nodoefeMirnbe ozbtbpu lono niz riaybimto` fdfjfetdfroe

aofr nor fdfjfet vfyi iye fdfjfetoe ìrmd nor ozbtbpueu tfmod ftjfm zfrf tinoo bdirimnbzueaumdir nodoejfmtfaor. Nbzuejiee hz lfefddomdf jfvgut fdfjfet jomtirydi brietd

bdirimafoor,abdiysydiriaybimto`nbzuejisrfgoafyumiraimoafemdfjdftiejdier;

3, ( ) m taJ mJ vfyi J t J fat

: :

Nu oìafaf m 5 3 bdup nbzueji sinoto iae idr. Nor irmfbdblue, nudauu mfjom midets

zfroeaf, nor gied hiyvieai nudueie jomtire %8o brieeai G91 ozbtbpueue bdauueu

siptijtr. G91 ozbtbpueue nbzueji sinoto 9.>1931/yd bdauuei lrf mfjooe yiehfsipdiyez.

zj

_umiraivfrodfeafemdfjafetyofmfrsfm,3

)(de

9

J

tJ

mt : nuduruz.


31/90

-79 -


at

taJ )(odfvfrodfenbzuejihz,jomtirsrfmdoafeJ(t)odfbrietdbdauueaienidirai

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3>

9)( JtJ : idersi,yumiraimoafemdfj,

mtt H

>de::

bdur. îaybimto` jiaafdfroe yir jrdfro lfdojo dinbrituirdirai ddr. îaybimto`

nbzuejieeefjdonoruyludijiidie,G91 ozbtbpueuenbzueujufsiseiaiyieieîaybMirnbe _i Qiyoeoaor. Nu jftbaue tfjfdoeaf, yiiyie hfrhielo nor giedaimo mirnbeitbjdiree mm nor brieee G91 ozbtbpu jfyaiei lftoraoo lzdfjo virar. Nu briegiedee yiij nbyuegi yimdim bdirim sinot midjimti, em nbzueie jomtir, giedee

fvrfsoeafe nfsoe vf sbdueuj ybduydi ida mirnbe itbjdirydi yfroef mbeur. Iegim dfe

giedyiirtmmirnbeloroobdjiyigiooesrfmdonornbzuejiszmbeusubdigimtr.Aofrnoroìafydf gied daeaf, ofrosoeaf G91 ozbtbpdiree ai nudueauu mirnbe loroo mfsodor,

giedeesihopbdauuG91srfmdonorizidjisrfgoeflorfr.G91eyirjr6608ydar.

Itjbs`frafmo G91 jomtir, izbtue itjbs`frafmo mbzjom edir efafeoydf G91 ozbtbpuei

aej ybduydi srfmdo bdirim yfeodfeor vf nydfgf G91e itjbs`frafmo G9> jomtirei

brie sinot midr. Nu imdijidire idteai irieie srf,

70?>3)38.3de(93>1.9

91

:

:

t ydbdigimtr.

refm>971 (Qh>71) ozbtbpu , edir yiyirim Si>71yfaejfmtfaor.Nu ozbtbpuenbzueji hz,fdfjfetoejfvgut jomtir odf abru brietdar.

Iyrgi933jl.Qh>71ozbtbpueainorhi`tiooeaflfroyf8>.31jl.midanodoeaooeflrf,

i) hfrhielonortieeailfroyfefmiairQh>71midae,n)jfvgutjomtireyiryioejfso

ooeefmiairzijielfjfsolfrfmtooeonudueuz.

zj(i) PfrodfedfrflrfJ3:933jl (t:3ieeai),J(t:.31jl.

3>87.3933

31.8>de

87.3933)( : bdigimtr.Aofryieaie(n) 6.>13>87.3

>de>de::::

mtt H le.

refm>98EfwtbeuesbujimieueuNidielti Qo : >3

3G bdie nimraie mm nor nodyf Q3: 9333Gaf miyeijimti bdie su

nieybsuei nrimdamtie >3 sieoyf sberi sgimd ef bdur. Wu nieybsu yftfroegf lfeo bdup

sgimdafojfjfmtfaor. 9.3::jg

hI9/s.


32/90

-7> -


zj

Aihi egf Efwtbeue sbuji yisiseaie ( )3aQ

Q Qat

: bdauueu lrjtm.

Afomfedfroef iyririm zajzaf, nimr nodyfeoe t ieeaimo sgimd

3 3( ) ( ) toQ t Q Q Q f : bdur.Pfrodfeaffrdfryfroefmbeursi ( >3 ) 8?.>Q t s: : 3Gbdur.

refm>9?Mirjprbndfjofrosoeaf nidielti 9333 D tfjoz su nudueie nor tiemi, ofrosoeafmo tuz mbesietrisybeu

3.9 ml/D bdie tuzdu su 63 D/aimomi afnosoeaf lorjfmtf, vf iye zijieai tiemtie yoef 63

D/aimomiafnosoeaftuzdusufmodjfmtfaor.Qiemofrosoeafmonormirtrg,tiemtimomirjehbjbkfedooeo sidijimtiar. Nfmdfeaoo lono, tiemtimo su sfvoyfso sinot midjisei mire

ofrosoeafmotuzdusuaimotuzmbesietrisybeusrfmdobdirimymsfdfgfmtor.Qiemtinfdordonort

ieeainudueietuzjomtirevfrfe odomoyolfdotoroeoz.Qiemtinudueinodfgfmjimsojujtuz

jomtirefaor.

zjJ(t) tiemti t ieeai nudueie tuz jomtire lstfrsoe. Qiemtimo tuz jomtirei mtdfeoembrueujuprfesonoeouyludiyidj;

Qiemtimotuzjomtireeafoojhz:QieminorojzijieailorfetuzjomtirQiemtienoroj

zijieaimietuzjomtir

( )( ) ( ) 636.3/9333

/63/9.3/63 :+

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at

aJDml

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at

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Afemdfjnoroegojfrtfnfafedoeffrnorao`friesoyfdafemdfjbdup 3)3( ::tJ mbuduydinornidielaffrprbndfjoaor.

6)(,36.3)( :: t^tS idejimsurftoydf tat

fft 36.336.3

)( :: .Afemdfj>.366)(

( ) ( )ttt ffftJ 36.336.336.3 99339933)( :: fdaf faodor. Aommit faodorsf t

aurujueaipirietfzoo9bdigieaie 933)( :tJ mlbdur.

refm>>3 Nortiem,t:3ieeaiooeafY3mltuzofrfe933Dtuzdusuzfdtosoodfabduaur.Dotrfsoeaf 9/7 ml tuz nudueie nimi nor tuz zfdtoso 6 D/aimdm nor hzdi tiemi imjimtiar.

Mirtrji odftiem ooeafsrfmdobdirimhbjbkfenortuzmirjfdaffaodjfmtfvfmirjiyehzai(6D/aim)tiemtieairmjimtiar.tieeaitiemtijfvguttuzjomtirevfrfe

Y(t)o`iafsoeonudueuz.

zj

Mtdfeoembrueujuodmfsoeafe=


33/90

-77 -


norojzijieailorfetuzmtdfsonorojzijieaimietuzmtdfso:tiemtimotuzmtdfsoeoezijiediafoojo

bdirim oìaffaodfnodor.Qiemtit:3ieeaiY3mltuzvirar(nidiel irt).Iegimtiemtit

ieeaimotuzmbesietrisybeuY/933bdigimtr(nmz.Mbesietrisybetiej).

6 66 6

7 933 933 7

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at at : + :

Doeffrao`friesoyfdafemdfjonudueur.Nuafemdfjoezjeafe

3.36933( )7

tY t Gf: + lfefdzjfdaffaodor.Iegim3( 3)Y t Y: : serirtuyludiersi,

3

933

7G Y: nudueur.Lfefdzjafyfroefmbyir,azfedfjfyipirsim,

( ) tt fYftY 36.3336.397

933)( +: fdaffaodor.

refm>>9Normojyisidjiaaf, zejfjo jomtirdi(x)vfabyju norzfdtoydfabyjij norzfdto

iriseaimombesietrisybeìrmodfabrubrietdbdirimzejfmtfaor.933ldmnorabyjuzfdtoaf 63 l jiaafeoe zea nodoeoybr. Ffr 73 l mojyisid jiaaf 933 l suydi

mirtrdrsi, 93 l > siittf zeybr. Nuei lrf 6 siit sbeueai ef miair zejf

bdigimtr.

zjx tsiitsberizejfyfejiaafjomtirelstfrsoe.Nuaurujaijfvgutzfdtoeoe

mbesietrisybeu 73973

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mbesietrisybeu933

63

933

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zejfjo jiaafjomtirvfnuomombesietrisybeiriseaimoìrmodfbrietnorhzai

zeaoìaffaodaooeflrfirieieao`friesoyfdafemdfj

63 73 0 973

ax x ax xm x m x

at at

: : +

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9/7 ml/D, 6 D/aim Y/933 ml/D, 6 D/aim


34/90

-71 -


0 973

axm at

xx

: +

fdaffaodor.Hfromotirièoetflridoideaeaiosf,

0 973

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m atxx : +

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< < < 76

xx x mt G

x + : : +

+ sbeugueivirdr.NudueieoìafafmvfG

nodoejfjfmtfaor.Iegimvfrodfeomoirtmuddiedirimnuedirnfdordfefnodor.

t : 3 x : 73 l (nidieltimotbpdijzejfjo jiaafjomtir)

t : > x:7393:>3l(>siitsberimidiezejfjo jiaafjomtir

)

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+

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< >3 76m m: :

+

ojaoafemdfj>0

de 3.11 >.8< 76

xt

x:

+ bdirimoìaffaodfnodor.Irtmt : 6siit

sbeueaizejfjo bdirimmidiejomtirx nudueinodor.

>0de 3.11 6 >.8

< 76

de 9.>06 3.>66 99.?8 9>76 76

x

x

x xx l l

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: +

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Nydfgf6siitsbeueaizefejomtir739>:98lbdur.

refm>>>

Wgimd9333Gbdienorgosoj633Gsgimdeainorbaiyinrimdybr93aimomisberigosjoesgimdee ?3 3Gyf at lrdybr. Nidiel ieeaie otonirfe ef miair zijie sberi

gosjoesgimdee033Gyfafgfoeonudueuz.Gfvip;996aimomi

refm>>7Nor sv ofrosoeafmo nise, svee yzfyoeafe iiyi abru oeodaomf irtir. Nu irt

hoarbstitooetfjfdafemdfjobdie lazap :/ ao`friesoyfdafemdfjoydfvfrodor.Nuoìafaf sveeybueduu,losfyfrfmojoovjfsovfzyzfyafeotonirfeafroedomtor.Bmyieuslonobmafroe sv ymsfmdooeoe sz mbeusu bdauu uyludijidirai )(p`: bdauueaie nuao`friesoyfd afemdfjoe oetflrf faodfnodjfso ooe ybuedum odf nise iriseai nor odomoeoe

murudjis lfrfmor.Nu odomosveesmjijbadeafefdaffaodor.Nu oìafyflrfsmji


35/90

-76 -


jbad

:a

apF odftiejdar.Wmjijbadeesinotmidaevfyzfyafmonisegs`r

vfybueduu3idirimnisegeafroedomdfafoojoeofdaffaoeoz.

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apF : sinot bdauueaie,

:

:

333

deFpa

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p

fdaf faodor. Nu oìafafe

ybueduufmfrsfm, Fp

f3: fdaffaodor.Nueuhoarbstitooetfjfdafemdfjoeafyizidj;

F

p

fllaz

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: + : :

vf hfr omo tiriè

abiddblirotjisideirim,

:

F

zlFzp 39de)( nudueur.

refm>>1Nfdordo mbuddirai sodoeaorom nor auviraie rabrudtusueai noroj zijieai lffe s jomtir

( ) ( ) ( ) /xY m I Q r Q r aQ r ar : : odf vfrodor. Nu oìafaf I yzfy idie, m s odftojmitsiysar. >3 gj ipeaimo nor nuhir nbrusu 0 gj midedeai ozbdisybe jidzfjfso odfmipdiejtr. zbdisybe jidzfjfso s odftoj mitsiys m : 3.31 \/jM aor. Nuhire sgimd

>333G,ozbdisybea yzfysgimdosf733Gaor.Nbrueuenorjftrfsoeafesiittfmiynbdigim

s

y

vfnbrugovir

eaimo(r593gj)s

gimd

mai

d

j

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zj

Nbrujfrmfzoeafermiairuzimtimosodoeaoromyzfyidie(vfyiryiripdnorsodoeaoroeyzfy

idie) >rI rD: bdirimoìaffaodfnodfgfoeafe,sbruaivfrodfeafemdfj

r

zbdisybe mided0 gj

93 gj

r : nbru jfrmfzoeafe uzimdm>33

3G af nuhir


36/90

-70 -


( ) >raQ

Y m I Q r m rDar

: :

Bdirimyizdinodor.NuriaiDsodoeaoroenbyuaur.Afemdfjoafomfedfroefiyrpoetflridididj;

( ) ( )

3

3

3.90 j 73 G

3.93 j >33 G

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r Q

ar m ar m aQ aQ

r Y D r Y D

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3.90 3.>697de 73 >33 ( ) ?3.? \/j

3.93Y D

Y D

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(gfvipi)

Iyeafemdfjo,nusf`froetflridoestserdirehfrhielonormbeuj(r)vfnumbeuaimo

sgimdm(Q)bdirimoetflrffafrsfm,

( )3

7

3.93 j >33 G

> 3.31de >.33?3.? 3.93

r Q

r Q

ar raQ Qr

: :

: :

709.00 de(93 ) >33r Q: vfyi >33 709.00 de(93 )Q r:

zbdisybejidzfjfsoooeafmosgimdmaidjiiaimolstfrodjotor.

refm>>6Mtdfsoohjidfaodfnodornoryiyafybdirimnoryzfyftutturudju,aofrugueiosfmtdfsoj

bdienorgosojisdjtr._iyafelfmbeujueaiymfe(x:3)isdnudueiemtdfeoehz 3P osf,

mtdfeoehzeyiyeyfrafotorjfsoef(x)nidbdirimo`iaffaoeoz.

zj

Q(3G)


37/90

-7< -


Efwtbeue>.yisiseilrf xx ij@ : bdauueaiemtdfyfftmofafexabrudtusueaimoeft

muvvft nfdordfejfdoaor. mxjl@x : . Aofr tiri`tie gosjoe ovjfso ooeax

aPPix:

yizdinodfgfoeafe gosjoe hirfmftoeo tiejdiyie ao`friesoyfd afemdfj, xmjlax

aPPj :

bdigimtr.Afomfedfroefiyririmoetflrffafrsfm=

+::

:x xP

P

xP

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xj

mlxPxPx

j

mlx

Paxx

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mlaPP

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>

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33

refm>>0Mtdfsoj bdie nor gosoj yfrafe bdaumi ymsfmtf nudueie nor ebmtiaie odm hzsz bdirim

sfrnfst ajfyf nrimdybr. Gosjf ftmo fafe yfrfmojo muvvfto sinot vf hivi aorfegoeoe

gosjoe hz odf brietd bdauu minud faodaooef lrf, hfrhielo nort ieeai gosjoe nidiel

ebmtiseaie hielo uzimdmti bdauueu vf b ieai hielo hzi hirfmft ftjfmtf bdauueu

nuduz.Gfvip; )9()( / jtmfm

ljtP : , ( )jtmf

m

ljt

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refm>>

>

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tieeai,gosjoe yfrmrfsoeafebdieuzimdxvf hzvbdsue.Nueilrfhirfmftafemdfjo

>>

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)(

.

^x

JjL

at

xaj

+: vfyi

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>

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at

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+: fdaf faodor. Afemdfjoe si yiexf nid

bdauueaiesbdyieeaiaixfnidovjfsoeoemuddiedjisyfroeafbdigimtr.Nueilrf,

+:+:xP

Pax^x

l^

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ax

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3 o`iafsoeafehzeymsfmdomdfafoojo`iafso,

^x

l^l^PP

++:

>>

3> >

> bdirimfdaffaodor.

Gosjoeyfrmrfyflfroaejfjfsoooesrfmdobdirimpbzoto`norhzisihopbdjislfrfmor.Nu

osfiegim 3>3 l^P irtsidiersijjmebdinodor.Nuaurujai

99>36931.089.?>> 03 :: l^P j/s.


38/90

-78 -


refm>>8

jiaafso, vf lonoomomojyisidjiaafeoerfimsoybeusbeugubdujimtiar.Abdiysydii

lr odf,n lr rfimsoybe sbeugu (i+n) lr

vfrjfmtfaor. Nidieltix3 lr vfy3 lr

virar= ho nuduejijimtiar. jiaafso bduuj hzee tfpmojfyf lorjfjo vf

jiaafdfroeoe irpj odf abru brietd bdauu nodoeaooef lrf, bduie jomtire, z(t),

zijiee`bemsoybeubdirimfdaffaoeoz.

zj

t ieeai bduie z(t) lr ofrosoeafmo ee brieni

i

+ , ee brie

ni

n

+ bdigimtr. Mtdf

goesoeafe osfz(t)lr ofrosoeaf )(tzni

i

+ lr, )(tz

ni

n

+ lrnudueigimtr. Abdiysydit

ieeai tfpmojfyf lorjfyfe vf jomtirdir srisydi, )( 3ni

zix

+ vf )( 3

ni

zny

+ bdur.

Nu aurujai irieie ao`friesoyfd afemdfj= ))(( 33

ni

zny

ni

zixm

at

az

+

+: hidoeo idr.

Azfedfjf yipirsim ( )( )zNzIMzyn

nizx

i

ni

ni

nim

at

az:

+

+

+: 33>

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faodor.Nuriai 33>)(y

n

niNx

i

niI

ni

nimM

+:

+:

+:

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9. I5NooezjtmNI

tmNI

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)(9)(

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+:

9)(

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rfegodfroenusbeudirfdafftjfdfrotivsoyffaodor.

refm>>?> lr _ odf 9 lr X tfpmojfyf lorfrfm 7 lr [ bduturjimtiar. 933 lr _ 63 lr X oyogf

mirtrdaeai 93 aimomi sberi 63 lr [ jiaafso bdutuu lzdfejotor. (i) >3 aimomisbeueaimilr[bduur,(n)03lr[fdafftjfm ooeefmiairnfmdfejfdoaor.Gfvip;73

dmfdfroelfdfgfmtfmoeùstihjoedfroeafmuddiedie >NEIEat

aE: ao`friesoyfdafemdfjo

bdaumi minud lrjfmtfaor. Nu afemdfjaf, I vf N pbzoto` dmf sinotdfro, E osf eùsu

lstfrjfmtfaor.Nidielieeaimo(t:3)eùsuE3idirimeùszijiee`bemsoybeubdirim

fdaf faoeoz. INA ooe I:7.97938 N:9.688363ydeaimoeùsdirnudueuz.


39/90

-7? -


refm>79Wfro bdirim nid nbnoe vf aorfetfe murudu nor idtfreito` imj afvrfsoeafe lffe imj

)(tFoât

aoD :+ ao`friesoyfd afemdfjo uyiregi afojfmtfaor. Nuriai o afvrfafe lffe

imj (ijpfr), D nbnoeoe oeamties (Hfery), ^ aorfe (bhj) vf F(t) osf vbdt bdirim

fdfmtrbjbtbr muvvfttor. t:3 ieeai afvrfafe lffe imje o3 vf fdfmtrbjbtbr muvvftoeF(t):F3bdjisaurujueaiafvrfafelffgfmbdieimjo(t)zijiee`bemsoybeubdirimfdaf

faoeoz.

zjFdojozafmo afemdfj noroego jfrtfnfafe doeffr nor ao`friesoyfd afemdfj bdauueaie

abruaieoetflridirpiemuddiedirimzdfnodor.

] Z +: Gatt^to )(9

)( vf Dtât

D

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idirim,

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+:

nudueur. t:3 ieeai o:o3 bdauueaie

yfroefmbyirim^

FoG 33: fdaffaodfrfmao`friesoyfdafemdfjzjeafyizdrsi,

( )Dt^f^

Fto /3 9)( : bdauulrdr.

refm>7>_umirai vfrodfe >>0 ebdu sbruai fdfmtrbjbtbr muvvftoeoe ttF : 9>3soe993)( uyiregiafotoo nodoejfmtfaor.. D:7 hfery,^:96 bhj bdirim vfrodaooef vf t:3 ieeai o:3 ijpfr

bdauueilrfafvrfafelffeimjzijiee`bemsoybeubdirimnudueuz.

zj

Afvrfafe lffe imj )(tFoât

aoD :+ ao`friesoyfd afemdfjoydf tiejd bdauuei lrf

zjirieieafemdfj, toât

aoD :+ 9>3soe993 vfyi t

Do

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at

ao:+ 9>3soe

993bdigimtr.

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ao:+ 9>3soe

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doeffrbdienuafemdfjoetflridirpiejftbauodfzdr.

tat fft 66

)( :: fdaffaodfrfmlfefdzjoìafsoeaf(>.3soe7933

)(9

)( 66 fdaffaodor.etflridodfjoooe

msjooetflridyetfjomuddiedjidar(numsjrfegoyfnrimdjtr).Nuriainoz

( ) +: tnntninif

attnfti

tigbssoesoe

>>


40/90

-13 -


midne muddieigiz. Srbndfjojozaf i:6 vf n:9>3 bdauueaie yfroef yizp niz

siafdftorjfafesberi=

( )( ) +: tt

fattf

tt 9>3gbs>19>3soe

63soe

>

66 fdaffafroz.Abdiysydiirieie

zj=

( )( ) tfGttto 6

> 9>3gbs>19>3soe

6>)( +

+: bdigimtr. Pfrodfe nidiel

affrdfroeafe oetflrid sinoto nfdordfefnodor. t:3 ieeai o:3 ijpfr bdauuei lrf, lfefd

zjafyfroefmbeupGfmodorsf,( )>61>>

+

:G fdaffaodor.Nydfgfirieiezfdzj=

( )( )tfttto 6

> >19>3gbs>19>3soe

6>)( +

+: fmdoeafbdigimtr.

refm>77Wfro nid aorfe vf mbeaiesitrafe bduie nor idtfreito` imj afvrfsoeaf, mbeaiesitraf

afpbdieirim fdfmtrom ym zijie )(tFg

q

at

aq^ :+ uyiregi afoor. ^:93 bhj, g:937

ìria, ttF : 9>3soe933)( bdirim vfrodaooef vf t:3 ieeai q:3 bdauuei lrfmbeaiesitrafafpbdieiefdfmtrom ymeezijiee`bemsoybeubdirimfdaffaoeoz.

zj

Ao`friesoyfdafemdfj,vfrodfedfroeyfroefmbejisydi tqat

aq:+ 9>3soe9339393 7 vfyi

tqat

aq:+ 9>3soe93933 hidoeo idigimtr. Nu osf noroego jfrtfnfafe doeffr nor ao`friesoyfd

afemdfjaor.

tat fft 933933

)( :: ideirimlfefdzjoìafsoeafyfroefmbeursi,

+: Gattfftq tt 9>3soe93)( 933933 fdaffaodor.etflridoidjimzfrf

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ni

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>>

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t 9>3gbs09>3soe6)70>6(>3

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vfrodfembudsiditdrsi,

>> 70>67

6303

+:+

+: GG nudueur.Nydfgfirieiezj=


41/90

-19 -


( )>

933

>70>6

79>3gbs09>3soe6

63

9)(

+

+

+:

tftttq

refm>71refm>71tft:3ieeailffeimjo:6Ibdauueilrfimjezijiediafoojoeofdaf

faoeoz.

zj

Nor egfmo refmtfe ( ) tfGtttq 933>

9>3gbs09>3soe663

9)(

++

: fdaf faodjoto.

at

aqo: bdauueaietrfvidjiodfjoyipdirim,

tfGtt

to 933

>

>

93363

9>3soe39>3gbs033)(

+

+: nudueur. Pfrodfe mbud yfroef

yizdrsi=

G93363

0336

>

+

: vfyi

>70>6

7336933

+

:G fdaffaodor.Nydfgfirieiezfd

zj=

tftt

to 933>>

>

70>6

7336

70>6

9>3soe7039>3gbs733)(

+

+

+: bdigimtr.


42/90

-1> -


Ndj>odfodlodoprbndfjdfr

9. LfefdaffrdfeaorjfsbrudirNoroegojfrtfnfafeao`friesoyfdafemdfjefafjfmtor.

Noroegojfrtfnfafenorao`friesoyfdafemdfj yyy ,, > lonotfrojdfrnudueaurinodor

jo2Noroegojfrtfnfafenorao`friesoyfdafemdfjhieloirtdiraiabruaieoetflridybduydi

zdfnodor.

>. Doeffrnoroegojfrtfnfafeao`friesoyfdafemdfjdfr

>.9.Iiaimoao`friesoyfdafemdfjdfroehielodfronoroegojfrtfnfafedoeffraor;

(i) xyxy soe7 > :+ (n) xfyxy :+ 7> (g) >:+ yxfy (a)

9>> :+ yxyx

(f) xyyyy :+ gbs> (`) xxyyy :+ (l) >>> xyy : (h)

xyy :+

>.>.Iiaimoao`friesoyfdafemdfjdfro,tjodfjiajdirelstfrfrfm,zez.

(i) xxyy >7 : (n) 3>)9( > : yyx (g) >> xyyx :

(a) >)/97( :+ yxy (f) 9>> :+ xyyx (`) xxyy soetie :+

(l)x

fy

xy

x>>:+ (h) >)9( > :++ yxyx (o) xy

xy soe

1:+

>.7.Iiaimonidielaffrprbndfjdfroeo,tjiajdirlstfrfrfm,zez.

(i) 9)3(,3 ::+ yyy (n) 9)(,>gbs> :: yxxyy (g)

1)9(,> :: yfxyy x

(a) 8)>(,1 1> ::+ yxyxy (f) ) 99> : xyyx>.1.Iiaimosbrudiraixonijd,yyonijszafomfeidirimzjyipez.

(i)x

y

ax

ay

:

9

>

, 9)3( :y (n) >)9(,>

:

:

yxf

f

ax

ayy

y

>.6. 3)( :+ yxSy noroego jfrtfnfafe ao`friesoyfd afemdfjoeoe nor zj )(9 xy

bdsue. )(9 xyG oìafsoeoeafiyeafemdfjoezjbdauueulstfroeoz.

>.0.Noroegojfrtfnfafeao`friesoyfdafemdfjdfroeuyludijidir

(i)

Jidthusoieeùsirt yisisefaor2Efafelfrfmotihjoedfrafaihiuylueaur.(n)

îaybimto`mirnbeodfyi tiyoeoeoefsisefaor.Nuo ooeefafeG91uylueaur.

(g)

Noryijuraijdiseedojothzetiejdiyez.Nuhzaijdieeatymsfmdof

nidjar2

(a)

_ftfroegf nfsoeoe nudueauu nor nidm ldeaf, ffr ho nidm ivdiejizsi, hfr yd

nidm siysee omoyf mitdia lzdfejotor. Nidieltimo nidm siyse E3ar.Jithusoie irt yisise muddieirim vf hfr le brtidiji 3.33>E3 iaft nidm

ivdieaeminudfafrfm,nidmsiysezijiee`bemsoybeubdirimfdaffaoeoz.

(f)

Nfdordo nor nimtfro mbdbeosoeafmo nimtfro siysee hfr 7 siit omo mit irtt

nodoejfmtfaor. Jithusoie irt yisise muddieirim nidieltimo nimtfro siysee 1miteiminodjfsoooeefmiairsrflfjfsolfrfmtooeonudueuz.

(`)

_ddmeùsirt hz%3.>bdienordmfeoemiydsberieùsueuomoyfmitdiyigie

nfdordfyoeoz.


43/90

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(l)

Nidieltimo sgimdQo:733G bdie nimraie yipdj mm nor mrf t: 3 ieeai

nuzdu suyi nrimdjtr. Mrfeoe sgimd 9 aimomi sberi >3 3Gyf atef lrf,

Efwtbe sbuji mieueueu muddieirim > aimomi sberi mrfeoe sgimdee ef

bdigienfdordfyoeoz.

(h)

Nortiemti,93mltuzmuddiedirimfdaffaodjo nudueie>33dotrfsidijuri(tuzdusu)

nuduejimtia

r. Qiemi, nor mir

t

r

g

afvrfafymfe, aimomiai 6 dotrf si` su odivffaodormfe,iyehigojaftuzdusuaitiemeidteaimoafdomtfeairnbidtdjimtiar.

73aimomisberitiemtiefmiairtuzmidr.Quzjomtiree9mlimiairajfso ooe

efmiairsrflfjfsolfrfmor.

(o) j mtdfdo nor gosoj nfdordo nor ymsfmdomtfe aurlue hidafymfe sfrnfst ajfyfnrimdjimtiar. Gosjf ftmoyfe hivi aorfegoeoe gosjoe hzydi brietd bdauu

nodoeaooeflrf,gosjoehzvfmbeujueuzijiee`bemsoybeubdirimfdaffaoeoz.

7. Noroegojfrtfnfafedoeffrbdjiyieao`friesoyfdafemdfjdfr

7.9.Afomfedfroefiyr

dinodortoptfmoao`friesoyfdafemdfjdfrozez.(i) 97 +: xyy (n) >)>( > +:+ yyx (g) 3> :+ +yxfy

(a) 7)(xyy : (f) 9++: yxfyxy (`) >9 yyxy :

(l) 9)3(,3> ::+ yxyy (h) 9)>/(,>gbs> :: yxfy y (o) >> 9 yyx : , 3)3( :y

(k) 9)1/(,soe

gbs:: y

yf

yfy

y

x

7.>. Iiaimo ao`friesoyfd afemdfjdfraf afomfe aej yipirim afomfedfroef

iyrdinodoraurujilftoropzez.

(i) ( ) )(9>

yxyxy +++: (n)yx

fyxy +

+: )( (g) 7> +: yxy

(a) >)( yxy : (f)y

x

f

yxfy

)( :

7.7.^yiripdmrfsfdnortiemsuydiabduaur.Qiemtinieeiidieiyiripdnorafdomtfesu

nbidtdigimtr.Qbrofddomieueueilrftiemtfrmfafesuyuebrtidijihz lyP >: bdup

nuriai y lz eef ideie aurujaimo su ymsfmdooaor. Qiemtimo su sfvoyfsoeo zijiee

`bemsoybeu bdirim fdaf faoeoz. Qieme tijijfe nbidinodjfso ooe lfrfmdo zijie vfrfe

oìafyolfdotoroeoz.

7.1._umiraimoprbndfjoyitiybdirimmbeujdieardj ^yiripdvfDnbyueaimosodoeaorom

tiemooetfmrirdiyez.

7.6. Dbkostom eùs irt EnEiat

aE)( : ao`friesoyfd afemdfjoydf tiejdier. Nu

ao`friesoyfdafemdfjoelfefdzjenudueuz.

7.0.Dbkostomeùsirtmieueueilrfnorizaihimirjim,iegimaihilfrfmonorjbafd

EgEnEiat

aE)9()( : ao`friesoyfd afemdfjoydf vfrodor. Nu afemdfjoe lfefd zje

nudueuz.

7.> (n) mxyx >>> :+ (g) Gyxyx :++ 91 >>


44/90

xy

-11 -


7.8.Wuabdunortiemeidteaienorafdomidrsisunbidjiyinidiyigimtr.Iegimtiemtimo

su sfvoyfso atmf noroj zijieai tiemtie nbidie su jomtir ai izidigimtr, abdiysydi

zijiei nidar. WinotIQ mfsotdo nor tieme tinieei idieI3 nymdeafmo nor afdomtfe

norojzijieai miesuhigjo lhMI >3 bdirimvfrodaooeflrf,(M:sinot,h:tieeaimosu

ymsfmdoo) tieme ofrosoeaf nidielti (t:3), H jftrf ymsfmdooeaf su nudueauueuvirsiyirsim,tiemtimosusfvoyfsoeozijiee`bemsoybeubdirimfdaffaoeoz.

7.?.Srbndfj7.8o,tfrsfvrodjo vftinieipAbdiembeommfsotdonortiemooezez.

7.93. Srbndfj 7.?u tfrs fvrodjo, idt vf st tinie ipdir srisydia vfA bdie mfsom mbeo

noojdonortiemooezez.

7.99. Azdfjsfd nor auviraimo tfm nbyutdu s odftojo, .xaQ

Y m Iax

: bdirim vfrodor. Nuriai x

see im(odftoj) abrudtusueu, m s odftoj mitsiyse I auvir idiee vfax

aQ, x

abrudtusueaimosgimdmfojoeo(lriayfeoeo)vfror.Ffrmjfvgutjidzfjfooesinotsfnu

afemdfjauvirofrosoeafmosgimdmprb`odoeo ] Z)(xQQ: nudjimzfrfmbdiygioetflrffaodor.Iegim m lfefdaf sgimde `bemsoybeuaur vf ).9(3 Qmm +: odomosoydf vfrodor. Nuriai

vfm3 sinotdfraor ( 335m ). Nuei lrf sinot norYx s tries`fro ooe auvir ofrosoeafmosgimdmaidjefdaffaoeoz.

7.9>. _itiydi 37< i yipie nor fom azdfj zfroeaf nudueie nor gosoj, fom azdfj

nbyuegi vf yumir abru :3P 9> j/sdom nor odm hzdi `rditdybr. Gosoj odf fom azdfjiriseaimosrtejfmitsiys >6.3: tor.Nueilrf(i) gosojfomazdfjnbyuegihielo

uzimdi loafnodor vf (n) gosoj itda ebmtiyi lfro aeaeaf hz ef bdur2Gfvip; ? j,8.18j/s.

1. Noroegojfrtfnfafehbjbkfetoptfao`friesoyfdafemdfjdfr

1.9.Hbjbkfeao`friesoyfdafemdfjefaor,eisdiedidr.

1.>.Iiaimoao`friesoyfdafemdfjdfroehbjbkfebdupbdjiamdireoegfdfyoeoz.

(i)yx

yxy

+

: (n)yx

xy

+

+:

>

9(g)

xy

xyxy

>7 >: (a)

x

yxy

7> :

1.7.Iiaimohbjbkfe(vfyihbjbkfefoeaorlfefnodor)ao`friesoyfdafemdfjdfrozez.

(i)yx

xy

+: (n)

7

>7 1

y

yxxy

: (g)

yx

yxy

+

: >

(a)xy

yxy

>

0 >> :

(f)yx

yxy

>

17

+

: (`)y

yxxy

>> ++: (l)

>

11>

x

yxyy

:

(h) 3)9(,9 :: yx

yy (o) 3)3(, :

+

: yyx

yxy (k) 3)9(,

>

>:

+

: yyx

yxy

(m) 0)>(,> >>

:+

: yy

yyxxy (d) >)3(,

>

>>

:

: yxy

yxy

(j)7>1>

+ ++: yxyxy (e) yxy +: 7> (b)

yyxy 9> +:


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(p)1

+:

x

yxy (r)

81>

>>

:yx

yxy

6. Qijao`friesoyfdafemdfjdfr

6.9.Qijao`friesoyfdefafjfmtor. xyxyyxu : >>),( `bemsoybeueuetijao`friesoyfdoeoidez.

6.>.Iiaimoao`friesoyfdafemdfjdfroetijao`friesoyfdbdupbdjiamdireoegfdfyoeoz.Qijao`friesoyfdbdiedirzez.

(i) ( ) ( ) 39797 :++ yyx (n) ( ) ( ) 39797 :+ yxy

(g) ) ) 3>>> :+ yfxyxy y (a) 3>> : yxyy(f) ( ) 3gbs>soe :+ yxfxf yy (`) ( ) ( ) 3>)> :+++ yyxyx

(l) ( ) ( ) 3gbssoe >> :+ yyyxx (h)9

9

++

:yx

yxy (o)

9

>> +

:

y

y

fx

xfy

(m)y

x

fyx

xfyxy>> >gbs

>soe>

+: (d) ( ) ( ) 3>> :++ yyxyx

(j) ( ) ( ) 31>> >> :++++ +++ ayfxaxxxffx yxyxyx

6.7.Iiaimozjdfrfsihopbdieao`friesoyfdafemdfjdfrofdaffaoeoz.

(i) 9.>soe),( > :: +yxfyyxyx` (n) 16tie7),( 7 :+: yxyxyx`

6.1.Iiaimonidielaffrprbndfjdfroeoezez

(i) ( ) ( ) 9)3(,39>19> 7> ::++ yyyyx

(n) ( ) ( ) >)9(,39gbssoe7 >7>

::+++ yyyyxxfyx x

(g) ( ) ( ) 3)3(,37>797> ::+++ yyyxyx

(a) ( ) ( ) 3)>/(,3gbssoe7 7> ::+++ yyyfxyfyx xx

(f) 1)3(,9

9>

>

:

+: y

fy

fxy

y

x

(`) 7)>(,9>7

97>:

+

: yyx

yxy

0. IiaimoNfrebuddotopoao`friesoyfdafemdfjdfrozez.

(i) 3)9(,1 :: yyyy (n) 9)3(,1> 7 ::+ yyyy (g) 3)9(,> :: yyx

yy

(a) xyyyx de>:+


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7. NDJ

MEG PFAIHI_MWFMJF^QFNFAFEDEFFÂ@FÎEW_FDAFEMDFJDF^

7.9L^

Noroego jfrtfnfafeao`friesoyfdafemdfjdfr,nor oetflridirpiemuddiedirimsostfjitomnor

yimdijdihfrzijiezdfnodordfr. zdfgfmafemdfjoeoetflridoideinodaoosrfgf,sinotvfyi afomfe mitsiyd bdjis nu auruju afotorjfz. Iegim omoego vfyi aihi ymsfm

jfrtfnfdo afemdfjdfr ooe iye fyo sydfyfjfyoz. em nu afemdfjdfroe zj, nym

brieai, mitsiydire sinot bdjisei vfyi nfdordo irtdir sidiyie trafe afomfe bdjiseinidar. moegovfaihiymsfmjfrtfnfdo ao`friesoyfd afemdfjdfr ooe lfefdnor zjybdu

ybmtur. bu jhfeaosdom prbndfjoeaf sinot mitsiyd omoego jfrtfnfafe ao`friesoyfd

afemdfjdfroydf mirdidr. Nu efafedf nu tr afemdfjdfroe zj ybddire oyo mivrijimlfrfmor.

moegojfrtfnfafedoeffrnorao`friesoyfdafemdfjfelfefdhidaf

)()()( x^yxYyxSy :++

`brjueai vfrodor. NuriaiS,Y vf^,x nijsz afomfedfroef nid `bemsoybedirar. ^(x)

tfrojoofrosoeafyvftrfvdfronuduejiyietjoìafdfrotfjsodfafrvfnuyzafehbjbkfebdjiyie tfroj iae idr. ^(x) :3 osf nu aurujai afemdfj hbjbkfeaor afeor. moego

jfrtfnfafeao`friesoyfdafemdfjdfrozfrmfehbjbkfemsjiyrfdfidjimlfefddomdfaihi

uylueaur. Nueue ooe odm ftipti afemdfjoe si yie s`rj lono hirfmft faodor. Doeffrafemdfjdfriyrgisinotvfafomfemitsiydbdirimaise`dieardrdir.

}7 >> 8 9xy y y x f + : + sinotmitsiyd

}>> 8y xy y x + : afomfemitsiyd

Qfbrfj79zjPirdvfQfmeoo

x94x4x>irideaiS(x),Y(x)vf^(x) xf nid srfmdo `bemsoybedir vfx3nu iridmti nor

ebmtiosf,nuaurujai=

( ) ( ) ( )y S x y Y x y ^ x + + :

ao`friesoyfdafemdfjoeoey(x3):y3 vf y(x3):y3omoiaftnidiel irtesidiyieiridmti

tfm norzj virar. Iegimy(x3):3 vf y(x3):3nidiel irtdire sidiyietfmzj

y:3zjar.

Ao`friesoyfdafemdfjoestieairt`brjaibdjisirttr.refoe=

>> 8 0xy x y : afemdfjo

71y xy

x : hidoeaf yizdjiami stieairt `brji lfdjo

siydjiz.


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refm79

Iiaimoao`friesoyfdafemdfjoetfmnorzjeebdauueuvfnuzjezjiridelstfroeoz.

76 gbs >

9

xy y y x

x

+ :

(6) 7y : vf (6) 9y : zj;Nunornidielaffrprbndfjoaoremhfromombudaiiyenijszafomfeaffroeaf

vfrodjotor. Afemdfj omoego jfrtfnfafe bdauu ooe fe ymsfm jfrtfnf >aor. Afemdfjafy

vfyinueuetrfvdfroeoeirpjssvfdoeffrbdjiyie`bemsoybedirbdjiaeaie,vfrodfe

afemdfj doeffraor. Afemdfjoe si yie s`r afod, abdiysydi hbjbkfe afodaor. Iyrgi

afemdfjstieairt`brjaiar.Nutihdoddfroyiptmtiesberi=

9

7

)( : xx

xp Y(x):6,^(x):gbsx>

yizidj. Imi lrdybr moY(x) vf,^(x) srfmdo `bemsoybedir,S(x) osfx:9af srfmsozaor.

Abdiysydi hfr o`iafeoe af srfmdo bdauu 4x 4 +9 vf 9 4x 4 iridmdireai

zjebdupbdjiairiejidar.Nidielmbudubdiex3:6ebmtis omoegoiridmtiarvf

nuiridmtizjvirdvftfmdoolirietoidteiidej bdur.

Doeffr sinot mitsiyd hbjbkfe ao`friesoyfd afemdfjdfroe mitsiydir zitfe 4x 4 +

irideai srfmdo bdauueaie zj tj x affrdfro ooe lffrdo bdigimtr. Iegim norao`friesoyfd afemdfjo zjf odf nor nidiel affr prbndfjoeo zjf iriseaimo iyrj

lrjfjoz lfrfmor. moego jfrtfnfafe nor ao`friesoyfd afemdfjoe zjeafe g9 vf g>eoe

idinodfgfosbesuzaffrfmirdmsbesuziaftzjvirar.Pfrodfeomombudooenusinotdfrnfdordfeor. Noroego bdisdm nu omo irt iye ebmtiai (iyex3 affro ooe) vfrjfmtor. Nu nozo

nidielaffr prbndfjoef ltrr vf ofrosoeaf x3e yfr ida nor iridmti zje

lirietosovirar.Iegimomoirt`irmdxaffrdfroooevfrodjosf mo norseraffrprbndfjojoz

virafjfmtor._umiraivfrodfeQfbrfj79,nutrnorprbndfjoezjeebdupbdjiambeusueai nor lirieto vfrjfz. Wiafgf g9 : g> fdaf faodfnodfe irtdirai seraffr

prbndfjoeoezjeevirdeaievftfmdooeafeszfaodfnodor.

refoe=> 7> 7 xy y x y x f

+ : y(3):>, (3) 6y : tfmnorzjlirietobdjisemire,

xfxyxyy :+ 7>7'>' y(3):>, (8) 7y : afemdfjo ooe zj tfm bdjiyinodor, hitti honor

zjnuduejiyinodor.

refm7>

Mirird rfkojaf D midedeaimo azdfjsfd nor auvir ofrosoeafmo sgimdm aidj 3y : afemdfjoydfvfrodor.Nuriaiy,xebmtiseaimosgimdtfjsodfaoybr.Pfrodfeafemdfjoelfefd

zjevfiiaimoaurujdirooezfdafemdfjdfroeofdaffaoeoz.

i) y(3):93, (3) 6y :

n) y(3):93,y(D):3

g) (3) 6y : , ( ) 93y D :


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a) (3) 6y : , ( ) 6y D :

zj;zjiridjz Dx3 bdigimtr.Irtiraiomomfzoetflrffafrsfmlfefdzj=

>9 gxgy +: bdur.Nuosffojog9bdienorabruafemdfjoaor.

i. y(3):6 }g9:6y(3):93 }93:6.3+g> ,g>:93

93.6)( +: xxy fdaffaodor.Aofrhonormbudnuirtdirsidiyijiz.Abdiysydivfrodfenidielaffrprbndfjoeoetfmzjar.

n. Winotdfrnfdordfeorsfg>:93,D

g 93

9 : nudueur.Nuaurujai 9393

)( +: xD

xy

Nuaivfrodfembuddirooefdaffaodfnodfgfmtfmzjbdauueaieseraffr

prbndfjotfmnorzjfsihoptor.

g. Pfrodfembuddirooey(3):6 }g9:6y(D):93}g9:93

nudueurmonuojmieszar.Abdiysydivfrodfembuddirooeprbndfjoezjybmtur.

@ozomsfd bdirim prbndfj auvire hfr omo yieeaie s vfrodjfsoeo vf mirird rfkoj

bdujiseelrjfmtfaormonuojmieszar.

a. Pfrodfembuddirooey(3):6 }g9:6y(D):6}g9:6

abdiysydi >6)( gxxy +: fdaf faodor mo nu zj g> yf nid bdauueaie tfmzj afodaor. @ozomsfd bdirim nu prbndfjoe auvire nor tiri`eaie vfrodfe see

aofr tiri`eaie iye hzdi uzimditrda nor auruji mirdm lfdor. Nu osf auvirofrosoeafmosgimdmaidjenudjiooeyftfrdonornodloafodaor.

Diferansiyel Denklemler - Doç.dr. Tahsin Engin Ders Notları

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Transcript of Diferansiyel Denklemler - Doç.dr. Tahsin Engin Ders Notları