diferansiyel denklemler - doç.dr. tahsin engin ders notları
TRANSCRIPT
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9. NDJ
N^EG JF^QFNFAFEA@F^IEW_FDAFEMDFJDF^
9.9.L^
Ao`friesoyfd afemdfjdfr uzue yddirar, aeyiai bu `ozomsfd 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 `ozomsfd bdiy aihi oyo iediji bdiei lftoror.
Nodoj vf jitfjitom rfegodfroef yefdom jitfjitom rftojo uzue srfaor nodoj vf
jhfeaosdom `imdtfdfro iriseai nor iedijizdm mbeusu bdjutur. Nodoj vf jhfeaosdom
`imdtfdfro rfegodfroe tfbrom jitfjitomdf iridiree bm ai oyo bdjia mbeusueaihfj`omordfraor. [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 `irmd ofromdo
afrsdfroe bdujisydi sbeudiejtr. Nydfgf abi nodojdfro vf jhfeaosdom `imdtfdfrorfegodfroef 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,`irmdirvfirtrjdirodfoetflrid
lonotfjfdmbeudirlzafelforjfdfromfsoedomdfefrodor.
9.>.NI[OQFJFDQIEOJDIJIDI
Nor yi ai aihi `izdi `bemsoybeue trfvdfroeo ofrfe afemdfjdfrf ao`friesoyfd afemdfjaoybruz.Aofrnoro`iafydfao`friesoyfdafemdfjnortimj`bemsoybedirodfnuediretrfvdfroiriseaimoodomoyotfjsodfafr.Numivrijodmbdirim90
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faodjfmtfaor.Abdiysydiao`friesoyfdafemdfjoefdaffaodjfsoprbndfjhimmeaiyftfrdonodlosihono bdjiy, prbndfjf aihod bdie afomfedfronfdordfyfnodjfyo,uylue nisotdftorjfdfr vf
virsiyjdiryipinodjfyo,muddiedigim`ozomsfdprfesopvfmieuedirnodjfyovfafaommitdonor
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 o`iaffaodaooeafe
>
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saj)t(ij)t(@ ::
yizdinodor.Azfedfjfyipdrsi
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at
sa>
>
: 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 o`iaf ftjfmtfaor. Ao`friesoyfd afemdfjdfr `ozomsfd 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 `izdi nijd afomfeoe tfm nor afomfef lrf iao trfvdfroeo ofrfeao`friesoyfd afemdfjdfrfIaoAo`friesoyfdAfemdfj (IAA) afeor. Nueue yieeai ofrosoeaf
noryiaiaihi`izdinijdafomfeoe,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> treafeo`iafdfrofrfnodor.
Aihilfefdnoro`iafydfffrnorao`friesoyfdafemdfj
)x(^y)x(`...y)x(`y)x(`y e)>e(
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9)e( :++++
`brjueai o`iaf 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@F^IEW_FDAFEMDFJDF E[JDF
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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 af`i 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`friesoyfdafemdfjozjfmnorafe`izditfmeooenfrinfrmuddiedjiseeyie 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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x>
f>y
fyideirim ao`friesoyfd afemdfjaf yizdrsi 3f>f> x>x> : bdur vf afemdfj
sidier.
refm96Norao`friesoyfdafemdfjoezj>
x9y : o`iafsoeoe 3xyy :+ ao`friesoyfd afemdfjoeoe nor zj bdauueulstfroeoz.Pfrodfezjehfrxaffroooelffrdobdupbdjiaeorafdfyoeoz.
zj
Pfrodfe zj ao`friesoyfd afemdfjaf yizdrsi >x9 3xx9>
x>
>:+
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.
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refm90Norao`friesoyfdafemdfjoelfefdzj
7x>Gxfy x> +: o`iafsoeoe >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,
3 3( ) ( )
hIt
jgQ t Q Q G f
:
bdirim fdaf faodor. Nu o`iafafG nor mfy`o 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 `ozomsfd bdiydi odlodo pfm bm `irmd prbndfj iye ao`friesoyfd
afemdfjdf o`iaf 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,nijszafomfeoenorafe`izdiaffroooenfdordfejosfnuirtdiriserirtdiraoyfgfoz.
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xfyyy xnidielaffrprbndfjo
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xfyyy x seraffrprbndfjo
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lat
za:
>
>
ao`friesoyfdafemdfjoydfnfdordoaor.Gosojodmhzszbdirimnrimdaeaie
3
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:::
tat
aztP yizdinodor. Nor aofr irt osf gosjoe nidielti h ymsfmdooeaf
nuduejisirtar.Aofrnoro`iafydf,
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@F^IEW_FDAFEMDFJDF EAB^VAIEEQFL^ID_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
mfy`osinotfdaffaodor.
refm98AbruaieoetflrisybeodfzjIiaimoao`friesoyfdafemdfjdfroeabruaie oetflridybduydizdpzdfjfyfgfmdfroeo
nfdortoeozvfzdfnodorbdiedirzez.
(9) 37y6y :+
(>) 3x0y > :
(7) 31yy> :
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zj(9)
Nu afemdfj, omoego tfrojoeoe nodoejfyfe `bemsoybeu (yieo nijd afomfeo)
ofrjfsoeafeabdiyzdfjfz.
(>)
Nu afemdfj doeffraor vf trfvdo tfm tfrojo nuduejimti vf aofr tfrojdfraf
nodoejfyfe `bemsoybe y nor irpie vfyi `imtr aurujueai afodaor. Nu efafedf afemdfj
zdfnodor.Afemdfj>.jfrtfnfafebdauueaieirtirai>mfzoetflridid
e
phfrsf`froeafnorsinotoefmdfejfsolfrfmor.
ax
ay
ax
a
ax
yay
>
>
:: yizdinodor.
Afemdfjafyfroefmbeursivfhfromotiri`axodfirpdrsi
3axx0)ax
ay(a
> : afemdfjonudueur.etflrididrsim 97 Gx>axay
: fdaffafroz.Normfz
aihioetflrididjimsurftoydf axGaxx>ay 9> : vfyi >91 GxGx>
9y +: yizdirim
>91
GxGx>
9y ++: irieiezjnudueju bdur.
(7)
Nuafemdfjdoeffrafodaorvfabruaieoetflridybduydizdfjfzlonolreybr.
Iegim aommitdo nor mbetrbd odf yy> tfrojoeoe >y eoe trfvoeafe nimi nor fybdjiaimilrdjfmtfaor.Bhidafafemdfjufmodafazfedfefnodor.
31)y(ax
a > : .Afemdfjoehfromotiri`axodfirpdirim 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>iaftmfy`osinotonuduejimtiarvfnuedirnodjfmsozoegosjoe7ssberimo
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 `irmd nor yfraf
bdigimt. Lfefd zj afemdfjoeo nozoj prbndfjojozf uylue hidf lftorfnodjfm ooe
ofrosoeafmosinotdfroeprbndfjafvfrodfembuddirilrfnfdordfejfsolfrfmor.Numbuddir
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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).
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-9> -
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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, ^iayuj 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 o`iaf faodor.
Nuriaimbrietmitsiysiaeidr.
>.9. Nijd vf nijsz afomfe efaor, nor `bemsoybeai nuedir nornoroeafe eisd iyrt
fafrsoeoz.>.>.Qrfvoelfbjftromiedijeimdiyez.
>.7.Iaovfmsjoao`friesoyfdafemdfjdfririseaief`irmvirar.>.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
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(f)7
x
6
x9
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: , 2
t
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6 :
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t
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>.8.
Iiaimooetflridodfjdfroeoyipez.
(i) ( ) ++ axx6soefx x7> (n)
+6
>
ax
x
6x7de (g) ( )axxt7t>soex >t> ++
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8/10/2019 Diferansiyel Denklemler - Do.dr. Tahsin Engin Ders Notlar
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(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
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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.).
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>. NDJ
N^EG JF^QFNFAFEDEFF^A@F^IEW_FDAFEMDFJDF^
>.9.L^
Sfmbmuyludijiainornymdeafoojhz(noroegotrfv),nunymdemfeaosoefvf
nijsz afomfef nidar. Nu tr prbndfjdfr lfefdaf )y,x(`y : `brjueai o`iaffaodordfr.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 vfhfrnorse`ooe`irmdyetfjdfrlfdotorodjotor.Nundjafnoroegojfrtfnfafe
afemdfjdfroeeisdse`dieardaieditdigim,iraeaiesostfjitomnoryimdijdihfrzijie
zj jjme bdie noroego jfrtfnfafe doeffr afemdfjdfr vf uyludijidir
odfefgfmtor.Iraeaiedoeffrbdjiyietrdfrooevfrodfenorzjirideaizjevirbdupbdjiatirtdigimtr. Nu se`i lorfe afomfedfroef iyrdinodor top, hbjbkfe vf tij ao`friesoyfd
toptfmoafemdfjzjdfrozfroeafaurudigimtr.
>.>.N^EG JF^QFNFA@F^IEW_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(`yfmdoeafo`iaffaodfnodfgfmtor.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.
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>.7.N^EG JF^QFNFDEFF^A@F^IEW_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 o`iaf faodfnodorsf, sriaie nor odfjf aefgfmtor.Nueue ooe afemdfjoe sbd tiri`e 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 tiri`ee ] Z y)x( eoe idj bdinodjfso ooe)x(S)x()x( : irt sidiejidar. Nudjiyi idtjz irpiee s`raie `irmd 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 o`iaf faodfnodor. (>.6) afemdfjoydf tiejdieie `bemsoybei oetflrid irpieaoyfgfoz.(>.9)afemdfjoeoesbdtiri`irtmtfmnortfrojoetrfvofmdoeafo`iaffaodaooeafe
] Z )x(^)x(y)x( : (>.0)yizdirim
+: Gax)x(^)x(y)x( vfyinijdafomfeoyideznrimirimao`friesoyfdafemdfjoelfefdzj,
] Z +: Gax)x(^)x()x(y 9
(>.
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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.(>.
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] Z
9)9(9>
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+:
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7
:+
++
+: G
G, nydfgfirieiezfdzj=
)9(7
)17(6 7
+
++
:xx
xxy bdigimtr.
>.1DEFF^BDJI_IEN^EG JF^QFNFAFEA@F^IEW_FDAFEMDFJDF
Doeffr vf noroego jfrtfnfafe ao`friesoyfd afemdfjdfrdf o`iaf faodfe nidiel affrprbndfjdfro eospftfe zjdfro mbdiy bdie prbndfjdfraor. Nueue efafeo (>.
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>.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.
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8/10/2019 Diferansiyel Denklemler - Do.dr. Tahsin Engin Ders Notlar
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Nu iajdir nor uyludiji odf lrfdoj. refoe xgy: fro iodfsoef aom bdie fro iodfsoeo
nudjim ostfyfdoj. Pfrodfe o`iafafe gx
yyx@ ::),( bdauu iedidjimtiar. Abdiysydi
gax
ay: fdaffaodor.Hfr omoafemdfjafgyffotbdauueilrfnufotdomdfroesbdyiedirai
nornordfroeffotbdjidar,aofrnoro`iafydf=
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
>
77 :
hbjbkfe toptf nor ao`friesoyfd afemdfjaor. em afemdfjoe si yie siafgf veoe
`bemsoybeubdirimo`iaffaodfnodor.
yx
yxy
>
77 : o`iafsoeafy:vxaejuyludiersi,
>7
>
777 99
)(v
vv
v
vxx
xvxy :
:
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ueu ebt ftjfjoz lfrfmor mo nuriai muddieajz hbjbkfe tfrojo, 9. Ndjafmoiedijeaie `irmdar. Sfm bm ao`friesoyfd afemdfjoe hbjbkfe bdup bdjia nisot nor
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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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Gv nudueur. vxy: vfyi vxy :/ bdauu
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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.
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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 +: o`iafdfroao`friesoyfdafemdfjafyizdirim=
3)9(>))(>9( :+++ ayvfyavvayf vv fdaf faodor. Afomfedfroef iyririm azfedfjfyipirsim,
:+
++
3>
>9av
fv
f
y
ay
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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 `brjueaiyizdienoro`iafeoetijao`friesoyfdbdinodjfsoooe afemdfjoe sbd tiri` ),( yxu fmdoeaf nor `bemsoybeue tij ao`friesoyfdo, si tiri`e 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.
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8/10/2019 Diferansiyel Denklemler - Do.dr. Tahsin Engin Ders Notlar
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93),( 7 :+:y
xxyxu
Nizfevfrodfenorao`friesoyfdafemdfjnor oetflridirpieyirajyditijao`friesoyfdhidf
lftorodfnodor. Ffr 3),(),( :+ ayyxEaxyxJ afemdfjo nueaie egfmo yetfjdfrdfzdfjoybrvfiyezijieaiaitijao`friesoyfdafodsf,
),(
),(),(
)(yxE
x
yxE
y
yxJ
xl
: 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
:+:
:+:
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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
99
ayaymyy
axaxhxx
:+:
:+:
( ) ( ) 39119 999999 :+++++ aymhyxaxmhyxFdaffaodfeafemdfjohbjbkfehidflftorfnodjfmooe
3,9391
39 ::
:+: mh
mh
mhbdjidar.Nydfgf afemdfj
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( ) ( ) 31 999999 :++ ayyxaxyx fmdoeafhbjbkfebdur.Nuafemdfjozjfmzfrf=
+:
:
999
99
vaxavxay
vxyaejuyludirsimafemdfj,
3)19()91( 99> :+++ avvxaxv hidoeo idr.Afomfedfroef iyrrsim,
391
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9
9 :+
++ av
v
v
x
ax fdaffafroz.
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avav
v
vx :
++
++
91
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91
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>>9 vfyi
gvvex :+++ )>(tie>991>9de 9>9 fdaf faodor. Niti itiajz afomfedfrafe lfro
aefrsfm,
93
99
9
9
999
999
::
:+:+:
:+:+:
x
y
x
yv
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> > 99 >de | 1 > 9| tie> 9
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
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9
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,
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->0 -
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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
o`iafsoeafyizdrsi,
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 mfy`o 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
tiri`ee 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 tiri`e gyf lrf trfvoeo ididj;
>
93>9
+::+
xggx nudueielfefdzjafyfroefmbeursi=
>)9(
1
9)9(
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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.
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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
>
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.^oggitoao`friesoyfdafemdfjo
Nuafemdfj >( ) ( ) ( )y S x y Y x y ^ x : + + bdirimvfrodorvf S(x)s`raie`irmdaffrdfridr.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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uy S x y Y x y ^ x
u uu
: + + + +
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>9 9 9( ) ( ) ( )y S x y Y x y ^ x: + + nuduruz. (>)
(>)afemdfjo(9)afemdfjoeafyfroefyizdpazfedfjfyipdrsi,
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zfdauruj;
Ffr vfrodfe ^oggito afemdfjoeoe y9 vf y> lono omo zfd zj nodoeoybr ai lfefd zj
ostfeoybrsinuaurujailfefdzjbdirimiiaimoo`iaffdaffaodor.
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Lfefdzj;( )> 9( )9
>
S x y y axy yGf
y y
:
refm>91
>
>
9 13y y y
x x + + : ao`friesoyfdafemdfjoeoenorzfdzj 9
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x: bdauueilrfnu
afemdfjoelfefdzjenudueuz.
zj
9 9 > > >
9 > 9 >u uy y y y
u x u u x u
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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( )
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: : : : :
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6
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x
: + : + : +
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y yu
: +
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xy:
nudueur._umiraimoo`iafafyfroefyizdaeai,
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Gx+ sbeugueiudidr.
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>.6.N^EG 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,`imitnorao`friesoyfdafemdfjoenuyetfjdfroenoroydfzdfnodfgfoeoelirietoso
ybmtur.Nizfenoryiaiomoyetfjyftfrdobdinodormfe,nizfehonoroeoeofyirijia
lononorauruj brtiyi minodor. Hfr omo aurujai ai prbndfjf sostfjitom nor jietm frfvfsoeaf
yimdijyipjim`iyaidbdinodor.
),( 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 `izdi jfrtfnfafe trfv tfrojo nuduejijimtiar.
Iraeaie abruaie oetflrisybe odf zdfnodordooef nimidj. Afemdfjoe si tiri`,nodoejfyfe`bemsoybe,emomovfyiaihi`izdijfrtfnfafetrfvtfrojonuduejijimtiar.
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
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:: de
fdaffafroz.Afemdfjafomfedfroefiyrdinodoraurujilfdjotor.
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www.ffjafrsebtdiro.gbj
Gxffaxxfayf xyxy +:: )9( vfhfromotiri`eabiddblirotjisideirim=( )Gxfy x +: )9(de
>.0.N^EG JF^QFNFAFEDEFF^A@F^IEW_FDAFEMDFJDF EV_LVDIJIDI^O
@ozomtf, noybdbkoaf vf sbsyid nodojdfraf mirdidie bm siyai prbndfjaf nor nymde
afoojhzeenymdemfeaosoydfbrietdbdauulzdfejotor.Aofrnoro`iafydf,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 `izdiysi afooj zijiee srfmdo nor `bemsoybeu
bdirim aedfnodor vf nu efafedf yipdigim hfsipdiji hitis lfefddomdf jimud serdir
ofrosoeafmidr.
refm>96E`usirt;JidthusoieyisisNfdordo zijie pfroybtdireai oesie tbpdudumdiree, hiyvie trdfroeoe, ngfmdfroe vf nimtfrombdbeodfroeafmonimtfrodfroemfeaodfroydfbrietdnoojafirttlzdfejotor.E(t),tieeaimo
siy bdjim zfrf vf t:3ieeaimo siyyE3 idirim, e`usue afooj hz (irt vfyi izidji
hz)=mtfEtE 3)( :
bdigimtr. Nuriai m abuj vf dj hzdir iriseaimo `irm tiejdiyie eft e`us afooj
hzar.refoem:3.396/yd,9333af96domnore`usirtafjfmtor.
refm>90^iaybimto`nbzuejivf^iaybimto`Mirnbe_i QiyoeoSdtbeyuj, ^iayujvfG91 bdirim nodoefeMirnbe ozbtbpu lono niz riaybimto` fdfjfetdfroe
aofr nor fdfjfet vfyi iye fdfjfetoe `irmd 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
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Nu o`iafaf 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
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at
taJ )(odfvfrodfenbzuejihz,jomtirsrfmdoafeJ(t)odfbrietdbdauueaienidirai
nbzuejihzbmymsfmtorvfzijielftomfnuhzafr.Jfvgutnorriaybimto`jiaafeoeyirsee nbzuejis ooe lfjfso lfrfmdo srfyf yir jr srfso aoybruz. Nu aurujai
3>
9)( JtJ : idersi,yumiraimoafemdfj,
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bdur. ^iaybimto` jiaafdfroe yir jrdfro lfdojo dinbrituirdirai ddr. ^iaybimto`
nbzuejieeefjdonoruyludijiidie,G91 ozbtbpueuenbzueujufsiseiaiyieie^iaybMirnbe _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.Aofrnoro`iafydf 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
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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
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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.
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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
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Afemdfjnoroegojfrtfnfafedoeffrnorao`friesoyfdafemdfjbdup 3)3( ::tJ mbuduydinornidielaffrprbndfjoaor.
6)(,36.3)( :: t^tS idejimsurftoydf tat
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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.
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Mtdfeoembrueujuodmfsoeafe=
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norojzijieailorfetuzmtdfsonorojzijieaimietuzmtdfso:tiemtimotuzmtdfsoeoezijiediafoojo
bdirim o`iaffaodfnodor.Qiemtit:3ieeaiY3mltuzvirar(nidiel irt).Iegimtiemtit
ieeaimotuzmbesietrisybeuY/933bdigimtr(nmz.Mbesietrisybetiej).
6 66 6
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3.36933( )7
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3
933
7G Y: nudueur.Lfefdzjafyfroefmbyir,azfedfjfyipirsim,
( ) tt fYftY 36.3336.397
933)( +: fdaffaodor.
refm>>9Normojyisidjiaaf, zejfjo jomtirdi(x)vfabyju norzfdtoydfabyjij norzfdto
iriseaimombesietrisybe`irmodfabrubrietdbdirimzejfmtfaor.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
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zejfjo jiaafjomtirvfnuomombesietrisybeiriseaimo`irmodfbrietnorhzai
zeao`iaffaodaooeflrfirieieao`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
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0 973
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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: :
+
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de 3.11 >.8< 76
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sbeueaizejfjo bdirimmidiejomtirx nudueinodor.
>0de 3.11 6 >.8
< 76
de 9.>06 3.>66 99.?8 9>76 76
x
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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.Nuo`iafaf 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 o`iafyflrfsmji
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jbad
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vfybueduu3idirimnisegeafroedomdfafoojoeofdaffaoeoz.
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>333G,ozbdisybea yzfysgimdosf733Gaor.Nbrueuenorjftrfsoeafesiittfmiynbdigim
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idie) >rI rD: bdirimo`iaffaodfnodfgfoeafe,sbruaivfrodfeafemdfj
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( ) >raQ
Y m I Q r m rDar
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zbdisybejidzfjfsoooeafmosgimdmaidjiiaimolstfrodjotor.
refm>>6Mtdfsoohjidfaodfnodornoryiyafybdirimnoryzfyftutturudju,aofrugueiosfmtdfsoj
bdienorgosojisdjtr._iyafelfmbeujueaiymfe(x:3)isdnudueiemtdfeoehz 3P osf,
mtdfeoehzeyiyeyfrafotorjfsoef(x)nidbdirimo`iaffaoeoz.
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Efwtbeue>.yisiseilrf xx ij@ : bdauueaiemtdfyfftmofafexabrudtusueaimoeft
muvvft nfdordfejfdoaor. mxjl@x : . Aofr tiri`tie gosjoe ovjfso ooeax
aPPix:
yizdinodfgfoeafe gosjoe hirfmftoeo tiejdiyie ao`friesoyfd afemdfj, xmjlax
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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
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Gosjoeyfrmrfyflfroaejfjfsoooesrfmdobdirimpbzoto`norhzisihopbdjislfrfmor.Nu
osfiegim 3>3 l^P irtsidiersijjmebdinodor.Nuaurujai
99>36931.089.?>> 03 :: l^P j/s.
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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
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ni
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goesoeafe osfz(t)lr ofrosoeaf )(tzni
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+ lr, )(tz
ni
n
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ieeai tfpmojfyf lorjfyfe vf jomtirdir srisydi, )( 3ni
zix
+ vf )( 3
ni
zny
+ bdur.
Nu aurujai irieie ao`friesoyfd afemdfj= ))(( 33
ni
zny
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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
dmfdfroelfdfgfmtfmoe`ustihjoedfroeafmuddiedie >NEIEat
aE: ao`friesoyfdafemdfjo
bdaumi minud lrjfmtfaor. Nu afemdfjaf, I vf N pbzoto` dmf sinotdfro, E osf e`usu
lstfrjfmtfaor.Nidielieeaimo(t:3)e`usuE3idirime`uszijiee`bemsoybeubdirim
fdaf faoeoz. INA ooe I:7.97938 N:9.688363ydeaimoe`usdirnudueuz.
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refm>79Wfro bdirim nid nbnoe vf aorfetfe murudu nor idtfreito` imj afvrfsoeafe lffe imj
)(tFo^at
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 Dt^at
D
^
fft /)( ::
idirim,
Dt^Dt^Dt^fG
^
FGat
D
Fffto /33//)( +:
+:
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 )(tFo^at
aoD :+ ao`friesoyfd afemdfjoydf tiejd bdauuei lrf
zjirieieafemdfj, to^at
aoD :+ 9>3soe993 vfyi t
Do
D
^
at
ao:+ 9>3soe
993bdigimtr.
Pfrodfeaffrdfryfroefmbeursiafemdfj, toat
ao:+ 9>3soe
7
9936 bdur.Noroegojfrtfnfafe
doeffrbdienuafemdfjoetflridirpiejftbauodfzdr.
tat fft 66
)( :: fdaffaodfrfmlfefdzjo`iafsoeaf(>.3soe7933
)(9
)( 66 fdaffaodor.etflridodfjoooe
msjooetflridyetfjomuddiedjidar(numsjrfegoyfnrimdjtr).Nuriainoz
( ) +: tnntninif
attnfti
tigbssoesoe
>>
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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
`iria, 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
)( :: ideirimlfefdzjo`iafsoeafyfroefmbeursi,
+: Gattfftq tt 9>3soe93)( 933933 fdaffaodor.etflridoidjimzfrf
( )
+: tnntni
ni
fattnf
titi gbssoesoe
>>
bdauueaiehirfmftdfi:933,n:9>3ideirim
( ) +: ttf
attft
t 9>3gbs09>3soe6)70>6(>3
9>3soe>
933933 nudueur.zj
o`iafsoeafyfroefyizdirim=
( )
+
+: Gtt
fftq
tt 9>3gbs09>3soe6
)70>6(>)(
>
933933 vfyi
( ) tfGtttq 933>
9>3gbs09>3soe663
9)(
++
: fdaffaodor.Gsinotoeonudjimzfrf
vfrodfembudsiditdrsi,
>> 70>67
6303
+:+
+: GG nudueur.Nydfgfirieiezj=
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( )>
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.
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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 o`iafsoeoeafiyeafemdfjoezjbdauueulstfroeoz.
>.0.Noroegojfrtfnfafeao`friesoyfdafemdfjdfroeuyludijidir
(i)
Jidthusoiee`usirt yisisefaor2Efafelfrfmotihjoedfrafaihiuylueaur.(n)
^iaybimto`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.
(`)
_ddme`usirt hz%3.>bdienordmfeoemiydsberie`usueuomoyfmitdiyigie
nfdordfyoeoz.
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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
o`iafyolfdotoroeoz.
7.1._umiraimoprbndfjoyitiybdirimmbeujdieardj ^yiripdvfDnbyueaimosodoeaorom
tiemooetfmrirdiyez.
7.6. Dbkostom e`us irt EnEiat
aE)( : ao`friesoyfd afemdfjoydf tiejdier. Nu
ao`friesoyfdafemdfjoelfefdzjenudueuz.
7.0.Dbkostome`usirtmieueueilrfnorizaihimirjim,iegimaihilfrfmonorjbafd
EgEnEiat
aE)9()( : ao`friesoyfd afemdfjoydf vfrodor. Nu afemdfjoe lfefd zje
nudueuz.
7.> (n) mxyx >>> :+ (g) Gyxyx :++ 91 >>
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xy
-11 -
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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^QFNFAFEDEFF^A@F^IEW_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)
tfrojoofrosoeafyvftrfvdfronuduejiyietjo`iafdfrotfjsodfafrvfnuyzafehbjbkfebdjiyie 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
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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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www.ffjafrsebtdiro.gbj
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.