Diferansiyel Denklemler - Arzu Erdem Ders Notları

8/10/2019 Diferansiyel Denklemler - Arzu Erdem Ders Notlar

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Amj mjhiradsjyi` midbìfìr detàrj

Arzu Irmif

l >33:/>313 Kuz medifj fugidmjs`jb detàr1

1

Baydabàr?

1. Bayìy îlterys- \urviy eh App`jlanì Ada`ysjs>. [j``jaf Neyli adm ^jlgarm MjSrjfa - Iìfidtary mjffiiridtja` iquatjeds adm neudmary vaùi prenìfs9 Akustes >33:


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Ledtidts

@jst eh Hjkuris v

@jst eh ]anìs vjj

Lgaptir 3. Kjrjs 11. Fatifatjbsi` femi` ir 1>. Ikrj ajìsjdjd mjhiradsjyi` midbìfìrj 9

Lgaptir 1. Mjhiradsjyi` midbìf ir vi edàrd lezufìrj ;. Njrjdlj firtinimid AMM 1y+l>

herfudmabj AMM 1:

13. @jdiir AMM >111. Nirdeu``j midbìfj >=1>. ^jllatj Midbìfj >8

171=. Jdtikrasyed L arpad >. @jdiir gefeoid AMM jljd tifi` tierifìr 9;>=. \anjt batsay` gefeoid `jdiir AMM 83>9. Gefeoid e`fayad AMM 8>>8. Laulgy-Iuìr midb ifj ;3

Lgaptir 8. \anjt batsay`Jbjdlj Firtinimid Mjhiradsjyi` Midbìfìrjd Yykuàfaàr ;, , 6 sanjt 6 3

mjhiradsjyi` midbiìfj jì jhami imj`fjstjr.

1.>. Ibeèoj? âmyeabtjh atb urudìr. âmyeabtjvjtj, yubsib atef akr`b` (uradyuf fjdira`j kjnj)iìfidtìrjd br`fas sedulu i`mi imj`jr. Zapay ramyeabtjvjtj bjfya, tp vi dubìir idiroj kjnj aàdàrma lebbu`àds`mr. Habat dubìir idirojdjd idmustrjsi` bu`àdf sed mirili mjbbat kiribtjrjyer. L udbu ramyeabtjh atbfammiìr pepuàsyed alsdmad tig`jbi eùsturfabtamr. âmyeabtjh nezu`fadd fatifatjbsi` jhamisj, nezu`fajì eradt` eàrab jhami imj`jr vi mjhiradsjyi` midb ifj

mY

mt 6 bY, b6 sanjt 43

nuramaYfammidjd biyfit addmabj rayeabtjjb eradd kestirfibtimjr. Nu midbìfjd lezufu

Y(t) 6 Y3ixp (b (t t3))sib`jdmi jhami imj`jr.

1.. bura`-àr eàrab ma amàdmr`fabtamr. Bipìr kizikidìrjd das` garibit ittjkj serusudud livand virfjstjr adlabdimid serusudud livan maga sedra Ka`jìe Ka`jìj vi Diwted tarahdmad virj`fjstjr. Diwtedud yirlibjfjbura`da keri, kudis vi kizikidìr arasdma libjf buvvitj

H 6

r 3. KJ^J\

eàrab jhami imj`fjstjr. Nurama K kidi` yirlibjfj sanjtj f vi F srasyà kizikid vi kudisjd butììrjmjr.Neyìli kizikidìrjd libjfìrj a`tdma kudisjd garibitjdj jgfa` imirsib

fm>x

mt> 6

r. bura` eàrab jhami imj`fjstjr. Nu prenìfjd jdtikra` a`dfas jì i`mi imjìd prenìf mi Bipìrprenìfj eàrab amàdmr`fstr.

1.=. Mudya yabddma sirnist musfi garibitj. Mudya uzirjdmi, yirlibjfjdjd sanjt e`mukudu varsa-yarab sirnist musfi garibitjdj iì aà`f. f 6 sanjt butìdjd akr`k g yubsib`jb t zafad k :71 lf/sd>surtudfi jvfisjdj kestirfib uziri, surtudfi buvvitj

H 6 fkeàrab kestirj`jr vi Diwted midbìfj

m>g

mt> 6 k

eàrab yaz`r. Nudud lezufu

g6 k

> t> + l1t + l>

eàrab nuùdur. Nurama l1,, l> biyfi sanjtìrmjr.

1.9. \ekutfa (sdfa) jljd Diwted femi`j. \ekutfa (sdfa) eày, gayatfzd gir aàddma bu`-àdàd njr jsìfmjr. Gavay sekutfa jsìfìrj, hrd stfa vn. Diwtedud sekutfa buraf eàrab amàdmràdfemi`

m

mt 6b (] t) , b6 sanjt 43

jì jhami imj`jr. Nurama ](t) sekutuàd disdidjd gjlnjr itbj kestirfimjkj slab`btr(meyuf debtas). (t) biyfit addmabj slab`b vit zafad kestirfibtimjr.Njr njdadd b`jfa tarahdmad st`mkd varsaya`f. G(t) jì slab`kd artf eraddA (t) jì slab`kd mikjsjferadd kestirjrsib, yubarmabj midbìfj

mmt

6b (] t) + G(t) + A (t)eàrab muzidìyinj`jrjz.

1.8. Fibadjb tjtrisjf vi \arbalàr. Zaprabàrd ruzkarma gsrts, suyud ma`kaàdfas nu femi`ìrjljd naz fizjbsi` eàyàrmr. Id tifi` sa`df garibitj, njr yiri sanjt as` eàd njr nenjdmid akr njr ljsjfjd jìrjkirj garibitjmjr. Geebi yasasda keri nasàdkl addmad bars taraha eàd garibitj

H1 6 by, b6 sanjteàrab yaz`r. Nuramay yirmikjstjrfiyj kestirjr. \urtudfi buvvitjdj mi

H> 6 ` mymt

eàrab yazanj`jrjz. h(t) jì tepàf ms buvvitj (ruzkar) kestirjrsib, Diwtedud >. bura`da keriH6H1+ H>+ h(t)

vi femi`

fm>y

mt> + `

my

mt + by 6 h(t)

1.;. Asha`tàrd lebfisj. Asha`tàrd pirjyemjb njljfmi lebfisj asakmabj mjhiradsjyi` midbìf jì jhamiimj`jr

m=u

mx= 6h

nurama u yeùd muz pezjsyedudmad lebfisj sedulu i`mi imjìd yir mikjstjrfisj vi h firbizbal buvvitjdjdyekudùkumur.


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1. FA]IFA]JB\I@ FEMI@@I^ vxx+ (a+ n) vt+ anv 6 3,l> 6

1

L@, a6

K

L, n6

^

@

L6 bapasjti, @6 bidmjdj jdmubìfi, ^6 risjs tad s, K6 balab.

1.13. Faxwi`` midbìfj. Iìbtrefakditjb aàd > njìsidi sagjptjr. Iiìbtrjb aàd kestirid vibter viGfakditjb aàd kestirid vibter. Nuda keri femi` njr ljht midbìfmid eùsur?

I

t 6 l ( G) =o, I6 =

G

t 6 l ( I) , G6 3

o vi iìbtrjb abf vi yubìdfi yekudùbàr, l < 1313 sb gzd kestirfibtimjr. Nu midbìf sjstifjdjfizjbti kidi`mi

1

l

I

t 6 lur`G =

l o, mjvI6 =

1

l

G

t 6 lurÌ, mjvG6 3

sib`jdmi kestirj`jr.

1.11. Davjir-\tebis Midbìfj. Zapsbad fammiìrjd abf nu tur midbìfìr jì jhami imj`jr

vt+ (v.) v+1p6 vv

p nasdl, yekudùb, v absbadd gzd kestirfibtimjr.

1.1>. \uàfa(jrrjkatjed) sjstifìrjdjd femi`ìfisj. \uàfa sjstifìrjdjd fatifatjbsi` femi`j

L() t6 (B() x)x+ (B() (z 1))z \()sib`jdmimjr. Nurama teprabtabj dif nasdl, L() teprakd su bapasjtisj, B() gjmre jb jìtbid`jkjd meyuferad, \() baydab hedbsjyedu, t zafad, x yatay ibsid, z mjbiy ibsidj kestirfibtimjr.

1.1


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= 3. KJ^J\

1.19. Hjdadsta fatifatjbsi` femi`. Fatifatjbsi` fidadstabj tifi` la`sfaàr ma`kaàdad steb fiyat-àrmr vi asakmabj midbìf jì jhami imj`jr

ut+1

>A>x>uxx+ Nxux Lu 6 3, A , N , L 6 sanjt.

1.18. Nuyuyid tufer femi`j. Nu femi` `jdiir e`fayad mjhiradsjyi` midbìf jì jhami imj`jr

ut 6 h(u) (ulx)xlt 6 k (l, p)

pt 6 g (u, l) Bpu, l,psrasyà gasta`b` gulridjd bedsadtrasyedu, gulri ms svs, pretiaz (idzjfìrjd parlaàdfasd sakàyadidzjf krunu).

1.1;. Ma`ka midbìfj. ]i` vn. kjnj fammiìr uzirjdmibj ma`kaàdfa garibitj

utt b> (x) u6 H(x, t)jì jhami imj`jr.

1.17. Mjhiradsjyi` Midbìfìrjd ]arjgj. Mjhiradsjyi` midbìfìrj tadfamad vi edàrd lezufìrj gabbdnj`kj sagjnj e`famad edli njraz tarjgjdmid nagsimi`jf. Mjhiradsjyi` midbìfìr bedusu j`b eàrab Jsaal Diwted(18=

1;>;) vi Ketthrjim [j`gi`f @ijndjz(18=81;18) tarahdma 1; yuzy`ma la`s`faya nasàdfstr. Diwted, Jdkj`tirimi nuyuyup, ]rjdjty beìoj- Lafnrjmki mi ikjtjf a`fstr vi 188: ma @ulasjad (tud prehissetùb gjzfitìrjdjdnak` e`muku ) prehiseru e`fustur. Mivrjf yaratad la`sfaàr gisapàfama vi fibadjb prnìfìrmi, 1889y`dma kirlibìsfjstjr. Fatifatjb lafjsa tarahdmad banu` kerfisjdi rakfid, Diwted iìstjrjìr gabbdma lebgassas e`mukudmad la`sfaàrd 187; yi bamar nasfafstr. 187; y`dma leb udù bjtan Sgjèsepgjai Datura`jsSrjdljpja Fatgifatjla na`fstr. Diwted mjhiradsjyi` midbìfìr jì la`sfaàrd yuruturbid,gisapàfamavi fibadjbtibj tifi` ki`jsfiìrj Iuìr tarahdmad sakàdm. Diwted, 1. firtinimid mjhiradsjyi` midbìfìrjmy/mx6 h(x), my/mx6 h(y), vimy/mx6 h(x, y) herfudma sdadmrm. \edrasdma jsi h(x, y)x vi y djd pe`j-defu e`mukudma, sirjìrj bu`àdarab lezuf yedtifj ki`jstjrmj. Diwtedud abtjh la`sfaàr 18:3 àrd nasàrdmased nu`mu vi maga edli i`mi itfjs e`muku sedulàrd yaydàdfas vi muzidìdfisj la`sfaàrd kirlibìstjrmj.

18:8 ma Nrjtjsg Fjdt ti tibrar prehiser e`mu. 1;39 mi seva`yi eàrab jàd imj`mj vi [istfjdstir Anniy mikefu`mu.@ijndjz, >3 yasdma @ijpzjk mi A`tmerh udjvirsjtisjdmi, fièzefi aàddma mebtera la`sfasd tafafàm Gayatneyudla, njrbal aàdmabj la`sfaàr jì fisku` e`mu. Edli`jb`j aàdàr arasdma fatifatjb varmr ludbu >3`j yasàrdma nu aàdma la`sfaàr kirlibìstjrfjstjr. Diwted mad njraz sedra e`fasda rakfid mjhiradsjyi`midbìfìr jì j`kj`j tifi` sedulàra uàsfstr habat 187=ti Diwted mad edli nas`fstr. Fatifatjbsi` detasy-edàr bu`àdfa bedusudma leb jyjmjr vi turiv jljd my/mx vi jdtikra` sifneùdud bu`àdfàr eda ajttjr. 18:1y`dma mikjsbidìri ayrfa yedtifjdj virfjstjr vi gefeoid midbìfìrj, mikjsbidìrjdi ayrànjìd mjhiradsjyi`midbìfìri jdmjrkifjstjr. 1. firtinimid mjhiradsjyi` midbìfìrjd lezufudu 18:= y`dma virfjstjr. Gayatdnjr i`lj kjnj yasafstr vi A`fa bra`jyit ajììrjdi tavsjyiìrmi nuùdfustur. Nu kerivj sayisjdmi, leb saymakizj muzidìfjs vi yazsfaàrd mjkir fatifatjbljìri tasyanj fjstjr. Ezi``jbì mi Nirdeu``j barmisìri. Nujsnjr jkj sayisjdmi pib leb prenìf 1; yuzy`ma lezuìnj`fjstjr.Oaben(189=1;39) vi Oegadd(188;1;=7) Nirdeu``j barmisìr mjhiradsjyi` midbìfìrmi yedtifìr ki`jstjrjp nud-

àrd uykuàfa aàdàrd kidjsìtfjsìrmjr. Oaben, 187; mi Nasi` mi prehiser e`fustur. Oegadd, barmisjdjd1;39 y`dma e`fisjdmid sedra ayd kerivi kitjrj`fjstjr. Gir jbj amafma bavkal bsbadl vi ezi``jbì mi bidmjtartsfaàrdma sbla barstr`ràrm. Zjdi mi gir jbjsj mi fatifatjbti leb edif`j ki`jsfiìri jfza atfsàrmr.Gisapàfaàrd ma yarmf jì fibadjbti mjhiradsjyi` midb if eàrab jhami imjìd prenìfìr jljd lezuf yedtifìrjki`jstjrfjsìrmjr. 18:3 ma Oalen y 6 Ray a mjhiradsjyi` midbìfjdj lezfustur vi fabaìsjdmi j`bmiha jdtikra` tirjfjdi yir virfjstjr. 18:= ti Oegadd my/mx 6 y/ax. mjhiradsjyi` midbìfjdjd lezufudu i`miitfjstjr. Ki`jstjrmjbìrj id edif`j prenìfìrmid njrjsj mi nralgjstelgredi prenìfjmjr.Oegaddd ekù eàd Madji` Nirdeu``j(1;331;7>), maga giduz yidj buru`fus eàd \t. Sitirsnurk abamifjsidkel ittj adlab 1;


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>. IK^J AJ@I\JDJD MJHIÂD\JZI@ MIDB@IF@I^J 9

17 yuzy`d id edif`j fatifatjbljìrjdmid njrjsj mi Oegadd Nirdeu``jdjd ekridljsj eàd vi Nasi` yabdàrdmayasayad @iedgarm Iuìr(1;3;1;7; mi arbamas Madji` Nirdeu``j yj tabjp imirib \t. Sitirsnurk akjrfjstjr. 1;>;1;=1 vi 1;881;7< y`àr arasdma \t. Sitirsnurk abamifjsjdmi 1;=11;88 mi Nir`jd abamifjsjdmila`sfstr. Iuìr nutud zafadd id virf`j fatifatjbljìrjdmi njrjmjr vi tuf la`sfàr tepàfma ;3 mirkjyj

kilir. J`kj aàd fatifatjkjd vi uykuàfadd tuf aàdàrd bapsar. Zasafdd sed 1; y`d ber eàrabkiljrfisjdi rakfid, eìdi bamar la`sfàrd mivaf ittjrfjstjr. Ezi``jbì mi fibadjkj fatifatjbmi leb jyjbu`àdrm. @akradki, Iuìrjd fibadjb uykuàfa`r jljd ada`jzmibj id edif`j la`sfa garibitjd nj`jfjdi uyku-àdm tanjrjdj bu`àdm. Mjkir la`sfaàr jì njr`jbti 1;. Ikrj ajìsjdjd mjhiradsjyi` midbìfìrj

(x, y) muzìfjdmi

y6 h(x, l1, l>,...,ld) (>.1)

ikrj ajìsjdj musudi`jf. Bapa` herfama

(x,y,l1, l>,...,ld) 6 3 (>.>)

sib`jdmi yazanj`jrjz. Nurama l1, l>,...,ld uykud parafitriìrmjr. Ikrjdjd bapa` herfudma bapa` hedbsjyedàrdturivj bura`d uykuàyarab d miha turiv aànj`jrjz.

x+

yy 6 3, (>.

x> + >

>

xy+

>

y >y> +

yy 6 3,

. . .d

xd + . . . +

yy(d) 6 3.

d nj`jdfiyid`j sjstfimi l1, l>,...,ld parafitriìrjdj yeb imirsib,

H

x,y,y, . . . , y(d)

6 3


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LGAS]I^ 1

Mjhiradsjyi` midbìfìr vi edàrd lezufìrj


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7 1. MJHIÂD\JZI@ MIDB@IF@I^ XI ED@A^JD LE_YF@I^J

ya ma

y(d) 6h

x,y,y, y, . . . , y(d1)

mdy

mxd 6h

x,y,

my

mx,m>y

mx>, . . . ,

md1y

mxd1

(=.=)

]adf =.. firtinimid

y 6h(x,y,y)midbìfj jljd y yadj lezuf ikrjsjdjd ikrj`jkj nuùdfa`mr. < vi maga yubsib firtinimid AMM jljd nidzirjkieftrjb yerufàr yebtur.

]adf =.:.

y1 6 k1(x) , y> 6 k>(x) , . . . , yd6 kd (x) (=.8)

hedbsjyedàr

y1 6 h1(x, y1, y>, . . . , yd)y> 6 h>(x, y1, y>, . . . , yd). . .yd6 hd(x, y1, y>, . . . , yd)

(=.;)

sjstifjdj ezmis eàrab sak`yersa, (=.8) hedbsjyedàrda sjstifjd lezufu (jdtikra`j) midjr.

Nu murufma ma lezufìrj bapa` hedbjyed kjnj musudinj`jrjz. (Nbz Yyar=.8).

Yyar =.13. d6 > murufu jljdy1, y> hedbsjyedàr yirjdi nj`jdfiyidy, z hedbsjyedàrd iì aà f. Nuda

keri(=.8) hedbsjyedàr

y6 k1 (x) , z 6 k>(x)

kiefitrjbsi` eàrab njr ikrj tadfàr (< neyutù uzayma). Nu siniptid eturu(=.8) hedbsjyedàr(=.;)sjstifjdjdjdtikra` ikrjsj mjyi amàdmr`r.

Yyar =.11. Kidi` eàrab(=.;) sjstifjdmi nj`jdfiyid hedbsjyed says jì midbìf says isjt e`fab zerudma

mikj`mjr. Habat lezuf tadf aydmr. Zjdi mi lezufud var`k vi tib`jkj gabbdma njr kidi`ìfi yapfa`yz.


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=. ]IFI@ BAXÂF@A^ :

]ierif =.1>. (=.;) sjstifj vi

S(a, n1, n>, . . . , nd) (=.7)

debtasd iì aà f. (=.;) sjstifjdmibj h1, h>, . . . , h d d+ 1 mikjsbid j vi mikjsbidìrj x, y1, y>, . . . , yd e`fabuziri, S debtasdd njr E befsuùkudma, s urib`j vi y1, y>, . . . , yd mikjsbidìrjdi k eri surib`j bsfj turivìrisagjp hedbsjyedàr e`sud. E befsuùkudma(=.;) sjstifjdj vi

k1(a) 6 n1, k> (a) 6 n>, . . . , kd(a) 6 nd (=.:)

besuùdu (nasàdkl besuù) sakàyadk1(x) , k> (x) , . . . , kd (x) hedbsjyedàr varmr vi nu hedbsjyedàr tibtjr.

]adf =.1 mikirj jì j`kj`j jsi sdr mikir prenìfjmidjr.

Erdib =.1=.

y + y 6 3,y (1) 6

6 9

sdr mikir pren ifjmjr (\MS)- (neudmary vaùi pren if NXS).

Yyar =.18. Yyar =.13i keri h1, h>, . . . , h d hedbsjyedàr E befsuùkudma yubarmabj besuàr sak`yersa

(=.;) sjstifjdjdS debtasdmad kilid njr tib jdtikra` ikrjsj varmr.

Ezi` murufma

h(x, y) ,h

y(x, y) (=.13)

hedbsjyedàrS debtasddE befsuùkudma surib`j jsi

y 6h(x, y)midbìfjdjd njr tib jdtikra` ikrjsj fivluttur vi nu ikrj S debtasdmad kilir.

Yyar =.1;. Ikir girx Ra, nV , y1, y> Rl, mV jljd|h(x, y1) h(x, y>)| B|y1 y>| (=.11)

isjtsjz`jkjdj sakàyalab sibj`mi pezjtjh njrBsays nuùdanj`jrsih(x, y).hedbsjyeduy mikjsbidjdi k eri^

(a x n, l y m) ne`kisjdmi @jplgjtz besuùdu sakàr midjr. Ezi` eàrabh(x, y).hedbsjyeduy mikjsbidjdikeri^ ne`kisjdmi t urivj fivlut vi hy

B (=.1>)besuù sakàdyer jsi(=.11) besuù sakàdr. Habat tirsj mekru mikj`mjr. Zadjy mikjsbidjdi k eri turivj fivlute`fayanj`jr habat (=.11) besuùdu sakàyad hedbsjyedàr ma varmr. Asakmabj erdib nududà j`kj`jmjr.

Erdib =.17. h(x, y) 6 |y| hedbsjyeduy mikjsbidjdi k eri bsfj turivj yebtur adlab @jpslgjtz besuùdu sakàr.Kirlibtid

||y1| |y>| | |y1 y>| , B6 1sakàdr.


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=. ]IFI@ BAXÂF@A^ 11

midbìfjdjd kidi` lezufu mikj`mjr. L udbu

l1ixp (x + l>) 6 l1ixp (x)ixp(l>) 6 Bixp x

eàrab yazanj`jrjz.

Yyar =.>:. Kidi` murufma

H

x,y,y, y, . . . , y(d)

6 3 (=.18)

midbìfj jljd kidi` jdtikra`mid nagsitfib leb fufbud mikj`mjr. L udbu lezufud tib`jkj serusudud edli`jbìlivap adfas kiribfibtimjr. Erdikjd

y> x=y> 6 3 (=.1;)midbìfj asakmabj mikirìr jljd sakàdr?

y 6x>y, y 6 x>yIkir ni``j njr ne`kimid vi ya nasàdkl besu`àrdmad nagsimjyersab, (=.18) midbìfjdjd kidi` lezufudmid nag-sitfib fufbudmur.

]adf =., 3) debtas neyudla

y6 1

>;(x >), 3) , (>, 3) debtaàrdma njì AMM djd jdtikra` ikrjsjmjr. L udbu nu debtaàrma(=.>3) hedbsjyedumx

my 6 y

x

AMM yj sakàr.


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LGAS]I^ >

Njrjdlj firtinimid AMM

9. y 6h(x) herfudmabj midbìfìrh(x) virjìd J ara`kdma surib`j njr hedbsjyed e`fab uziri

y 6h(x)AMM djd kidi` lezufu

y6 h(x) mxsib`jdmimjr vi ni`jrsjz jdtikra` biyfi sanjtj jlirjr.y (x3) 6 y3

nasàdkl besuù virj`mjkjdmi lezuf

y6 y3+

xx3

h(t) mt

sib`jdmimjr.

Erdib 9.1.

y6

AMM djd lezufudu nuùduz.

L ezuf

y 6

mx y 6 x< + l

Erdib 9.>.

y 6

y (;

NMS djd lezufudu nuùduz.L ezuf

y6 mx y 6 x< + l >; 6 >; + l l6 3 y 6 x. NJ^JDLJ FI^]INIMID AMM

Erdib 9.=.

y 6 8x + 1y (>) 6 3

NMS djd lezufudu nuùduz.L ezuf

y 6

8x + 1mx y6 x< + x + l 3 6 7 1> +l l6 >3 y6 x< + x + >3

Erdib 9.9.

y 6 ixp (x)y (3) 6 1, y (3) 6 1, y (3) 6 y 6

(ixp (x) + l1x + l>) mx y 6 ixp(x) + l1 x

>

> + l>x + l< 6 1 + l1

y 6 ixp(x) + >x>

8. y 6h(y) herfudmabj midbìfìrh(y) 6 3 surib`j njr hedbsjyed e`fab uziri (Nbz. Yyar=.

AMM djd lezufu

x61

y + l

ya ma

y6 1

l x


21/133

;. MIKJ\BID@I^JDI AZ^J@ANJ@ID AMM 19

mjr. Ikir lezuf(

y 1 6

mx

y 1 6 x + l y6 1 + (x + l)>

;. Mikjsbidìrjdi ayrànjìd AMM

]ierif ;.1. h(x) , Ra, nV ara`kdma s urib`j, k (y) , Rl, mV ara`kdma s urib`j hedbsjyedàr vik (y) 6 3 e`sud.

y 6 h(x)k (y)

AMM djd lezufu h(x) mx6

k (y) my

herfudmamr. Ikir lezuf(x3, y3)debtasdmad kiljyersa

xx3

h(x) mx6 yy3

k (y) my

Yyar ;.>.

y 6h(x) k (y)midbìfj jljdmi nidzir tierif kilir`jmjr. L ezuf

h(x) mx6

1

k (y)my

sib`jdmimjr vi lezuf(x3, y3) debtasdmad kiljyersa xx3

h(x) mx6

yy3

1

k (y)my


22/133

18 >. NJ^JDLJ FI^]INIMID AMM

mjr. Ikirk (y3) 6 jsi lezuf

y6 y3

mr.

Erdib ;.

1

y> 6x les x sjd x + lviy (3) 6 1 besuùdmadl 6 1/> i`mi imj`jr. Neyìli lezuf

y6 1

>x les x >sjd x + 1 .

y (3) 6 +1 besuùdmad meày pezjtjh beb a`dfstr. Ayd midbìfjd y (3) 6 3 nasàdkl besu` u lezufuy 6 3 mr.

Erdib ;.=.

y 6 yx < , x 6 y

(y> + =)my 6 3

>

x + >+

1

x + >y

y> + =my 6 3


23/133

;. MIKJ\BID@I^JDI AZ^J@ANJ@ID AMM 1;

midbìfjdjd lezufu

`d (x + >)>

+ `d (x + + => 6 `d l`d(x + >)

>(x + + =)> 6 `d l(x + >)

>(x + + =)> 6 l

(x + >)>

(x + + =>

Erdib ;.8.

x (y+ 1)> mx +

x> + 1

yiymy 6 3

midbìfjdj lezuduz.

L ezuf Midbìfj

x

(x> + 1)mx +

yiy

(y+ 1)> my6 3

sib`jdmi yazanj`jrjz. Nuda keri x

(x> + 1)mx +

yiy

(y+ 1)>my 6 3

1

>`d

x> + 1

+

(y+ 1) iy

(y+ 1)> my

iy

(y+ 1)>my 6

iy 6umy

(y+1)> 6mv v6 1(y+1)

l

1

>`d

x> + 1

+

iy

(y+ 1)my

i

y

(y+ 1)+

iy

(y+ 1)my

6 l

midbìfjdjd lezufu1

>`d

x> + 1

+ iy

(y+ 1)6l

Erdib ;.;.

my

mx6

x

y (1 x>)1/>midbìfjdj lezuduz.

L ezuf Midbìfj

ymy x

(1 x>)1/> mx6 3sib`jdmi yazanj`jrjz. Nuda keri

ymy

x

(1 x>)1/>mx 6 3

1

>y> +

1 x>1/> 6 l

midbìfjdjd lezufu1

>y> +

1 x>1/> 6l


24/133


25/133

:. y 6 ha1x+n1y+l1a>x+n>y+l>

HE^FYDMABJ AMM 1:

L ezuf Midbìf gefeoid AMM mjr. u6 y/x y 6 xu y6 u + xu jhamisjdj midbìfmi yirjdi yazarsab

y 6

y+ x +

y x

y+ x y x

u + xu 6 xu + x + xu xxu + x xu x

u + xu 6

u + 1 +

u 1

u + 1 u 1

u + xu 6

u + 1 +

u 1>

u + 1 (u 1)

6 u + 1 + u 1 + >u> 1

> 6u +

u> 1

xu 6

u> 1

muu> 1 6

mx

x ().jdt

`d

|u +u> 1| 6 `d x + `d l u +u> 1 6lx u6y/x

y

x+

yx

> 1 6 lx

Erdib 7.8.

+ :xy+ 9y>

mx 8x> + =xy my6 3, y (>) 6 8 NMS djd lezufudu nuùduz.L ezuf Midbìf gefeoid AMM mjr. u6 y/x y 6 xu y6 u + xu jhamisjdj midbìfmi yirjdi yazarsab

y 6 + :xy+ 9y>

8x> + =xy

u +xu 6 + :ux> + 9u>x>

8x>

+ =ux>

u +xu 6

< + :u + 9u>

8 + =u xu 6 < + :u + 9u

> 8u =u>8 + =u

xu 6 + ) 6 8 (: + : + 6 >l l6 :

>

+

6 :

>x

:. y 6ha1x+n1y+l1a>x+n>y+l>

herfudmabj AMM

a1 n1a> n> 6 3 (:.1)

e`fab uziri

y 6h

a1x + n1y+ l1a>x + n>y+ l>

(:.>)

AMM djd lezufu jljd

x6 W+ g, y6 Z + b (:.


26/133

>3 >. NJ^JDLJ FI^]INIMID AMM

mikjsbid mikjstjrfisj yap`r. Nudu (:.>) midbìfjdmi yaza`f?

mZ

mW 6h

a1W+ n1Z + a1g + n1b+ l1a>W+ n>Z + a>g + n>b+ l>

(:.=)

mjhiradsjyi` midbìfjdmia1g + n1b+ l1 6 3 (:.9)

a>g + n>b+ l> 6 3

eàlab sibj`mi silirsibmZ

mW 6h

a1W+ n1Z

a>W+ n>Z

(:.8)

midbìfjdj i`mi imirjz bj nu midbìf gefeoid midbìfmjr vi gefeoid midbìfìr jljd kilir`j eàd lezufyedtifj bu`àd`r.Ikir (:.1) mitirfjdadt shr jsi yadja>x + n>y6 b (a1x + n1y) jsi (:.>) midbìfj beàyla lezuùr.

z6 a1x + n1y a>x + n>y6 bzyazarab

z 6a1+ n1y y 6z

a1

n1jhamiìrjdj (:.>)midbìfjdmi yazarsab

zn1

6a1n1

+ h

z+ l1bz + l>

mz

a1+ n1h z+l1bz+l>

6mxmikjsbidìrjdi ayrànjìd AMM i`mi imirjz.

Erdib :.1.

y 6 >x y+ :x

AMM djd lezufudu nuùduz.L ezuf > 11 < 6 3

e`mukudmad (:.9) midbìfjdjd lezirsib g 69, b 61 nuùruz. x 6 W 9, y 6 Z 1 yazmkfzma vimidbìfmi yirjdi yazmkfzma

mZ

mW 6

>W ZW z1

>z+ 1

z+ mz 6

mW

W

1>

`d

> >z+ 6 `d (W) + `d l `d

> >z+ 6 > ` d (lW) 6 `d 1l>W>

6 `d l1W>

, l1 6 1

l>

> >z+ 6 l1W>

,

z 6 ZW kirj medusufudu viW6x + 9, Z 6y + 1 kirj medusufudu yaparsab

>W> >WZ + 6l1


27/133

13. @JDII^ AMM >1

ya ma> (x + 9)

> > (x + 9) (y+ 1) + < (y+ 1)> 6l1lezufudu i`mi imirjz.

Erdib :.>.(>x + x + x+ z medusufudu bu`àdarab z 6 > + ) /< i`mi imirjz.Nudàr midbìfmi bu`àdmkfzma

z >z+ 1

z 6 < (z+ 1)>z+ 1

+ >

z 6 z 1>z+ 1

mikjsbidìrjdi ayrànjìd AMM i`mi imirjz. >z+ 1

z 1 mz 6

mx> +

z+ < `d(z 1) 6 x + l

lezufudu i`mi imirjz. z 6 >x + (>x + x + )midbìfjdi mi (13.1) midbìfjdi bars`b kiìd gefe oid `jdiir midbìf midjr.Gefeoid tirjfj gefe oid AMM jì barstr``fafa`mr. Nuda rakfid `jtiraturmi nu tirjf bu`àd`fabtamr.

a (x) , n (x) J ara`kdma surib`j hedbsjyedàr e`sud. Yyar =.>=j bu` adarab (13.1) vi (13.>) AMM djd tiblezufudmid nagsiminj`jrjz. Edli`jbì (13.>) midbìfj mikjsbidìrjdi ayrànjìd AMMmjr vi lezuf

my

y 6

a (x) mx

`d (Ly) 6

a (x) mx, Ly 43

Ly 6 ixp

a (x) mx

,

y 6 l ixp

a (x) mx

, L6 1/l (13.


28/133

>> >. NJ^JDLJ FI^]INIMID AMM

\jfmj (13.1) midbìfjdjd kidi` lezufudu i`mi imi`jf. L ezufu parafitriìrjd mikjsjfj eàrab amàdmràdyedtifj bu`àdarab nuà`f. (13.1) jd kidi` lezufudu (13.xy6 x< (13.;)

AMM midbìfjdjd lezufudu nuùduz.L ezuf (13.1) midbìfjdi keria (x) 6 >x, n (x) 6x< mjr. Nuda keri (13.8) herfuda keri

y 6 i

>xmx

xxmx)mx + l1

y 6 ix

>

x

mx + l1

y 6 ix

>

x>xix

>

mx+ l1

,

x> 6u, xix>

mx6 mv

>xmx6 mu, v6 1> ix>

y 6 ix> 1>

x>ix> 1> >xix>mx+ l1

y 6 ix>

1

>x>ix

> 1>

ix>

+ l1

y 6

1

>

x> 1+ l1ix> (13.7)

vi (13.;)midbìfjdjd lezufu(13.7) sib`jdmimjr.

Yyar 13.>. (13.1)midbìfjy3 6 3nas adkl besuùdu sakàsa njì(13.8)lezufu tibtjr. Erdikjd(13.;)midbìf

lezufu(3, 3) debtasdmad kiljyersa(13.7) a keri lezuf

y61

> x> 1

+

1

>ix

>


29/133

13. @JDII^ AMM >x + 1

x

y6 i>x

AMM midbìfjdjd lezufudu nuùduz.

L ezuf (13.1) midbìfjdi keria (x) 6 >x+1x , n (x) 6 ixp(>x) mjr. Nuda keri (13.8) herfuda keri

y 6 i >x+1

x mx

ixp(>x) i(

>x+1

x mx)mx + l1

y 6

i>x

x

ixp(>x)ixp(>x) xmx + l1

y 6

i>x

x

x>

> + l1

,

y 6 xi>x

> + l1

i>x

x (13.1x

> + l1

i>x

x

sib`jdmimjr.

Erdib 13.9. x> + 1

mymx

+ =xy 6 x

y (>) 6 1.

AMM midbìfjdjd lezufudu nuùduz.


30/133

>= >. NJ^JDLJ FI^]INIMID AMM

L ezuf (13.1) midbìfjdi keria (x) 6 =x(x>+1) , n (x) 6 x

(x>+1) mjr. Nuda keri (13.8) herfuda keri

y 6 i

=x

(x>+1)mx

x

(x> + 1)i

=x

(x>+1)mx

mx + l1

y 6

x> + 1> x

(x> + 1)

x> + 1

>mx + l1

y 6

x> + 1

> x==

+x>

> + l1

,

y(>) 6 1 1 6 (9)>

18

= +

=

>+ l1

l1 6 1: (13.1=)

vi midbìfjdjd lezufu

y6

x> + 1>x=

= +

x>

> + 1:

sib`jdmimjr.

11. Nirdeu``j midbìfj

y + a(x)y6 n(x)yd (11.1)herfudmabj AMM ìri Nirdeu``j AMM midjr.a (x) , n (x) J ara`kdma surib`j hedbsjyedàr e`sud. d 6 3 ya ma d 6 1 murufu jljd srasyà (13.1) vi (13.>)`jdiir AMM yj i`mi imirjz. Nuramad saysdd 3 vi 1 mid harb` murufàrd iì aàlakz. (11.1) midbìfjdj yd

yi ne`mukufuzmiyyd

+ a(x)

yd1 6n(x) (11.>)

midbìfjdj i`mi imirjz.

z6 1

yd1 6yd+1

medusufudu bu`àdmkfzma

z 6 (d + 1) ydy yyd

6 z

(d + 1)nuùruz. Nu jhamiìrj (11.>) midbìfjdmi yazarsab

z(d + 1)+ a(x)z 6 n(x) (11.x

mz

(z 1) 6 >xmx

`d (z 1) 6 x> + l


31/133

11. NI^DEY@@J MIDB@IFJ >9

viz 6 y> medusufudmid

`d

1 y>

y>

6 x> + l

(11.=) midbìfjdjd lezufumur.

Erdib 11.>.

xmy

mx+ y6 >x8y=

AMM lezufu nuùduz.L ezuf midb ifjxy = ya neì`jf?

y=my

mx+ x1y< 6 >x9

i`mi imirjz.

z 6 y< mzmx

6 l (x) 6 >x< + l1

z6 >x< + l1x< y6 z1/< 6 >x< + l11/x< + l1

1/ mxx

z 6 lx> z 6 l (x) x>

lx> >xx

l (x) x> 6 =x

l 6 =x< l (x) 6 x= + l1 z 6 x> + l1x> y6 z 1/= 6

x> + l1x

>1/= > 6 (1 + l1)1/= l1 6 19

viy= 6x> + 19x>


32/133


midbìfjdjd lezufumur.

1>. ^jllatj Midbìfj

y 6a (x) y> + n (x) y+ l (x) (1>.1)herudmabj midb ifìri ^jllatj midb ifìrj midjr. a (x) 6 3 murufudma (1>.1) midbìfj `jdiir midbìfmjr vil (x) 6 3 murufudma (1>.1) midbìfj Nirdeu``j midbìfjmjr.Kidi` eàrab ^jllatj midb ifjdj ada`jtjb eàrab lezfib fufbud mikj`mjr. Ikir (1>.1) midbìfjdjd njr lezufunj`jdjyersa kidi` lezufu i`mi itfib fufbudmur? y1(x) (1>.1) midbìfjdjd njr lezufu e`sud, yadj

y1 6 a (x) y>1+ n (x) y1+ l (x) (1>.>)

yaz`r.y6 y1+ z (1>...1) midbìfjdmi bu`àdrsab

y1+ z 6 a (x) (y1+ z)> + n (x) (y1+ z) + l (x)y1+ z

6 a (x) y>1+ >a (x) y1z+ a (x) z

> + n (x) y1+ n (x) z+ l (x)

(1>.>) mid haymaàdrsabz 6 (>a (x) y1+ n (x)) z+ a (x) z> (1>.=)

Nirdeuj``j midbìfjdj i`mi imirjz.

Erdib 1>.1.

xy 6 =x> =x (1>.9)AMM jljd njr ezi` lezufy1 6 Ax + N e`fab uziri kidi` lezufudu nuùduz.

L ezuf Ezi` lezuf y16 Ax +N (1>.9) ma yirjdi yazarsab

xA + >ANx + N> 6 =x> =xjhamisjdmid

A6 >, N6 3i`mi imirjz. Neyìli midbìfjd njr ezi` lezufu

y1 6 >x

mjr. (1>.1) midbìfjdi keri (1>.9) ma

a (x) 6 1x

, n (x) 6 .x + z

(1>.=) midbìfj

z 6

= +

Nirdeu``j midbìfjdi medusur. (11.1) Nirdeu``j midb ifjdjd jhamisjdi keri

a (x) 6 =

mjr. Nuda keri

u61

zmedusufudmid

u +

= ;

vi lezufud (13.8) herfuda keri

u (x) 6 i

( .1. ^jllatj midbìfjdjd ezi` murufàr.1>.1.1. A..

y 6ay> + ny+ lturudmibj midbìfìr. Nuramaa, n, lsanjtìrmjr. Midbìf Neùf8 vi ; tibj yedtifìr bu`àdàrab lezuùr.

1>.1.>. N..

y 6ay> + nx>

turudmibj midbìfìr. Nuramaa, n biyfi sanjtìrmjr.

y61

z

medusufu jì

y 6 zz>

nuùdur. Nu jhamiìrj midbìfmi yirjdi yazmkfzma

zz>

6 a

z>+

n

x>

vi

z 6 a n z

x

>gefeoid midbìfj i`mi imj`jr vi Neùf7 tibj yedtif jì lezuf i`mi imj`jr.


34/133


1


35/133

1:

e`mukudmad (1

H(x, y) 6 x> y> mx + (y) H(x, y) 6

x + (y) (1xy+ (y) 6 y> >xy

(y) 6 y> (y) 6 y


L ezuf

>x les y+ y

y 6

x< x> sjd y yx

6 >x sjd y

e`mukudmad (1x les y+ y

mx + (y)

H(x, y) 6 x> les y+ x sjd y y

(y) 6 y (y) 6 y>

> + l

sed (y) jhamisjdj midbìfjdmi yirjdi yazarsab kidi` lezufu

H(x, y) 6 x> les y+ x

>

+ l

sib`jdmi i`mi imirjz. \ed eàrab nasàdkl besuùdmad,

3 6 =>

+ l l6 >

x> les y+ x

> 6 >

lezufmur.

Erdib 1x >ixy sjd>x + >x) mx + (xixy les>x


36/133

. NJ^JDLJ FI^]INIMID AMM

L ezuf

(yixy les>x >ixy sjd>x + >x)y

6(xixy les>x x >x sjd>x + xy les>x)

e`mukudmad (1ixy sjd>x + >x

H

y 6 k (x, y) 6 xixy les>x <

H(x, y) 6

(xixy les>x x) ixy x) ixy x >ixy sjd>x + >x

y (les>x) ixy

>(sjd>x) ixy +

(x) 6yixy les>x

>ixy sjd>x + >x

(x) 6 >x (x) 6 x> + lsed (x) jhamisjdj midb ifjdmi yirjdi yazarsab kidi` lezufu

H(x, y) 6 (les >x) ixy + lsib`jdmi i`mi imirjz.

1=. Jdtikrasyed L arpad

Ikir (1) midbìfj taf mjhiradsjyi` midbìf mikj` jsi yubarmabj yedtif uykuàdafaz. Neyì njr murufma(1) midbìfjdj taf AMM yapalab sibj`mi shrmad harb` njr hedbsjyed jì larpfa`yz. Zadj,

f (x, y) h(x, y) mx + f (x, y) k (x, y) my6 3 (1=.1)

midbìfj taf AMM eàlab sibj`mi f (x, y)6 3 hedbsjyedu nu`fa`yz.

]adf 1=.1. Ikir (1) midbìfj taf mjhiradsjyi` midbìf mikj` habat (1=.1) midbìfj taf mjhiradsjyi`midbìf jsif (x, y) hedbsjyeduda(1) midbìfjdjd jdtikrasyed larpad (jdtikratjed halter) midjr.

Erdib 1=.>.

mx+

>x + y

my6 3

AMM taf mikj`mjr. L udbu

h(x, y) 6 , k (x, y) 6 >x + y h

y 6 < + 7xy,

k

x6 > + 8xy

h

y 6

k

xhabatf (x, y) 6 x>y jljd

y> + =xx + 1>x 6

(fk)

x

besuùdu sakàmkdmad tafmr. Nu murufmaf (x, y) 6 x>y jdtikrasyed larpadmr.

Yyar 1=.


37/133

1=. JD]IKÂ\ZED L A^SADJ )

i`mi imirjz.

Kidi`mi jdtikrasyed larpadd nu`fab e`mubla zermur. Nu yuzmid (1=.>) jhamisjdj bu`àdarab f (x, y) jdti-krasyed larpad samilix vi samili y djd hedbsjyedu e`muku ezi` murufàrma jdliìyilikjz.

1=.1. J.Muruf. f (x, y) jdtikrasyed larpad samili x jd njr hedbsjyedu e`sud. Nu murufma (1=.>)jhamisjdmi fy 6 3 mr vi (1=.>)jhamisjdmid

k (x, y)f

x 6

h

y k

x

f

mf

f 6

hy kx

k (x, y)

mx (1=.)jhamisjdmi fx 6 3 mr vi (1=.>)jhamisjdmid

h(x, y)fy

6

h

y k

x

f

mf

f

6

hy kx

h(x, y)

my

(1=.9)

`d f 6

hy kx

h(x, y)

my

f (x) 6 ixp

hy kx

h(x, y)

my

(1=.8)Yyar 1=.9. (1=.9) midbìfjdmimy jd batsaysy jd njr hedbsjyedu e`fa`mr.

Erdib 1=.8.(x y) mx my6 3


L ezuf

h(x, y) 6 (x y) , k (x, y) 6 1 h

y 6 1, k

x6 3

h

y 6 k

x

e`mukudmad midbìf taf AMM mikj`mjr. Nuda keri jdtikrasyed larpadd nuà`f. Edli f (x, y) jdtikrasyedlarpad samilix jd njr hedbsjyedu e`fa murufudu jdliìyi`jf?

hy kx

k (x, y)

6 1


38/133

>. NJ^JDLJ FI^]INIMID AMM

vi samilix jd njr hedbsjyedumur. (1=.=) lezufudmid jdtikrasyed larpad

f (x) 6 ixp

hy kx

k (x, y)

mx

6 ixp

mx

6 ix

eàrab nuùdur. Nuda keri

ix (x y) mx ixmy6 3 (1=.;)midbìfj taf AMM mjr?

h(x, y) 6 ix (x y) , k (x, y) 6 ix h

y 6 ix, k

x6 ix

h

y 6

k

x

vi lezufu

Hx

6 h(x, y) 6 ix (x y)

H(x, y) 6

(ix (x y)) mx+ (y) (=:) vi (91) deù herfu`ìrmid

H(x, y) 6 ix (x 1) yix + (y) (1=.7)H

y 6 k (x, y) ix + (y) 6 ix

(y) 6 3 (y) 6 lsed (y) jhamisjdj (1=.7) midbìfjdmi yirjdi yazarsab (1=.;) midbìfjdjd kidi` lezufudu

H(x, y) 6 ix (x 1) yix + l H(x, y) 6 ix (x

y

1) + l

sib`jdmi i`mi imirjz.

Erdib 1=.;.

ymx + (< +


39/133

1=. JD]IKÂ\ZED L A^SADJ

ymy

6 i> `d y 6i`dy

>

6y>

eàrab nuùdur. Nuda keriy (< +


40/133


jdtikra` larpadmr. Nuda keri

+ y+


41/133

1=. JD]IKÂ\ZED L A^SADJ ) herfuùdi keri

k (x, y)f

x h(x, y) f

y 6

h

y k

x

f

h

yk

x 6 =xy+ > f

x 6 3,(>xy> + y) f

y 6 (=xy+ >) f

mf

f 6

>

ymy

`d f 6 > `d y f 6 y>

jdtikra` larpadmr. Nuda keri

(>xy> + y)y>mx + (>y< x)y>my6 3midbìfj taf AMM mjr. h(x, y) 6 (>xy> + y)y>, k (x, y) 6 (>y< x)y>

hy

6 (>xy> + y)y>y

6 1y>

,

k

x 6

(>y< x)y>x

6 1y>

h

y 6

k

x

H

y 6 k (x, y) 6 (>y< x)y>

H(x, y) 6

(>y< x)y> my

6 y> +x

y

+ (x)

H

x 6

y> + xy + (x)

x

6 1

y+ (x)

6 h(x, y) 6 (>xy> + y)y> 1

y+ (x) 6 (>xy> + y)y>

(x) 6 >x (x) 6

(>x) mx

(x) 6 x> + l H(x, y) 6 y> +

x

y + (x) H(x, y) 6 y> +

x

y+ x> + l6 3

lezufmur.

1=.


42/133


]adf 1=.13. Ikirk (x3, y3) 6 3, h(x3, y3) 6 3

besuù sakàdyersa(x3, y3) debtasda(1. Ikir AMM jd lezufu tibj` debtamad kilirsi nu tibj` debtaya muk uf debtas midjr. Erdib1=.11mi(3, 3) debtas njr muk uf (demi) debtasmr.

Erdib 1=.1 6l

eàrab i`mi imj`jr. L ezufmi(3, 3) debtasdmad kilfifibtimjr. vi

y6 3, x6 3

lezufud asjfptetàr mikj`mjr.

]adf 1=.18.Ikir AMM jd tibj` debtasdmabj lezufu asjfptetàr mikj` jsi vi lezuf tibj` debtamad kilfjyersa

nu tibj` debtaya firbiz (lidtir) debtas midjr. Erdib1=.19mi(3, 3) debtas njr firbiz debtasmr.


43/133

LGAS]I^ )

eàrab i`mi imj`jr. Ezi` eàrab mjb yerudki murufu jljd

`jf

>

y tad 1 +y tad 61y

i`mi imj`jr. Nuda keri(19.>) jhamisjdmid

h

x,y,

1y

6 3

sib`jdi medusur.

Yyar 19.>. Xirjìd ikrjdjd mjhiradsjyi` midbìfj

S(x, y)mx + Q(x, y)my6 3 (19.


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< 7 6l>, l 4 3 (19.9)

ikj ajìsj, aydE firbiz`j vil yarlap` majriìrmjr. Nu ikrjìri mjb eàd ikrj ajìsjdj nuùduz.

L ezuf (19.9) jhamisjdmi mjhiradsjyi` a`rsab

>xmx + >ymy 6 3 (19.8)

AMM i`mi imirjz. (19.x, Q (x, y) 6 >y

vi (19.=) i keri(19.9) ikrj ajìsjdi mjb ikrj ajìsj

>ymx >xmy6 3 (19.;)eàrab i`mi imirjz. AMM jd lezufu

>ymx >xmy6 3 myy

6 mx

x y6 fx

jì jhami ijìd mekru ajììrjmjr.

18. Fibadjb prenìfìrj

Nj`jdmjkj kjnj Diwtedud jbjdlj garibit badudu,

H 6 fa6 fmv

mt 6f

m>x

mt>

f 6 wk

sib`jdmi jhami imj`jr. Nurama f ljsfjd butìsj, Hljsfi itbj imid sanjt buvvit, a ljsfjd garibitjdjdJvfisj, vgz vi kjmjìd ye` x , w jsi ljsfjd akr`k jì jhami imj`jr.

Erdib 18.1. Butìsjf eàd njr ljsfjd yirmid e`mubla yubsibti nuùdad njr debtamad J`b gzsz eàrab sirnistmusfiyi nrab`yer. Ljsfi itbj imid yirlibjf buvvitj sanjt vi gava mjridljdjd ljsfjd gz ì eradt` e`mukubanu` imj`mjkjdi keri. girgadkj njr t addma ljsfjd nasàdkl debtasdmad gadkj uzab`bà e`mukudu vi e admagadkj gzà garibit itfibti e`mukudu nuùduz.


45/133

18. FIBADJB SÊN@IF@I^J k

b> + l l6 g f

>k

b>

x 6 fk

b

t +

f

bi

bf t

+ g f>k

b>

Erdib 18.>. Yzirjdmibj medadfàr jì njr`jbti njr parasutludud akr`k 183 `n (`jnri) virj`fjstjr. Saras utal`famad edli gava mjridlj gzd yarsda isjttjr. Sarasut 9 sd sedra al`mkdma gava mjridlj gzd barisjdjd9/7j bamar eùyer. Sarasut al`famad vi al`mbtad sedra parasutludud gzd nuùduz. (Zirlibjfj jvfisjdjk 6 hjt/sd> eàrab a`dz.)

L ezuf Sarasut al`famad edlibj gzd nuà`f.

w6 183`n f6 wk

6 183/ 6 9


46/133

= 3 .burafdmad

H 6 fmv

mt 6H1+ H> f mv

mt6 183 v

> 9 mv

mt6 183 v

>

9

mv

mt 6

3

v

> mv

3 v 6 mt

13 `d(3 v) 6 1

13 t `d l 3 v 6 li 113 t v (t) 6 3 li 113t

eàrab nuùruz. Sarasutludud j`b zafadma gzdd e`fafasdmad

v (3) 6 3

besuùdmad

3 6 3 l l6 3 v (t) 6 3

1 i 113t

parasut al`famad edlibj parasutludud gzd virfibtimjr. 9 sd. sedrabj gz

9>.93->.9-9

133

93

3

-93

-133

-193

->33

x

y

x

y

v (9) 6 3

1 i 1139

eàrab nuùruz. Nuda keri 9 sd sedrabj mjhiradsjyi` midbìfj

fmv

mt 6 183 9v

>

7

v (9) 6 3

1 i 1139

6 1>9.:1

? eàrab virinj`jrjz. Midbìfjd lezufu

mv

18> v> 6 mt

7 1

118 v + 118 + v mv6 mt7 `d

18 + v

18 v 6 =t + `d l 18 + v

18 v 6li=t v 1 + li=t6 18 li=t 1

v 6 18

li=t 1

(1 + li=t) v (9) 6 3

1 i 1139

3

1 i 1139

6 18

li>3 1

(1 + li>3) l6 i>3

v (t) 6 18

1131=>

i>3+=t 11 + 1131=> i

>3+=t , t 9

eàrab nuùdur.


47/133

18. FIBADJB SÊN@IF@I^J =1

) H>, surtudfi buvvitj, akr`kd mjbiy njìsidj jì surtudfi buvvitjdjd larpfmr?

H> 6 D6

1

==7les+ H

fmv

mt 6 >= 8

< v

>v (3) 6 3

mjhiradsjyi` midbìfjdj lezi`jf?

mv=7 1>< v 6

mt

< v

6 t

< v 6 li t< v (3) 6 3

=7 1>

< 6 l v 6

=7 1>

)


48/133

= > = e`fas murufudma sed debtamabj gzdd di e`muku serusuda kiì`jf?

mx

mt 6 v6

=7 1>

.8h t/ sil

1;. Erad Srenìfìrj

Erad prenìfìrj, njr fizjbsi` nuyubùbtibj, njrjf zafad jljdmi fiymada kiìd mikjsfi eàrab tarjh imj`jr.

Erdib 1;.1. âmyeabtjh njr iìfidtjd nezudfa gz iìfidtjd fivlut fjbtar jì mekru eradt`mr. âyeabtjhiìfidtjd erojda` akr`kdd yars 1933 y` jljdmi mak`mkda k eri

(1) =933 y` sedra, ramyeabtjh iìfidtjd akr`kd nuùduz.(>) Erojda` akr`kdd %13 uda varfas jljd di bamar zafad kilfisj kiribtjkjdj nuùduz.

L ezuf x ramyeabtjh iìfidtjd fjbtard kestirfib uziri mx/mt ramyebatjh nezu`fadd eradd ni`jrtjr. Nu

erad fivlut fjbtar jì mekru eradt` e`mukudmad

mx

mt 6 bx

fjbtarma aàfa e`mukudmad jsarit dikatjhmjr.

x (3) 6x3

vi 1933 y` sedrabj nezu`fa fjbtar

x (1933) 6x3

>mjr.

mx

mt 6 bx,x (3) 6x3

mjhiradsjyi` midbìfjdjd lezufu

x6 x3ibt x (1933) 6 x3

>e`mukuda keri

x3>

6 x3i1933b b6 `d >

1933 3.333=8 x6 x3i3.333=8t

ib 6 1>1/1933 x6 x3 ibt 6x3 1

>t/1933

(1) t6 =933 jljd nezudfa

x6 x3

1

>

=933/19336

x37

Nu jsi =933 sidi sedudma nezudfadd 1/7 viya %1>.9 e`mu kudu seyìr(>)

x6x313

x313

6x3

1

>

t/1933 t6 1933 `d13

`d > =:79

y` sedra kirlibìsjr.


49/133

17. SESY@A\ZED SÊN@IF@I^J =. Diwtedud sekufa burafda keri, sekuyad njr ljsjfjd slab`k, ljsfjd slab`k vi ljsfjbapàyad ertafd sanjt slab`kdd harb jì eradt`mr. t6 3mabj93 sanjt slab`b` njr ertafmabj slab`k73

e`mukuda k eri vi9sadjyi sedra ljsfjd slab`k;3i must uk udi keri ljsfjd slab`kdd, t yi nak` hedbsjyedudunuùduz vi 13 sd sedrabj slab`kd nuùduz.

L ezuf x,t zafadmabj ljsfjd slab`kd kestirsjd nuda keri mjhiradsjyi` midbìf

mx

mt 6 b (x 93)

x (3) 6 73

vi

x (9) 6 ;3

besu`àrfz varmr.

mx

x 93 6 bmt `d (x 93) 6bt + `d l x 93 6 libt x6 93 + libt x (3) 6 73

x 137x> 6mt 133

1

x+

138

1 138x

mx6 mt

`d x `d 1 138x 6 t133

+ `d l x1 138x 6li

t/133

x6 lit/133

1 + l138it/133, x (1:73) 6 139 139 6 li

1:73/133

1 + l138i1:73/133

l 6 138

:i::9

x (>333) 6138

:i::9

i>333/133

1 + 138

:i::9

138i>333/133 11:=:9


50/133

= = . sbbd lezufu jljd?

>.139 6

138

:i::9

it/133

1 + 138

:i::9

138it/133 i1:.7t/133 6 =

: t >381

1:. Barsf Srenìfìrj

Nurama barsfàrmabj eradàr iì a`dr. \ fammisj ni``j eradàrmabj barsfàr jlirjp, barsf, barstrl jìsanjt tutuùyer. x jì \ fammisjdjd fjbtard kestirjrsib mx/mt x mibj mikjsjfjd t yi keri eradd virjr. Nudakeri mjhiradsjyi` femi`j

mx

mt 6kjrjs virjìrj lbs virjìrj

eàrab virj`jr.

Erdib 1:.1. Nasàdklta 93 ka`(ka`èd, `n `jb tuz jlirid lezu`fus tuzù su ibìdfibtimjr. Barstrl jì barsf gefeoid tutuùrbid,nasba njr vadamad ayd gzà barsf nesa`t`fabtamr. Nuda keri, tadbta girgadkj t zafaddmabj tuz eradddimjr5

L ezuf Kjrjs virjìrj? < ka`/sd `jb gzà tadbd jljdi > `n `jb tuz jlirid lezu`fus tuzù sukjrjs virjìrj6 >(`n/ka`)


51/133

>3. I@IB]^JB MIXÎ SÊN@IF@I^J =9

L ezuf Krjsìr?

kjrjs virjìrj6 >(`n/ka`)9(ka`/sd) 6 13 `n/sd

kjrjs gz 9 vi lbs gz < e`mukuda keri tepàf gz

9 < 6 >vi neyìli girgadkj zafadmabj tuzù su fjbtar, nasàdklta ma 93 e`mukudmad

93+ >t

vi tuz bedsadtrasyedu jsix

93+ >teàrab virj`jr. Neyìli

ljbjs virjìrj6 x

93 + >t(`n/ka`)t x6 13

`jdiir midbìfjdj i`mi imirjz.

mx

mt +

tx 6 3 mx

x 6 tmt `d x6 `d(93 + >t) + `d l

x 6 l (93 + >t) x6 l (t) (93 + >t)

gefeoid e`fayad midbìfjd lezufu e`sud.

l (93 + >t) t)9/> + t

l (t) (93 + >t)

6 13

l 6 13(93 + >t) l6

13 (93 + >t) mt6 7

> (t + >9)

9/>+ l1

x6 7> (t + >9)9/> + l1 (93 + >t) x (3) 6 3besuùdmad

3 6

7

> (>9)9/>

+ l1

(93)

l6 >9333

>

x 6

7

> (t + >9)9/> >9333

>

(93 + >t)

sedul eàrab nuùruz.

>3. Iìbtrjb Mivri Srenìfìrj

Id nasjt iìbtrjb mivriìrj, oidirater viya pj` kjnj iìbtrjb baydak vi idirojyj bu`àdad njr rizjster (erdikjdiìbtrjb afpuù) (risjstadli) nuùdad njr sirj mivrimjr. Ikir mukfi bapat`rsa njr Jabf rizjsteri mekruabalab vi njr ve`tao musfisjdi sinip eùlabtr. Zadj rizjsterud jbj uludmabj petadsjyi` harb` eùlabtr. Nupetadsjyi` harb viya ve`tao mususu jsi Xe`tfitri midjìd njr i`t jì e`luìnj jr. Ibitrjb mivriìrjdmibj nasjt njrbura` Bjrlgeffi bura` eàrab amaàdmr`r, Nu buraà keri, iìbtrjb mivrisjdmibj tuf ve`taoàrd tepàf, tepàfbuvviti isjttjr. ]epàf buvvitj I(t) jì kestirjrsib (ifh-iìltrefetjvi herli)

X@+ X + XL6I(t)

^ rizjster (rijstadli), L bapasjter (lapaljter), J jdmubter (dmulter). J6J(t) iìbtrjb mivrisjdmibj abf viq6 q(t) bapasjtermibj adj iìbtrjb yubudu kestirfib uziri

q 6 J

sib`jdmi njr nakdt fivluttur. Egf badududa keri rizjster uzirjdmibj ve`tao musubùku abf jì mekrueradt`mr?

X 6 ^J


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= 8 .firtinimid midbìfj ^L@ midbìfj eàrab ama`dmr`r.Jdmubterud e`famk murufma mivriyi^Lmivrisj midjr vi midbìf

XL+ X 6 I(t) 1L

q+ ^J6 I(t) J6q

^q+ 1

Lq 6 I(t)

eàd 1. firtinimid AMM i`mi imirjz.


53/133

>3. I@IB]^JB MIXÎ SÊN@IF@I^J =;

Erdib >3.1. \ibj`mi virjìd ^Lmivrisjdmi (a) edli \ adagtardd 1 bedufuda kitjrj`fisj murufudma (n)\ adagtardd > bedufuda kitjrj`fisj murufudma mivrimid kilid abf sjmmitjdj nuùduz.

I(t)

\

1

>

^

L

L ezuf (a) ikir adagtar njr murufudma jsi ifh-iìltrefetjvi herli I(t) mjr vi midbìf

^q+ 1

Lq6 I(t) q+ 1

^Lq6

I(t)

^

`jdiir midbìfmjr vi lezuf

q 6 i

1^L mt

I(t)

^ i

1^L mtmt + l1

q 6 I(t)

^ 1

^Lq6 J

J 6

I(t)

^ 1

^Li t^L I(t)^ i t^L mt + l1(n) sbb jljd jsi I(t) 6 3 murufumur vi midbìfjd lezufudmi yirjdi yazmkfzma

J6 l1^L

i t^L

eàrab nuùruz. Nuramabj - jsaritj abfd tirs yedi mekru kjttjkjdj kestirjr.

Erdib >3.>. >3 ùb njr mjridl ti`j, 3.31H(Haram) `b njr bapasjtads eàd bedmadsater, ifh(iìltrefetjviherli)sj=3i>t+>3i=t `jb njr uritili sirj eàrab nakàdyer. Nasàdkl zafadma yubu e`famkda k eri girgadkjnjr admabj yubudu nuùduz5

L ezuf

^6 >3, L6 3.31H, I(t) 6 =3i>t

+ >3i=t

virjìrj vi midbìf?

q+ 1

^Lq 6

I(t)

^q(3) 6 3

`jdiir midbìfj jljd edli gefeoid midbìfj iì aà`f?

mq

q 6 1

^Lmt6 9mt q6 li9t

vi gefeoid e`fayad midbìfjd lezufu

q6 l (t) i9t


54/133

= 7 t + >3i=t

l6 =3i3it

l (t) 6

=3

) eàrab nuùruz. ^s6 ^1+ ^> jsitepàf risjstads virjr.

X 6 J(^1+ ^>) J6 X^1+ ^>

X1 6 J.^1 6 ^1

^1+ ^>X,

X> 6 J.^> 6 ^>

^1+ ^>X

L. besuùdma jsi

X 6 X 6 J1^1 6 J>^> J6 J1+ J>J 6 X

1

^1+

1

^>

X 6 ^1^>

^1+ ^>J

J1 6 X

^16

1

^1

^1^>^1+ ^>

J6 ^>

^1+ ^>J

J> 6 X

^>6

1

^>

^1^>^1+ ^>

J6 ^1

^1+ ^>J

sedulàrd i`mi imirjz.

Erdib >3.


55/133

>3. I@IB]^JB MIX^I S^EN@IF@I^J =:

JL6LmXmt

, Jr 6X

^>, J6

1

^1(X X )

J6 JL+ J 1^1 (X X ) 6 LmXmt + X^>

mXmt

+

1

L^>+

1

L^1

X 6

X

^1


56/133


57/133

LGAS]I^ =

1. firtinimid yubsib mirili`j AMM

1.firtinimid yubsib firtini`j AMM kidi` herfu

H(x,y,y) 6 3 (19.1)

sib`jdmimjr. Kidi` eàrab

y 6p

eàrab kestirj`jr. Nuda keri (19.1) midbìfj

H(x,y,p) 6 3 (19.>)

sib`jdmi jhami imj`jr.

Yyar 19.=. (19.1) midbìfjdj lezfib kidi`mi leb beày mikj`mjr. Xi gatta lezufud e`fas ma kiribfiz.

Nudud jljd id edif`j sedulàr nurama ni`jrtilikjz.

18. y6 h(x, p) herfudmabj AMM

(19.1) midbìfjy6 h(x, p) (18.1)

herfudma e`sud.

my

mx6p my6 pmx

vi (18.1) midbìfjdmid

h

xmx +

h

pmp 6 my 6 pmx mx i neìrsib

h

x+

h

p

mp

mx 6 p (18.>)

(18.>) midbìfj p vi x jd midbìfjmjr vi Neùf> mibj lezYf yedtifìrj jì lezuf i`mi imjìnj`jr.

Erdib 18.1.

y> 1 + y> 6 3 (18.


58/133

9> =. 1. FI^]INIMID ZYB\IB MIÎLI@J AMM

eàrab ma yazanj`jrjz. (18.>) jhamisjdmid

h(x, p) 6

1 p>h

x

+ h

p

mp

mx

6 p

p

1 p>mp

mx 6 p

p

1 +

11 p>

mp

mx

6 3

1 + 1

1 p>mp

mx 6 3, p6 3

11 p>

mp 6 mx, mymx

6 3

arlsjdp 6 x + l, y6 l p 6 sjd (

x + l) , y6 l

y 6 les (x + l) , y6 l(18.=) AMM djd lezufìrjmjr.

1;. x6 h(y, p) herfudmabj AMM

(19.1) midbìfjx6 h(y, p) (1;.1)

herfudma e`sud.mx

my 6

1mymx

61

p

vi (1;.1) midbìfjdmid

h

y+

h

p

mp

my 6

mx

my 6

1

ph

y+

h

p

mp

my 6

1

p (1;.>)

(1;.>) midbìfj p vi y jd midbìfjmjr vi Neùf> mibj lezuf yedtifìrj jì lezuf i`mi imjìnj`jr.

Erdib 1;.1.

x 1 + y> 6 3 (1;.) jhamisjdmid

h(x, p) 6 1 p>h

y+

h

p

mp

my 6

1

p

>p mpmy

6 1

p >p>mp6 my

>) lezufudmiyirjdi yazarsab leuzufud parafitrjb kestirjfj eàrab

y6 (p) x + (p) (17.9)

i`mi imirjz.

Erdib 17.1.

y6 >xy + y> ya may 6 >xp+p> (17.8)AMM djd lezufudu nuùduz.

L ezuf (17.1) jhamisjdi keri

(p) 6 >p, (p) 6 p>

vi (17.

p >p x + >p

p >p 6 >

px >

`jdiir midbìfjdj i`mi imirjz. (13.1) jhamisjdmid

a (p) 6>

p, n (p) 6 >

vi (13.8) lezufudmid

x 6 i p

pmp

>i

>i`dp>mp+ l16 p>

>p>mp + l

6 p>

>

p1

> (17.7)

i`mi imirjz. (17.;) vi (17.7) , (17.8) d kidi` lezufudud parafitrjb midbìfjmjr.


60/133

9= =. 1. FI^]INIMID ZYB\IB MIÎLI@J AMM

1:. Làjraut Midbìfj

@akradki midbìfjdmi (p) 6 p murufudma

y6 xy

+ (y) viyay 6 px + (p) (1:.1)

turudmibj midbìfìri Làjraut midbìfj midjr. (1:.1) midbìfjdj x i keri mjhiradsjyi`jdj a`rsab

p 6 p + xmp

mx+ (p)mp

mx

mp

mx(x + (p)) 6 3 mp

mx6 3

p 6 l (1:.>)

lezufudu i`mi imirjz. Nu (1:.>) yj (1:.1) midbìfjdmi bu`àdrsab (1:.1) midbìfjdjd kidi` lezufudu

y6 lx + (l) (1:.) jhamisjdi keri

(p) 6 p

p 1 (1:.:)vi (1:.>) medusufudmid

p6 l (1:.13)

i`mi imirjz. Nu (1:.13) yj (1:.7) midbìfjdmi bu`àdrsab (1:.;) midbìfjdjd kidi` lezufudu

y 6 lx + l

l 1 (1:.11)i`mi imirjz. Ikir (1:.=) sakàdrsa

x + (p) 6 x +p

(p

1)

p

p

(p 1)> 6x p

(p 1)> 6 3 mp

(p 1)> 6 xmx

mp

(p 1)> 6

xmx (8) jhamisjdmid

1p 1 6

x> + l

>

p 6 1 >x> + l

(1:.1>)

i`mi imirjz. Nu (1:.1>) yj (1:.7) midbìfjdmi bu`àdrsab (1:.;) midbìfjdjd kidi` lezufudu

y 6

1 >

x> + l

x + 1 +

1

1 >x>+l

1

(1:.1


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1:. L@AJ^AY] MIDB@IFJ 99

i`mi imirjz.


62/133


63/133

LGAS]I^ 9

Zubsib Firtinimid @jdiir AMM

>1. Kjrjs

]adf >1.1. d. jdlj firtinimid gefeoid e`fayad njr `jdiir AMM kidi` eàrab

a3(x)y(d) + a1(x)y

(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x) (>1.1)

a3(x)mdy

mxd+ a1(x)

md1y

mxd1+ + ad1(x)my

mx+ ad(x) y 6 H(x)

sib`jdmi yaz`r. Nuramaa3(x), a1(x), . . . , ad(x) mikjsbid batsayàrmr. Nu batsayàr viH(x) hedbsjyedu, x jd

njrJ6 Ra, nV ara`kdma tadfàdfs s urib`j hedbsjyed àrmr.

]adf >1.>. Ikir, H(x) 6 3 jsi, (>1.1) midbìfj

a3(x)y(d) + a1(x)y

(d1) + ... + ad 1(x)y+ ad(x)y6 3 (>1.>)

sib`jdj a`r. E zafad nu midbìfi, d jdlj firtinimid mikjsbid batsay` gefeoid `jdiir AMM midjr.

Erdib >1.. @jdiir gefeoid AMM jljd tifi` tierifìr

gefeoid e`fayadd.jdlj firtinimid mikjsbid batsay` njr `jdiir mjhiradsjyi` midbìfjd, yadj(>1.1) midbìfjdjdkidi` lezufudu nu`fab afalfzmr. Nudud jljd edli (>1.>) gefeoid mjhiradsjyi` midbìfjd kidi` lezufududnuùdfas jdliìdilibtjr. ((>1.1) mjhiradsjyi` midbìfjdjd lezufu jì j`kj`j var`b tierifj ]ierif (=.>>.1. d. jdlj firtinimid njr gefeoid mjhiradsjyi` midb ifjd njrjnjrjdmid harb` f sayma lezufuy1, y>,...,yf e sud. Nurama f dmr. Nu murufma l1, l>,...,lf batsayàr biyfi sanjt sayàr e`fab uziri,y 6 l1y1+ l>y>+ ... + lfyf hedbsjyedu ma ayd midbìfjd njr lezufu eùr.

]adf >>.>. y1, y>,...,yf girgadkj hedbsjyedàr vi l1, l>,...,lf girgadkj biyfi sanjt sayàr e`sudàr. Numurufmal1y1+ l>y>+ ... + lfyf jhamisjdiy1, y>,...,yf hedbsjyedàrdd `jdiir befnjdasyedu midjr.

Yyar >>.>.>i keri(>1.>) gefeoid mjhiradsjyi` midbìfjd lezufìrjdjd `jdiir befnjdasyedu ma

njr lezufmur.

Erdib >>.=. sjd x viles x hedbsjyedàry+y 6 3 AMM djd lezufumur. Yyar>>.1=a keriy 6 l1les x +l>sjd x mi midbìfjd lezufumur.

]adf >>.9. Njra

x

nara`kdma tadfàfs eàdy1, y>,...,yfhedbsjyed bufisj jljd, gipsj shr e`fayadl1, l>,...,lf kjnj sanjt sayàr nuùdanj`jrsi vi xjd nu ara`btabj nutud mikirìrj jljd,

l1y1+ l>y>+ ... + lfyf6 3

jsi, nu hedbsjyedàra araàrdma `jdiir nakf` hedbsjyedàr midjr.

Erdib >>.8. x vi>x hedbsjyedàr R3, 1V ara kdma jdiir nakf`mr. L udbu

l1x + l>(>x) 6 3 (l1+ >l>) x6 3,x R3, 1V l1+ >l> 6 3 l1 6 >l>Erdikjdl> 6 1, l16 > jljd jhami sakàdfs eùr.]adf >>.;. Njra x n ara`kdma `jdiir nakf` e`fayad hedbsjyedàra jsi, `jdiir nakfsz hedbsjyedàr

midjr. Zadj,l1y1+ l>y>+ ... + lfyf6 3 l16 l> 6 6lf6 3

9;


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97 9. ZYB\IB FI^]INIMID @JDII^ AMM

Erdib >>.7. x vi x> hedbsjyedàr R3, 1V ara`kdma `jdiir nakfszmr. l1x+ l>x> 6 3 jhamisjdj mjhirad-

sjyi`ìrsibl1+ >l>x6 3 i`mi imirjz vix jì nu midbìfj larparsabl1x + >l>x> 6 3 i`mi imirjz.

l1x + l>x> 6 3 vil1x + >l>x

> 6 3,x R3, 1V

midbìfìrjdj lbartrsabl>x>

6 3,x R3, 1V i`mi imirjz. Nuda keril> 6 3 6 l1 mjr.]ierif >>.:. (>1.>) gefeoid mjhiradsjyi` midbìfj d `jdiir nakfsz lezufi sagjptjr. y1, y>,...,yd (>1.>)

gefeoid mjhiradsjyi` midbìfjdjd `jdiir nakfsz lezufìrj jsi(>1.>) djd kidi` lezufu

l1y1+ l>y>+ ... + ldyd

jì jhami imj`jr. Nuramal1, l>,...,ld biyfi sanjt sayàr.

Erdib >>.13. sjd x viles x `jdiir nakfsz hedbsjyedàry+ y6 3 AMM djd lezufumur. Yyar >>.1=a vi]ierif>>.:keriy 6 l1les x + l>sjd x mi midbìfjd lezufumur.

\jfmj, jbj viya maga hazà hedbsjyedud gadkj besu`àrma `jdiir nakf` viya `jdiir nakfsz e`mukudu arastra`f.

]adf >>.11. Njra x nara`kdma tadfàfs eàdy1, y>,...,yd hedbsjyedàr(d 1) .firtinmid turivisagjp e`sudàr.

[(y1, y>,...,yd) 6

y1 y> . . . ydy1 y

> . . . y

d

......

y(d1)1 y

(d1)> . . . y

(d1)d

mitirfjdadtda [redsbjad midjr via x n debtasdmabj mikirj bsala[(y1, y>,...,yd) jì kestirj`jr.]ierif >>.1>. y1, y>,...,yd (>1.>) gefeoid mjhiradsjyi` midbìfjdjd lezufìrj `jdiir nakfsz e`fas jljd

kirib vi yitir besu`[(y1, y>,...,yd) 6 3 e`fa`mr.

Erdib >>.1>.1=. (>1.>) gefeoid mjhiradsjyi` midbìfjdjd lezufuy (x) e`sud vi

y (x3) 6 3, y (x3) 6 3, . . . , y

(d1) (x3) 6 3

besuùdu sak`yersa nu murufma lezufy (x) 6 3 mr.

]adf >>.19. Yyar>>.1=mibj lezufi asjbar lezuf midjr.

>1.>) gefeoid mjhiradsjyi` midbìfjdjd asjbar e`fayad h lezufudu nj`jyersab y 6 h vmedus ufu jì midbìfj(d 1) . firtiniyi jdmjrkiyinj`jrjz.]ierif >.

a3(x)y+ a1(x)y

+ a>(x)y 6 3 (>)

medus ufudu midbìfmi yirjdi yaza`f

a3(x) (h v)

+ a1(x) (h v) + a>(x)h v 6 3 a3(x) (h

v+ >hv+ h v) + a1(x) (hv+ h v) + a>(x)h v 6 3

(a3(x)h+ a1(x)h + a>(x)h)

63

v+ a3(x) h v+ (>a3(x) h

+ a1(x) h) v 6 3

jì(>a3(x) h

(x) + a1(x) h(x)V w6 3 (>


65/133

> mx (>xy + >y6 3 midbìfjdjd njr lezufu e`fab uziri, firtinidjdjdmjrkidfisj yedtifjdj bu`àdarab `jdiir nakfsz lezufu nuùduz.

L ezuf (> + 1, a1(x) 6 >x, a>(x) 6 > mjr. (> mx

6 ixp >xx>+1 mx

x> mx 6

x>+16u>xmx6mu

ixp muu x>

mx

6

ixp (`du)

x> mx6

u + l

x> mx 6

x>+16u

x> + 1

x> mx6

1 +

1

x>

mx

6 x 1x

+ l6 x 1x

+ l

(>) medusufudmid lezufu

y6 x

x 1

x+ l

6 x> + lx 1

sib`jdmi i`mi imirjz. ]ierif >>.1i keri

y 6 l>x + l< x> + lx

16 A x> 1+ Nx

kidi` lezufmur.

Erdib >y =xy+ =y6 3midbìfjdjd njr lezufu e`sud. Firtinidjd jdmjrkidfisj yedtifj jì kidi` lezufu nuùduz.

L ezufy 6 xv

medusufudu midbìfmi yirjdi yaza`f.

x> (xv) =x (xv)+ =xv 6 3 x> (>v+ xv) =x (v+ xv) + =xv6 3

x>

(xv >v) 6 3 xv >v6 3 w6v xw 6 >w mw

w 6 >

mx

x `d w 6 > `d x + `d l w6 lx>

w6vv 6 lx> mv6 lx>mx

v 6 lx=.1)

sanjt batsay` gefeoid midbìfjd lezufuduy6 ifx (>=.>)

eàrab araya`f. Nuda keri

mby

mxb 6fbifx, b6 1, . . ,d

turivìrjdj (>=.1) midbìfjdmi yirjdi yazarsab

ifx

a3fd + a1f

d1 + +... + ad 1 f + ad

6 3

midbìfjdj i`mi imirjz. ifx 6 3 e`mukudmada3f

d + a1fd1 + +... + ad 1f + ad6 3 (>=.=.1. (>=.,...,fd kjnj d tadi bebu e`fa mr. Nu bebìr njrir njrirmidbìf(>=.1)ma yirjdi bedursa gir mihasdma njr ezi` lezuf i`mi imjìlibtjr. Nuda keri midbìfjd bebìrjdikeri lezufu jrmiìyi`jf.

>=.1. 1. Muruf? Ayrb rii` bebìr. (>=.=.>. (>=.,...,fdkjnj harb` (ayrb) rii` bebu var jsi(>=.1)midbìfjdjdkidi` lezufu

y6 l1if1x + l>i

f>x + ... + ldifdx (>=.=)

sib`jdmimjr. l1,l,...,ld biyfi sanjtìrmjr.

Erdib >=. 6 3. Neyìli midbìfjd lezufu (f 1) (f >) 6 3 f1 61, f> 6 >. (>=.=)

mad

y6 l1ix + l>i

>x

lezufmur.

Yyar >=.=. Erdib>=.x hedbsjyedàr `jdiir nakfszmr.

[

ix, i>x

6

ix i>xix >i>x6 ix 6 3

mr.

Erdib >=.9. y

y

1>y6 3, y (3) 6 f 1> 6 3lezufìrf1,> 6


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>=. \ANJ] BA]\AZJ@J GEFEOID @JDII^ AMM 81

besu` ardmad

l1+ l> 6 6 >

y 6 i=.>. >. Muruf? ]ibrar bebìr.

]ierif >=.8. (>=. 6 6 fb 6 f) nu murufma(>=.1) midbìfjdjd tibrar imid bebìri bars`b kiìd lezufu

l1+ l>x + + lbxb1

ifx (>=.9)

sib`jdmimjr. fb+1, fb+>, . . . , fd njrnjrjdmid harb` rii` bebìr e`fab uziri(>=.1) midbìfjdjd kidi` lezufu

y6

l1+ l>x + + lbxb1

ifx + lb+1ifb+1x + lb+>i

fb+>x + ... + ldifdx (>=.8)

herfudmamr.

Erdib >=.;. yjv + 9y+ 8y+ =y 7y6 3 AMM djd lezufudu nuùduz.L ezuf Midbìfjd barabtirjstjb midbìfj

f= + 9f< + 8f> + =f 7 6 3vi midbìfjd bebìrj >, >, >, 1 mjr. Nuda keri tibrar imid f 6 > bebudi bars`b kiìd lezuf (>=.9) mad

y1 6

l1+ l>x + l

i>x

vif> 6 1 bebudi bars`b kiìd lezufy> 6 l=i

x

mjr. Nuda keri kidi` lezuf(>=.8) mid

y6 y1+ y>6 l1+ l>x + l i>x + l=ixmr.

Erdib >=.7. y(9) >y(=) + y 6 3 midbìfjdjd lezufudu nuùduz.L ezuf Mjhiradsjyi` midbìfjd barabtirjstjb midbìfj

f9 >f= + f< 6 3vi lezufìrf1,>,. Muruf? ]ibrar` bebìr mjr. Nuda keri

1, x, x>, ix, xix

`jdiir nakfsz lezufìrmjr. ]ierif >x + l + l=ix + l9xi

lezufmur.

>=. 61 e`fab uziri befpìbs sayàr a+ jn, n6 3 herfudmamr. (a, n rii` sayàr). a jnsaysda jsia + jn djd befpìbs isìdjkj midjr.

]ierif >=.13. (>=.les nx) (>=.;)

eàrab yaz`r.


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8> 9. ZYB\IB FI^]INIMID @JDII^ AMM

Sreeh. Kidi` lezufub1i

(a+jn)x + b>i(ajn)x

eàrab yazanj jrjz. Nuramaij 6 les + j sjd

Iuìr herfuùdu bu`àdrsab

b1i(a+jn)x + b>i

(ajn)x 6 b1iaxijnx + b>i

axijnx

6 iax

b1ijnx + b>i

jnx6 iax (b1(les nx + j sjd nx) + b>(les nx j sjd nx))6 iax ((b1+ b>)les nx + j (b1 b>)sjd nx)6 iax (l1sjd nx + l>les nx) , l1 6 b1+ b>, l>6 b1 b>

i`mi imirjz.

Erdib >=.11. y 8y+ >9y6 3 AMM djd lezufudu nuùduz.L ezuf AMM djd barabtirjstjb midbìfj f> 8f + >9 6 3 mr. Midìfjd bebìrj

f 6 8

68 7j

> 6 < =j

a 6 =.;)

y 6 iles =x)

lezufmur.

]ierif >=.1>. (>=.x + + lbxb1

sjd nx + iax

l1+ l>x + + lbxb1

les nx (>=.7)

eàrab yaz`r.

Erdib >=.1y+ y 6 3 AMM djd lezufudu nuùduz.L ezuf Mjhiradsjyi` midbìfjd barabtirjstjb midbìfj

f8 >f< + 1 6 3lezufìrf1,>,< 6 1, 1>+ j

, 1>+ j

, vi mjkirìrj isjt e`fab uziri 1. Muruf? Ayrb rii` bebìr & >. Muruf& x les

, i

1>x sjd

, xi

1>x les

, xi

1>x sjd

`jdiir nakfsz lezufìrmjr. ]ierif >xi

x + lx les

+ l=i

1>x sjd

+ l9xi

1>x les

+ l8xi

1>x sjd

lezufmur.

>9. Gefeoid e`fayad AMM

(>1.1) gefeoid e`fayad midbìfj iì aà`f?

a3(x)y(d) + a1(x)y

(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x)

]ierif >9.1. v hedbsjyedu (>1.1) gefeoid e`fayad midbìfjd njr lezufu viu hedbsjyedu(>1.>) gefeoidmidbìfjd lezufu jsiu + v hedbsjyedu ma(>1.1) gefeoid e`fayad midbìfjd njr lezufumur.

Erdib >9.>. y 6 x hedbsjyedu y+ y 6 x gefeoid e`fayad midbìfjdjd njr lezufumur. sjd x hedbsjyeduy+y6 3gefeoid AMM djd lezufumur. ]ierif>9.1i keriy 6 x+sjd xhedbsjyedu may +y 6 x midbìfjdjdlezufumur.


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>9. GEFEOID E@FAZAD AMM 81.>) gefeoid midbìfjd kidi` lezufu eàd yl (lefpìfidtary hudltjed)djd das` nuùdmukudu nj`jyeruz.\jfmj jsi afalfz, gefeoid e`fayad nu midbìfjd njr ezi` lezufu eàd vi njr biyfi sanjt say jlirfiyid yp(partjluàr jdtikra`) lezufudu vi sedul eàrab (>1.1) gefeoid e`fayad midbìfjdjd kidi` lezufudu nu`fabtr.yp djd nuùdfas jì j`kj`j eàrab njrbal fitet ki`jstjrj`fjstjr. Nu fitetàr, sras jì,

(j)? Ni`jrsjz Batsayàr Fitemu(jj)? Sarafitriìrjd Mikjsjfj Fitemu

>9.1. Ni`jrsjz Batsayàr Fitemu.

]adf >9.) iax, a 6 3(9.=. xx, sjd() , les(>x + /=) YL hedbsjyedàrmr.

]adf >9.9. @jdiir nakfsz YL hedbsjyedàrdd bufisjdi bsala YL bufisj midjr.

YL hedbsjyedu YL bufisj1 xd

xd, xd1, . . . , x , 1

> iax {iax}< sjd (nx + l)

viya les (nx + l){sjd(nx + l) , les(nx + l)}

= xdiax

xdiax, xd1iax, . . . , x iax, iax

9 xd sjd(nx + l)

viyax

d

les(nx + l)

{xd sjd(nx + l) , xd les(nx + l) ,xd1 sjd(nx + l) , xd1 les(nx + l) ,

. . . , x sjd(nx + l) , x les(nx + l) ,sjd(nx + l) , les(nx + l)}8 iax sjd(nx + l)

viyaiax les(nx + l){iax sjd(nx + l) , iax les(nx + l)}

; xdiax sjd(nx + l)viyaxdiax les(nx + l)

{xdiax sjd(nx + l) , xdiax les(nx + l) ,xd1iax sjd(nx + l) , xd1iax les(nx + l) ,. . . , x iax sjd(nx + l) , xiax les(nx + l) ,iax sjd(nx + l) , iax les(nx + l)}

Erdib >9.8. h(x) 6x> sjd x hedbsjyedudud YL bufisj

x> sjd x, x> les x, x sjd x, x les x, sjd x, les x

Zedtif(>1.1) gefeoid e`fayad midbìfjdmi

a3(x)y(d) + a1(x)y(d1) + ... + ad 1(x)y+ ad(x)y 6 H(x)

H(x) hedbsjyedudu YL hedbsjyedàrdd `jdiir befnjdasyedu e`fas murufudma nu yedtifj ukuàyanj`jrjz.

(1) H 6l1u1+ l>u>+ + lfuf e`sud. Nurama u1, u>, . . . , uf YL hedbsjyedàrmr.(>) u1, u>, . . . , uf YL hedbsjyedàrda bars`b kiìd YL bufiìrjdj\1, \>, . . . , \ f eàrab ni`jrìyi`jf.(9.;. y >y ix 13sjd x AMM djd lezufudu nuùduz.


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8= 9. ZYB\IB FI^]INIMID @JDII^ AMM

L ezuf

y >y

>f

< 6 3

barabtirjstjb midbìfjdjd lezufudmidf1 6 6 1 yl 6l1iix herfudmamr. \jfmj gefeoide`fayad tirjfi kiì`jf?

>ix 13 sjd xhedbsjyedu(1)

ix, sjd x, les x

YL hedbsjyedàrdd `jdiir befnjdasyedu eàrab yaz`fstr. Nuda keri YL bufiìrj(>)

\1 6 {ix}\> 6 {sjd x, les x}

(bufiìrjdjd iìfadàr jì gefeoid bsfd hedbsjyedàr njr njrj jì ayd mikj`mjr. Nu sbbma kili`jf.(9) < vi =. murufàr sakàdfamkdmad njr sedrabj muruf.

(8) Ezi` hedbsjyedu nu bufiìrjd eùsturfus e`muku iìfaàrd befnjdasyedu sib`jdmi yaz`mkd varsaya`f?

yp 6 Aix + N sjd x + Lles x

yp 6 Aix + N les x Lsjd x

yp 6 Aix N sjd x Lles x

midbìfmi yirjdi yazarsab

y >y ix 13sjd x Aix N sjd x Lles x > (Aix + N les x< (Aix + N sjd x + Lles x) 6 >ix 13sjd x

=Aix + (=N+ >L)sjd x + (=L >N)les x 6 >ix 13sjd x=A 6 >,

=N+ >L 6 13=L >N 6 3 A6 1

>, N6 >, L6 1

yp 6 1>

ix + > sjd x les xezi` lezuf vi

y 6 yl+ yp 6 l1ii

x 1>

ix + > sjd x les xkidi` lezufmur.

Erdib >9.7. y y6 >x

>

+ i

x

+ >xi

x

+ =i

6 3barabtirjstjb midbìfjdjd lezufudmid f1 6 1, f> 6 > yl 6 l1ix +l>i>x herfudmamr. \jfmj gefeoide`fayad tirjfi kiì`jf?

>x> + ix + >xix + =i, ix, xix, i


71/133

>9. GEFEOID E@FAZAD AMM 89


\1 6

x>, x, 1

\> 6 {ix}\< 6 {xix, ix}\= 6

iix +Ixix + H iAx + Iix + >xMix + xIix + A + >Mix + >Iix + =xMix + xIix + :H iMix

midbìfmi yirjdi yazarsab

y y 6 >x> + ix + >xix + =iA + >Mix + >Iix + =xMix + xIix + :H iAx + Iix + >xMix + xIix + Ax> + Nx +

+Ixix + H ix> + ix + >xix + =iA L) + (>N 8A) x + >Ax>

+>MiIxix

+ (>I H) ix

6 >x>

+ ix

+ >xix

+ =iA L 6 3

>N 8A 6 3>A 6 >

>M 6 =

>I 6 >>I H 6 1 A6 1, N 6 , H 6 , I6 1

yp 6 x> + + >x>ix xix x

+ x>

+ + >x>

ix

xix

les x AMM djd lezufudu nuùduz.L ezuf

yjv + y6 3

gefeoid midbìfjd lezufuf= + f> 6 3

barabtirjstjb midbìfjdjd lezufudmidf1,> 6 3, fx+ lles x


72/133


hedbsjyedu(1)

x>, sjd x, les x


\1 6

x>, x, 1

\> 6 {sjd x, les x}\< 6 {les x, sjd x}

( 6 \< e`mukudmad\1, \> bufiìrj iì a`dlabtr.(=) \1 6

x>, x, 1

bufisj x, 1 hedbsjyedàrd jlirmjkjdmid vi x, 1 hedbsjyedàr gefeoid midbìfjd lezufu

jljdmi e`mukudmad \1 bufisjdjd iìfadàrd x> jì larpp yidjmid muzidìrjz? \1 6{x=, x}. Neyìli nubufidjd iìfadàr turmis bsfd lezufudmibj hedbsjyedàrmad harb`mr. Zjdi\>bufisjdjd iìfadàr, turmisbsfd lezufudmi e`mukudmad, iìfadàr x jì larpa`f? \> 6 {x sjd x, x les x}(9) Zidjmid muzidìdid bufiìr srasyà?

\1 6 {x=, x}\> 6 {x sjd x, x les x}

sib`jdmimjr.(8) Ezi` hedbsjyedu nu bufiìrjd eùsturfus e`muku iìfaàrd befnjdasyedu sib`jdmi yaz`mkd varsaya`f?

yp 6 Ax= + Nx< +Lx> + Mx sjd x + Ix les x

yp 6 >Lx + Iles x + M sjd x + xM les x xIsjd x + =Ax< + yp 6 >L+ 8Nx + >M les x >Isjd x xIles x xM sjd x + 1>Ax>

yp 6 8N+ >=Ax les x >=A =M les x + =Isjd x + xIles x + xM sjd x+>L+ 8Nx + >M les x >Isjd x xIles x xM sjd x + 1>Ax>

6 + = sjd x >les x>=A + >L 6 3

8N 6 3

1>A 6 M 6 >>I 6 = A6 1

=, N6 3, L6

yp 6

1

= x=

+ x sjd x + >x les x

ezi` lezuf vi

y6 yl+ yp6 l1+ l>x + lx les x

kidi` lezufmur.

>9.>. Sarafitriìrjd Mikjsjfj Fitemu.

y + a (x) y6 n (x)herfudmabj midbìfìri `jdiir midbìfjdj gatràya`f. Midbìfjd

y + a (x) y6 3


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>9. GEFEOID E@FAZAD AMM 8;

turmis midbìfjdi bars`b kiìd lezufu

y 6 l ixp

a (x) mx

,

eàrab i`mi itfjstjb.(13.1) jd kidi` lezufudu

y6 l (x)ixp

a (x) mx

.

herfudma arafstb bj nu tierj eàrab parafitriìrjd mikjsjfj fitemu mur. Zedtifj >. firtinmid mikjsbidbatsay` njr AMM jljd adàta`f?

a3(x) y+ a1(x) y

+ a>(x) 6H(x) (>9.1)

midbìfjdj iì aà`f. y1 viy>

a3(x) y+ a1(x) y

+ a>(x) 6 3 (>9.>)

gefeoid midbìfjdjd `jdiir nakfsz lezufìrj e`sudàr. Nu murufma

yl 6 l1y1+ l>y>(>9.>) gefeoid midbìfjdjd njr kidi` lezufumur. Sarafitri irjd mikjsjfj yedtifjdmi l1, l> sanjtìrj yirjdihedbsjyedàr musuduùr. Zadj

yp 6 v1(x) y1(x) + v>(x) y>(x) (>9.9.1) midbìfjdjd njr lezufumur. Nurama v1(x) vi v>(x) hedbsjyedàrdd jhamiìrjdj nu`fafzkiribfibtimjr. I`jfjzmi > nj`jdfiyid var vi 1 besu` eàrabma (>9.. njr ib besuà jgtjyal muyu`fabtamr.

yp 6 v1(x) y1(x) + v>(x) y

>(x) + v

1(x) y1(x) + v

>(x) y>(x) (>9.=)

Nuramay p hedbsjyedudu nu`famad edli maga edli nagsitfjs e`mukufuz >. besu` eàrab

v1(x) y1(x) + v>(x) y>(x) 6 3 (>9.9)

aà`f. \jfmj hedbsjyedud >. turivjdj a`rsab

yp 6v1(x) y1(x) + v>(x) y

> (x) + v

1(x) y

1(x) + v

>(x) y

>(x) (>9.8)

i`mi imirjz. (>9.9.9.=) , (>9.8) jhamiìrjdj (>9.1) midbìfjdmi yirjdiyazarsab

a3(x) Rv1(x) y1 (x) + v>(x) y

>(x) + v

1(x) y

1(x) + v

>(x) y

> (x)V +

a1(x) Rv1(x) y1(x) + v>(x) y

>(x)V + a>(x) Rv1(x) y1(x) + v>(x) y>(x)V

6 H(x)

v1(x) Ra3(x) y1(x) + a1(x) y1(x) + a>(x) y1 (x)V +

v>(x) Ra3(x) y>(x) + a1(x) y>(x) + a>(x) y> (x)V + a3(x) Rv

1(x) y

1(x) + v

>(x) y

>(x)V

6H(x)(>9.;)

i`mi imirjz. y1 viy> gefeoid midbìfjdjd `jdiir nakfsz lezufìrj e`mukudmada3(x) y1 (x) +a1(x) y1(x) +a>(x) y1(x) 6 3 vi a3(x) y> (x) + a1(x) y

>(x) + a>(x) y>(x) besu`àr sakàdr. (>9.;) mi nudàr mjbbati a`rsab

a3(x) Rv1(x) y

1(x) + v

>(x) y

> (x)V 6H(x) (>9.7)

besuùdu i`mi imirjz.Neyìli nj`jdfiyid v1(x) viv>(x) hedbsjyedàr jljd v1(x) y1 (x) + v

>(x) y>(x) 6 3

v1(x) y1 (x) + v

>(x) y

>(x) 6

H(x)a3(x)

(>9.:)

midbìf sjstifjdj i`mi imirjz. Batsayàr mitirfjdadt

[Ry1(x) , y>(x)V 6

y1(x) y>(x)y1(x) y

>(x)


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eàrab i`mi imirjz. y1 viy> `jdiir nakfsz lezufìr e`mukudmad [redbjyid[Ry1(x) , y>(x)V 6 3 mr. Neyìli(>9.:) sjstifjdjd tib lezufu varmr. Xiv 1(x) viv

>(x) hedbsjyedàrd

v1(x) 6 3 y>(x)

H(x)

a3(x) y

>(x)y1 (x) y>(x)y1 (x) y>(x)

6 H(x) y>(x)

a3(x) [Ry1 (x) , y>(x)V (>9.13)

v>(x) 6

y1(x) 3y1(x) H(x)a3(x)y1 (x) y>(x)y1 (x) y>(x) 6

H(x) y1(x)

a3(x) [Ry1(x) , y> (x)V (>9.11)

Neyìli (>9.13) vi (>9.11) mi jdtikra` a`mkfzmav1(x) viv>(x) hedbsjyedàrd nuùruz.

Erdib >9.13.y+ y6 tad x

AMM djd lezufudu nuùduz.L ezuf Gefeoid bsfd lezufu

yl 6 l1sjd x + l>les x

e`mukudmad kidi` lezufuyp 6 v1(x)sjd x + v>(x)les x

e`sud.yp 6 v1(x)les x v>(x)sjd x + v1(x)sjd x + v>(x)les x

Nuramay p hedbsjyedudu nu`famad edli maga edli nagsitfjs e`mukufuz >. besu` eàrab

v1(x)sjd x + v>(x)les x6 3

aà`f. \jfmj hedbsjyedud >. turivjdj a`rsab

yp

6

v1(x)sjd x

v>(x)les x + v1

(x)les x

v>(x)sjd x

i`mi imirjz. Midbìfmi yirjdi yazarsab

v1(x)les x v>(x)sjd x6 tad xi`mi imirjz. Neyìli nj`jdfiyidv1(x) viv>(x) hedbsjyedàr jljd

v1(x)sjd x + v>(x)les x6 3

v1(x)les x v>(x)sjd x6 tad xmidbìf sjstifjdj i`mi imirjz. Xiv 1(x) viv

>(x) hedbsjyedàrd

v1(x) 6

3 les xtad x sjd x

sjd x les xles x

sjd x

6 sjd x v1(x) 6 les x + l(x) 6

sjd x 3les x tad xsjd x les xles x sjd x 6

sjd> x

les x 6

les> x 1les x

v>(x) 6

(les x sil x) mx 6

(;;).jdtksjd x `d (sil x + tad x) + l=

eàrab i`mi imirjz. Nuda keri kidi` lezuf

yp(x) 6 v1(x)sjd x + v>(x)les x

6 ( les x + l


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>9. GEFEOID E@FAZAD AMM 8:

e`arab i`mi imirjz.

y 6 yl+ yp 6 l1sjd x + l>les x + l + 11f

8 6 3

f

1,>,, i>x + lx + v(x) i

>x + x + v(x) i

>x + :v(x) i>x +


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;3 9. ZYB\IB FI^]INIMID @JDII^ AMM

midbìf sjstifjdj i`mi imirjz. Xiv 1(x) , v>(x) viv

x ii>x x xi x + l=

v>(x) 6

ix 3 ii>x 3

ix

=i>x

ix

ix i>x ii>x x

>ixi>xii>x vx + l8

v>(x) 6

(les x sil x) mx 6

(;;).jdtksjd x `d (sil x + tad x) + l=

? ? ? ? 3 ? eàrab i`mi imirjz. Nuda k eri kidi` lezuf

yp(x) 6 v1(x) ix + v>(x) i

>x + vx +

1

=i>x + l8

i

xix + l=ix + ix + l9i

>x

1

=

ix + l8ii

>x +lx 1

=ix + l8i

x + Lixix

nuùruz.

>8. Laulgy-Iuìr midbìfj

]adf >8.1.

a3xdy(d) + a1x

d1y(d1) +... + ad 1xy+ ady6 3 (>8.1)

midbìfjdi Laulgy-Iuìr midbìfj midjr.

]ierif >8.>. x6 it medus ufu jì(>8.1) midbìfj sanjt batsay` `jdiir midbìfi medus ur.

Erdib >8.y >xy+ >y6 xy

mx> 6

1

x

m>y

mt>mt

mx 1

x>my

mt 6

1

x>

m>y

mt> my

mt


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LGAS]I^ 8

\anjt batsay` Jbjdlj Firtinimid Mjhiradsjyi` Midbìfìrjd

Yykuàfaàr

>;. \a`df Garibitj

\jstif midki bedufudma njr fjbtar asak mekru libj`jr vi nrab`rsa kjmjp-ki`fi garibitj yapar. Nuda sa`dfgaribitj viy tjtrisjf garibitj midjr. Gava mjridlj jgfa` imj`jrsi, njr nasba miyjsì surtudfi buvvitj yeb banuìmj`jrsi, ljsjf yubarma vi asakma ayd jfjt ir arasdma njr kjmjp-ki`fi garibitj yaplabtr. Nudanasjt garfedjbgaribit midjr. Ikir surtudfi varsa ljsfjd sa`df kid`jkj kjttjbli aza`lab vi ljsjf njr suri midki bedufuda

ki`jlibtjr. Ljsfjd nu tur garibitjdisirnist sedufù garfedjb garibitj midjr. Nudud yad sra, ljsfi mivaf`vi sjdusejma` njr hedbsjyedà jhami imjìnjìd njr garfedjb itbj virjìnj`jr. Nu murufma ljsfjd garibitj zeràdfssedufù garfedjb garibittjr.

1? Nasjt Garfedjb Garibit

Geebi yasasda keri libjìd vi nrabàd njr yayd nuyubùku, yayd uzafa fjbtar jì eradt` njr kirj lakrlbuvvittjr. Nuda keri buvvitj,

H6 bxjì kestirjrjz. Nuda keri mjhiradsjyi` midbìf

fm>x

mt>

6

bx

m>x

mt>

+ 3x6 3, 3 6 b

feàrab virj`jr.

>? \irnist \edufù Garfedjb Garibit

Midki bedufudmadx uzab`kdma nuùda njr ljsfi itbj imid sedmurulu buvvitjd nuyubùku

nmx

mt

jì virj`jr. n 4 3 sanjttjr vi sedmurfi batsays eàrab amàdmr`r. \edmurulu buvvitjd yedu, ljsfjd garibityedudi zttr. Nuda keri mjhiradsjyi` midbìfj

fm>x

mt> 6 bx n mx

mt f m

>x

mt> + n

mx

mt + bx 6 3

eàrab virjrjz.

;


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;= 8. \ANJ] BA]\AZJ@J JBJDLJ FI^]INIMID MJHI^AD\JZI@ MIDB@IF@I^JD YZKY@AFA@A J


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>7. I@IB]^JB MIXÎ SÊN@IF@I^J ;9

(>)

> m>x

mt> 6 = mx

mt :7x + les t > m

>x

mt> + =

mx

mt + :7x6 les t

edli gefeoid midbìfj lezi`jf

> m>x

mt> + =

mx

mt + :7x6 3 >f> + =f + :7 6 3 f> + >f + ; 6 3

f1,>6 j

8 1 xl(t) 6 it

l1les

8t + l>sjd

8t

sjfmj kiì`jf gefeoid e`fayad midbìfjd ezi` lezufudi YL yedtifjdj bu`àdrsab?

u1 6 les t \1 6 {les t, sjd t} xp(t) 6 A les t + N sjd t xp 6 A sjd t + N les t xp 6 A les t N

> m>x

mt> + =

mx

mt + :7x6 les t > (A les t N sjd t) + = (A sjd t + N les t) + :7 (A les t + N sjd t) 6 les t

(:8A + =N)les t + (=A + :8N)sjd t6 les t :8A + =N6 1=A + :8N6 3 A6 8

9;;, N6

1

>sjd

8t

+ 8

9;;les t +

1

>

8 6 1> 6 11< 7=7

8

x (t) 6 it

8

9;;les

8t 1

1< 7=7

8sjd

8t

+

8

9;;les t +

1

>7. Iìbtrjb Mivri Srenìfìrj

Id nasjt iìbtrjb mivriìrj, oidirater viya pj` kjnj iìbtrjb baydak vi idirojyj bu`àdad njr rizjster (erdikjdiìbtrjb afpuù) (risjstadli) nuùdad njr sirj mivrimjr. Ikir mukfi bapat`rsa njr Jabf rizjsteri mekruabalab vi njr ve`tao musfisjdi sinip eùlabtr. Zadj rizjsterud jbj uludmabj petadsjyi` harb` eùlabtr. Nupetadsjyi` harb viya ve`tao mususu jsi Xe`tfitri midjìd njr i`t jì e`luìnj jr. Ibitrjb mivriìrjdmibj nasjt njr

îsjstr

Mfi

Baydab

bura` Bjrlgeffi bura` eàrab amaàdmr`r, Nu buraà keri, iìbtrjb mivrisjdmibj tuf ve`taoàrd tepàf, tepàfbuvviti isjttjr. ]epàf buvvitj I(t) jì kestirjrsib (ifh-iìltrefetjvi herli)

X@+ X + XL6I(t)


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;8 8. \ANJ] BA]\AZJ@J JBJDLJ FI^]INIMID MJHIÂD\JZI@ MIDB@IF@I^JD YZKY@AFA@A J

X@+ X + XL6I(t)

^ rizjster (rijstadli), L bapasjter (lapaljter), J jdmubter (dmulter). J6J(t) iìbtrjb mivrisjdmibj abf viq6 q(t) bapasjtermibj adj iìbtrjb yubudu kestirfib uziri

q 6 J

sib`jdmi njr nakdt fivluttur. Egf badududa keri rizjster uzirjdmibj ve`tao musubùku abf jì mekrueradt`mr?

X 6 ^J

nurama ^ rizjsterud mjridljmjr vi sanjttjr. Bapasjtermibj ve`tao mususu jsi bapasjtermibj iìbtrjb yubu jìeradt`mr vi

XL6 1

Lq

eàrab virj jr. NuramaLbapasjtadstr (lapaljmadli). \ed eàrab jdmubtermibj ve`tao mususu jsi abfd mikjsjfgz jì eradt`mr?

X@6 @J

@ sanjtjdi jdmjbterud jdmubtads midjr (gidry jì e`luùr) (jdmultadli). Bjrlgeffi bura da keri

@J+ 1

Lq+ ^J6 I(t)

nakdtsd i`mi imirjz. Nurama turiv a`rsab viq 6 J

jhamisjdi keri

@q+ ^q+ 1

Lq 6 I(t)

@J+ 1

Lq+ ^J 6 I(t)

@J+ ^J+ 1L

J 6 I(t) >.firtinimid midbìfj ^L@ midbìfj eàrab ama`dmr`r.

Erdib >7.1. Mjridlj 13 eàd njr mjridl ti`j vi ez jdmubsjyed batsays @ 6 3.> gidry eàd njr nenjd,iìbtrefeter buvvitj =3 ve`t vi bapasjtads e`fayad njr mekru abf uritiljdi sirj eàrab nakàdyer. Nasàdkltaabf vi iìbtrjb yubu e`famkda keri.girgadkj zafadmabj iìbtrjb yubudu vi abf nuùduz.

L ezuf Zubarmabj midbìf keri

^ 6 13, @6 3.>, I(t) 6 =3, XL yeb3.>q+ 13q 6 =3 f> + 93f6 3 f1 6 3, f> 6 93

ql(t) 6 l1+ l>i93t


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>7. I@IB]^JB MIX^I S^EN@IF@I^J ;;

ezi` lezuf jljd?

u1 6 1, \1 6 {1} \1 6 {t} qp(t) 6 At 93A6 =3 A6 =

9

qp(t) 6

=

9 t q(t) 6 l1+ l>i93t

+

=

9 t

q(3) 6 3 l1+ l> 6 3, J6 q 6 93l>i93t +=9 J(3) 6 3 93l> 6 =

9 l>6 =

>93 l1 6 =

>93

q(t) 6 =

>93

i93t 1+=

9t, J(t) 6

=

9

1 i93t


84/133


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LGAS]I^ ;

@jdiir Mjhiradsjyi` Midbìf \jstifìrj

\u ada bamar njr nj`jdfiyid`j njr mjhiradsjyi` midbìfìrj jdliìmjb. \jfmj jsi jbj nj`jdfiyid hedbsjyedu nuùdadjbj mjhiradsjyi` midbìfj vi kidi` eàrab d nj`jdfiyid hedbsjyedu nuùdad d mjhiradsjyi` midbìfj jdliìyilikjz.Maga edliMjhiradsjyi` midbìf sjstifìrjdj tadtfab jì nasàya`f.

>:. @jdiir sjstif turìrj (Jbj nj`jdfiyid`j jbj midbìf)

x vi y nj`jdfiyid hedbjyedàr e`fab uziri 1. firtinimid mikjsbid batsay` `jdiir mjhiradsjyi` midbìf sjstifjkidi` eàrab,

a1(t) x+ a>(t) y+ a(t) y+ n< (t) x + n= (t) y6 H>(t)

(>:.1)

sib`jdmi yaz`r. Nurama t nakfsz mikjsbidmjr.

Erdib >:.1.

>x+ x + y 6 t>x (t) y+ a(t) y+ n(t) (>:.>)

sib`jdmi yaz`r.(>:.1) sjstifjdj ezi` eàrab

x 6 a11(t) x + a1>(t) y+ H1(t)y 6 a>1(t) x + a>>(t) y+ H>(t)

(>:.:.>.

x6 t>x + (t + 1) y+ tx t1. firtinimid, sanjt batsay` njr sjstifmjr.

d sayma mjhiradsjyi` midbìfmid eùsad sjstifj kidi` eàrab,

x1 6 a11(t) x1+ a1>(t) x>+ + a1d(t) xd+ H1(t)x> 6 a>1(t) x1+ a>>(t) x>+ + a>d(t) xd+ H>(t)...xd6 ad1(t) x1+ ad>(t) x>+ + add(t) xd+ Hd(t)

(>:.=)

sib`jdmi yaz`r.

;:


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73 ;. @JDII^ MJHI^AD\JZI@ MIDB@IF \J\]IF@I^J

x6 < m> t + 9

m

tt< 6 17t + 19t> >t@ RkV

Erdib e`sud. h6t>, k6 sjd t jljd

@ RkV 6 @ RkV

e`mukudu k estirjdjz.

L ezuf

@ RkV 6

mt>

+ > sjd t

+ 9m

mt

+ > sjd t

> + > sjd t6 + 17

@ RhV 6

mt>

t>

+ 9m

mt

t> > t>

6 >t> + 13t + 8@ RkV 6

mt>(sjd t) + 9

m

mt(sjd t) > (sjd t)

6 9 les t 9sjd t @ RkV 6 t> + 13t + 8+ >(9 les t 9sjd t)6 + 17

@ RkV 6 @ RkV


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y+

`jdiir sjstifjdj

(>M My 6 t(>M+ M+ 7) y 6 >

jì kestirinj`jrjz. Nuda keri@1, @>, @M >M@M+ M+ 7

Zedtif(. midbìf jsi @> epirateru jì larp`p tarah tarahalbart`r?

@=@1x + @=@>y6 @=h1(t)@>@@=y6 @>h>(t)

(@=@1 @>@h> (t) ()sanjt batsay`, tib nj`jdfiyid`j `jdiir AMM i`mi imj`jr. Neùf9mabj yedtifìr bu`àdàrab lezuf nuùdur.Nidzir sibj`mi nj`jdfiyidy hedbsjyedudu nu`fab jljd jsi (. midbìf jsi @1 epirateru jì larp`p tarah taraha lbart`r?

@M My 6 t(>M+ M+ 7) y 6 >


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7> ;. @JDII^ MJHI^AD\JZI@ MIDB@IF \J\]IF@I^

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