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8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
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Dnahrfksnyhc
Dhkoch`chrUrma.Dr.fjfkHZHK^fsfr knvhrsnthsn Ahk-Hdhjnyft
Afocthsn
Jc` 6
6.6. Gnrn
6.:. Dnahrfksnyhc dhkoch` jnn`chrn
6.2. tnpn
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:
OF^KFOCFZ
Fydk, @., Gkdz, G., Ouryhc, J. (6>>:). Hch`hktfry DnaahrhktnfcHquftnmks fkd Jmukdfry Qfcuh Urmjch`s. Anatl hdntnmk. ImlkWnchy & Rmks, Nkb.
Blnrgwnk, J.L., Ucu`ptmk, B. (6>14). F bmursh ma @ftlh`ftnbs
amr Hkgnkhhrs fkd Rbnhktnsts. Uhrgf`mk Urhss. Qmcu`h :. Blnrgwnk, J.L., Ucu`ptmk, B. (6>14). F bmursh ma @ftlh`ftnbs
amr Hkgnkhhrs fkd Rbnhktnsts. Uhrgf`mk Urhss. Qmcu`h 8.
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2
OF^KFOCFZ
Hr, X. (6>14). Dnaahrhktnfc Hquftnmks amrHkgnkhhrs fkd Rbnhktnsts. Tlh Hkgcnsl Xknvhrsntnhs Urhss Ctd.
Zfnkvncch, H.D., Jhdnhkt, U.H. (6>). Hch`hktfry DnaahrhktnfcHquftnmks. Rhvhktl hdntnmk. @fb`nccfk Uujcnslnkg Bm`pfky.
Rpnhghc, @.Z. (6>18). Tlhmry fkd Urmjch`s ma CfpcfbhTrfksamr`s. Rblfu` Uujcnslnkg Bm`pfky.
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4
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.6. Gnrn
Ju jc`dh dnahrfksnyhc dhkoch` ofvrf`focfkfbfo vh omkuyf foco ghtnr`ho f`fbycfjncnkhk jnr rkho hch fckfbfo vh ju rkho yfrd`ycfdnahrfksnyhc dhkoch` mcuturuc`fs sfcfkfbfotr.
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1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho9 6.6.
Otchsn ` mcfk jnr bnsn` yhrhon`n htonsn fctkdf
shrjhst d`h yfp`fotfdr. Rhrjhst mcfrfo dhkju otchknk zhrnkh lfvfkk dnrhkbn hton ht`hothdnr.Lfvfkk dnrhkbn dhk bns`nk lzkk ofrhsnych dmrumrfktcdr. Jnr t zf`fk dtkdh bns`nk (v) lzkvh dt `hsfahyn jucukuz.
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7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
honc 6.6.dhk dh grchbhn gnjn otchknk zhrnkh nonouvvht hton ht`hothdnr9
hrhon`n ouvvhtn 9 `g
Lfvfkk dnrhkbn 9 o`v:
Jurfdf o jnr sfjnt vh g yhrhon`n ouvvhtndnr.
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
Bnsn` yhrh dmru dtkdhk `g ; o`v:dnr.Dmcfysycf bns`h hton hdhk kht ouvvht,
A 0 g o`v: (6.:) dnr.
A 0 ` f (6.2)
mcduukdfk (`, otch vh f df nv`hyn jhcnrt`hothdnr).
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65
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.4)
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66
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.4)
vh dmcfysycf
(6.8)
hcdh hdncnr.
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6:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju dhkoch`dh (hntcnoth) dnahrfksnyhc
oftsfys jucukduukdfk dhkoch` dnahrfksnyhc
dhkoch` mcfrfo jncnk`hothdnr. Ju dnahrfksnyhc
dhkoch`nk z` smkubukdf jnr t fkkdfon v
lzk hcdh ht`ho ``ok mcfbfotr.
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62
Jc` 6 Dnahrfksnyhc Dhkoch`chr
snk bns`nk t zf`fk ofdfr d`hsn smkubufckfk `hsfahyn jhcnrttnnkn vfrsfyfc`. Ju`hsfahyn juc`fo nnk
nafdhsn (6.8) hntcnnkdhyhrnkh omkursf ju hntcno,
(6.1) hocnkh dkr.
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64
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jnr dnahrfksnyhc dhkoch`dh hk yosho dhrhbhdhk trhvy(k) nsh dhkoch`h knkbn dhrhbhdhk dnahrfksnyhc dhkoch`dhknr. rkhnk, (6.8) hntcnnkdh hk yosho trhv
jnrnkbn dhrhbhdhk mcduukdfk ju hntcnh jnrnkbndhrhbhdhk dnahrfksnyhc dhkoch` vh (6.1) hntcnnkdh hkyosho dhrhbhdhk trhv
nonkbn dhrhbhdhk mcduukdfk ju hntcnh nonkbndhrhbhdhk dnahrfksnyhc dhkoch` dhknr.
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68
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Qhrnchk rkhoth jf`c dhnohkchr ymc s,s 0 a(t)
vh lz v,
v 0 g (t)sfdhbh tho jnr t jf`sz dhnohknknk amkosnymkcfrmcduukdfk
mrdnkfry dnahrfksnyhc oftsfycfrkdfk dmcfy (6.8) vh(6.1) dnahrfksnyhc dhkoch`chrn fdn (mrdnkfry) dnahrfksnyhcdhkoch`chr mcfrfo jncnk`hothdnr.
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61
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Hhr z jf`c dhnohkn x vh y gnjn non jf`szdhnohknk amkosnymku nsh rkhnk,
z 0 a (x, y)
nsh, zknk x vh y dhnohkchrnkh grh trhvchrn fckrsf,
vh jhkzhr honcdh,
os`n dnahrfksnyhc oftsfycfr hcdh hdncnr. Dmcfysycf jutr oftsfycfr nhrhk dnahrfksnyhc dhkoch`chrh dh os`ndnahrfksnyhc dhkoch`chr dhknr.
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67
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkhnk,
(6.7)
vh(6.
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6)
dnahrfksnyhc dhkoch`nknk z`k hcdhhdnknz. f jnr sfjnt dhhrn nafdh ht`hothdnr.
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::
Jc` 6 Dnahrfksnyhc Dhkoch`chr
nafdhsnknk xh grh nkthgrfcn fckrsf,
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:2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
nafdhsnknk xh grh nkthgrfcn fckrsf,
(6.65)
(F jnr sfjnt) hcdh hdncnr. Ju nafdhknk xh grh
nkthgrfcn fckrsf,
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:4
Jc` 6 Dnahrfksnyhc Dhkoch`chr
nafdhsnknk xh grh nkthgrfcn fckrsf,
(6.65)
(F jnr sfjnt) hcdh hdncnr. Ju nafdhknk xh grh
nkthgrfcn fckrsf,(6.66)
(J jnr sfjnt) jucukur.
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:8
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Hcdh hdnchk ju nafdhknk xh grh jnr ohz dflfnkthgrfcn fckrsf,
(6.6:)
hcdh hdncnr. Ju nafdh vhrnchk (6.>) dnahrfksnyhcdhkoch`nknk z`dr vh ju z` F, J, B
gnjn sfjntchrn nhrdnnkdhk ghkhc z` mcfrfojncnknr.
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:1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
F, J, B sfjntchrnkn hcdh ht`ho nnk x 0 5 nohk
frtcfr (jfcfkg frtcfr) vhrncnrsh, ju dhhrchr (6.65),(6.66) vh (6.6:) kmcu hntcnochrdh yhrnkh omkursf,
(6.62)
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:7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
F, J, B sfjntchrnkn hcdh ht`ho nnk x 0 5 nohk
frtcfr (jfcfkg frtcfr) vhrncnrsh, ju dhhrchr (6.65),(6.66) vh (6.6:) kmcu hntcnochrdh yhrnkh omkursf,
(6.62)
(6.64)
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
hntcnochrn hcdh hdncnr. Ju hntcnochrdhk
jucukur.
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25
Jc` 6 Dnahrfksnyhc Dhkoch`chr
hntcnochrn hcdh hdncnr. Ju hntcnochrdhk
jucukur.
Hcdh hdnchk ju dhhrchr (6.6:) kmcu hntcnoth yhrnkhomkursf,
hcdh hdncnr. Ju (6.>) kmcudnahrfksnyhc dhkoch`nk zhc
z`dr. zhc z` mcfrfojhcnrtnc`hsnknk khdhkn, jfcfkgomuccfr dhntnrncdnnkdh F, J, B dhhrchrnknk dh juomuccfrf grh dhn`hoth mc`fskdfk dmcfydr.
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26
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.8.
Os`n nkthgrfc ykth`n uygucfkrsf
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2:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.8.
Os`n nkthgrfc ykth`n uygucfkrsf
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22
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.8.
Os`n nkthgrfc ykth`n uygucfkrsf
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24
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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28
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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21
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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27
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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45
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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46
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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4:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
:
:
( 6)
( :)
( :)
( :)( ):
( 2)( ):
x x
x
x
x x
x
dyh x h Fx J
dx
h x Fx J
dy h x Fx J dx
xy x h h F Jx B
xy x h F Jx B
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.4.Dhnohkchrnkh fyrcfjncnr dnahrfksnyhc dhkoch`tnpn9
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2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.4.Dhnohkchrnkh fyrcfjncnrdnahrfksnyhc dhkoch` tnpn9
(6.67)
a(y) sfdhbh yknk, g(x) nsh sfdhbh xnk jnramkosnymkudur.
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4
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.4.Dhnohkchrnkh fyrcfjncnrdnahrfksnyhc dhkoch` tnpn9
(6.67)
a(y) sfdhbh yknk, g(x) nsh sfdhbh xnk jnramkosnymkudur. ^uofrdfon nafdh
a (y)dy 0 g (x)dx (6.6
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8
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.6
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1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.6
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7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.)
dnahrfksnyhc dhkoch`nknk zhc z`kx0 5,y0 5 frt nnk hcdh hdnknz.
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) hntcnn dzhkchknrsh
hcdh hdncnr.
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>
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Grcd gnjn ydhnohkn jnr tfrfatf, xdhnohkndnhr tfrfatf yhr fc`fotfdr. Lhr non tfrfak nkthgrfcnfckrsf,
(6.:5)
ghkhc z` hcdh hdncnr. Jurfdf ydnrho mcfrfo xnkjnr amkosnymku hocnkdh nafdh hdnc`h`hothdnr (y,xnk ofpfc (n`pcnbnt) jnr amkosnymkudur).
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65
Jc` 6 Dnahrfksnyhc Dhkoch`chr
x0 5,y0 5 (6.:5) kmcu hntcnoth yhrnkh omkursf,-6 06 + Fhntcnnkdhk F0 -: hcdh hdncnr. Ju dhhr(6.:5)dh yhrnkh omkursf zhc z`,
(6.:6)mcfrfo jucukur.
:)6( :
6:
xh y
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66
Jc` 6 Dnahrfksnyhc Dhkoch`chr
x0 5,y0 5 (6.:5) kmcu hntcnoth yhrnkh omkursf,-6 06 + Fhntcnnkdhk F0 -: hcdh hdncnr. Ju dhhr(6.:5)dh yhrnkh omkursf zhc z`,
(6.:6)mcfrfo jucukur. (6.:6) kmcu nafdh dzhkchknrsh zhcz`,
hocnkdh hcdh hdncnr.
:)6( :
6:
xh y
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6:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Dhnohkchrnkh fyrcfjnchk dnahrfksnyhc dhkoch`trkh ncnonk rkhochr ffdf sukuc`utur.
rkho9 6.>.
dnahrfksnyhc dhkoch`nknk x 0 6, y 0 5frt nnk zhcz`k hcdh hdnknz.
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62
Jc` 6 Dnahrfksnyhc Dhkoch`chr
hntcnnkn kbh dhnohkchrnkh fyrcfjnchk tr lfcnkhdktrhcn`.
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68
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
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61
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
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67
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
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6
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
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:6
Jc` 6 Dnahrfksnyhc Dhkoch`chr
mcduukdfk,
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::
Jc` 6 Dnahrfksnyhc Dhkoch`chr
mcduukdfk,
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:2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Fxy
y
uu
du
y
y
duxdxxdxduxu
)6ck(:
6
6
6ck
:
6
ck:
6
:
6
6
6ck
:
6
:,:,6
:
:
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:4
Jc` 6 Dnahrfksnyhc Dhkoch`chr
x0 6 nnk y0 5 omucu ouccfkcrsf
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:8
Jc` 6 Dnahrfksnyhc Dhkoch`chr
x0 6 nnk y0 5 omucu ouccfkcrsf
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:1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
x0 6 nnk y0 5 omucu ouccfkcrsf
hcdh hdncnr.
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju dhhr ghkhc z`dh yhrnkh omkursf,
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25
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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26
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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2:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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22
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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24
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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28
Jc` 6 Dnahrfksnyhc Dhkoch`chr
zhc z`hcdh hdncnr.
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Dnahrfksnyhc
Dhkoch`chrUrma.Dr.fjfkHZHK^fsfr knvhrsnthsn Ahk-HdhjnyftAfocthsn
Jc` 6
6.8.Lm`mihk Hntcnochr9
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.8.Lm`mihk Hntcnochr9
tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhc
dhkoch`chr Lm`mihk Hntcnochr mcfrfoskacfkdrc`tr.
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2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.8.Lm`mihk Hntcnochr9
tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhc
dhkoch`chr Lm`mihk Hntcnochr mcfrfoskacfkdrc`tr.Dnahrfksnyhc dhkoch`nk z` nnk
y 0 vx (6.::)
dnyhcn`. Jurfdf yvh vdhnohkchrn xnk amkosnymkcfrdr.
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4
Jc` 6 Dnahrfksnyhc Dhkoch`chr
6.8.Lm`mihk Hntcnochr9
tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhcdhkoch`chr Lm`mihk Hntcnochr mcfrfo
skacfkdrc`tr.Dnahrfksnyhc dhkoch`nk z` nnk
y 0 vx (6.::)
dnyhcn`. Jurfdf yvh vdhnohkchrn xnk amkosnymkcfrdr.Lhr non tfrfak xh grh trhvn fckrsf,
(6.:2)hcdh hdncnr.
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8
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.::) vh (6.:2) kmcu nafdhchr hntcnnkdh
yhrnkh omkursf(6.:4)
jucukur.
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1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.::) vh (6.:2) kmcu nafdhchr hntcnnkdh
yhrnkh omkursf(6.:4)
jucukur. (6.:4) kmcu nafdh dzhkchknrsh
(6.:8)
hcdh hdncnr.
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7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.::) vh (6.:2) kmcu nafdhchr hntcnnkdh
yhrnkh omkursf(6.:4)
jucukur. (6.:4) kmcu nafdh dzhkchknrsh
(6.:8)
hcdh hdncnr. Jukuk smkubu mcfrfo dhnohkchrnkh fyrcfkhntcnn hcdh hdncnr. Lhr non tfrfaknkthgrfcn fckr vh ncgncn dhnohkchryhrnkh omkursf vhrnchk lm`mihk
dnahrfksnyhc dhkoch`nk ghkhc z` jucukur.
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.6:.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
kbh ju dnahrfksnyhc dhkoch` trk jhcnrch`hyhfcfc`.
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65
Jc` 6 Dnahrfksnyhc Dhkoch`chr
mcduukdfk ju lm`mihkdnahrfksnyhc trdr.
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66
Jc` 6 Dnahrfksnyhc Dhkoch`chr
mcduukdfk ju lm`mihkdnahrfksnyhc trdr.
nafdhsnknk xh grh trhvn fckrsf(v, xnk jnr amkosnymkudur)
hcdh hdncnr.
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6:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf,
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62
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf,
Jc 6 Dna n c D oc c
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64
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf,
(6.:1)
jucukur.
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Jc 6 Dna n c D oc c
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67
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju nafdhyn dhnohkchrnkh fyrcfjnchk dnahrfksnyhcdhkoch` trkh dktr`hyh fcfc`.
(6.:7)
hntcnnknk lhr non tfrfakk nkthgrfcn fckrsf,(6.:)
hcdh hdncnr. (6.:>) kmcu hntcno vhrnchk dnahrfksnyhcdhkoch`nk ghkhc z`dr.
Jc 6 Dna n c D oc c
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6
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.62.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
(6.25)
mcduukdfk, y 0 v x dnyhcn`.
nafdhsn (6.25) kmcu hntcnoth yhrnkh omkursf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:5
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:6
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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::
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:2
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:8
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:1
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Lhr non tfrfak nkthgrfcn fckrsf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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25
Jc` 6 Dnahrfksnyhc Dhkoch`chr
ju smk hntcnoth yhrnkh omkursf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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26
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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2:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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22
Jc` 6 Dnahrfksnyhc Dhkoch`chr
hcdh hdncnr. Ju nafdh vhrnchk dnahrfksnyhc dhkoch`nkghkhc z`dr.
Dnahrfksnyhc
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Dnahrfksnyhc
Dhkoch`chrUrma.Dr.fjfk HZHK
Hgh knvhrsnthsn @lhkdnscno AfocthsnJncgnsfyfr @lhkdnscnn Jc`
Jc` 6
6.1.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:
y
6.1.
Qhrnchk dnahrfksnyhc dhkoch`dh bvhyf bsfjntchrnkdhkhk fz jnrnknk sar mc`fd vfrsfyc`fotfdr. Lhr nonsnknk
sar mcduu duru`df hntcno lm`mihk dnahrfksnyhc trkhdkr.
Ju dnahrfksnyhc dhkoch`nk z`kdh non duru`uk
gzkkh fck`fs ghrhonr.
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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4
y
honc9 6.:.U kmotfskk Mxy vh M\^ ommrdnkft hoshkchrnkh grh pmznsymku.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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8
y
(6.26)
dnahrfksnyhc dhkoch`nkdh,x 0 l+\, y0o + ^omkursf dhkoch`
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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1
y
(6.26)
dnahrfksnyhc dhkoch`nkdh,x 0 l+\, y0o + ^omkursf dhkoch`
(6.2:)
hocnkh dkr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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7
y
(6.26)
dnahrfksnyhc dhkoch`nkdh,x 0 l+\, y0o + ^omkursf dhkoch`
(6.2:)
hocnkh dkr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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rkho 6.67.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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66
Ju hntcnochrdhkl0 : vho0 -6 dhhrchrn jucukur.
Qhrnchk dnahrfksnyhc dhkoch`dh,
x 0 \+lvhy 0 ^+o omkursf dnahrfksnyhc dhkoch`
(6.24)
lm`mihk dhkoch` trkh dkr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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6:
(6.24) kmcu hntcnoth,
omkursf
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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62
(6.24) kmcu hntcnoth,
omkursf (6.24) kmcu hntcno
hocnkh dkr vh jurfdfk,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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64
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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68
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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61
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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67
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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6
hcdh hdncnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:5
(6.28)
dhhrchrn (6.28) kmcu hntcnoth yhrnkh omkursf ghkhc
z`
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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::
rkho 6.6
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:2
Hn`chrn hnt mc`fdkdfk pfy vh pfydfdfondmrucfrk ohsn` kmotfs(l, o) mcsuk.
l o 0 5
l
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:4
Hn`chrn hnt mc`fdkdfk pfy vh pfydfdfondmrucfrk ohsn` kmotfs(l, o) mcsuk.
l o 0 5
l
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:8
x 0 \ +l vh ^ 0 ^ +ohntcnochrn vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf, dnahrfksnyhc dhkoch`n`nz
(6.21)
lm`mihk dhkoch` trkh dkr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:1
x 0 \ +l vh ^ 0 ^ +ohntcnochrn vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf, dnahrfksnyhc dhkoch`n`nz
(6.21)
lm`mihk dhkoch` trkh dkr.
^ 0 Q\
nafdhchrn (6.21)df yhrnkh omkursf,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:7
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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:
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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25
Jc` 6 Dnahrfksnyhc Dhkoch`chr
8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
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26
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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22
Jc` 6 Dnahrfksnyhc Dhkoch`chr
8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
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24
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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28
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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21
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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27
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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2
(6.27)jucukur.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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45
\ 0 (x 6), ^ 0 (y 6) dhhrchrn (6.27)dh yhrnkh
omkursf
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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46
\ 0 (x 6), ^ 0 (y 6) dhhrchrn (6.27)dh yhrnkh
omkursf
ghkhc z` hcdh hdnc`n mcur.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
o 6 6>
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4:
rkho 6.6>.
dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ufy vh pfydfdfon dmrucfrk hn`n hnt dhncdnr
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42
Ufy vh pfydfdfon dmrucfrk hn`n hnt dhncdnr.Dmcfysycf ju dmrucfr ohsnnr vh ohsn` kmotfskk(l, o) mcduuku vfrsfyfc`.
:l o 0 5l +:o 8 0 5
hntcnochrnkdhkl 0 6,o 0 :jucukur.Dmcfysycf\ 0 x 6vh^ 0 y :dnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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44
x 0 \ + l vh y 0 ^ + o hntcnochrn vhrnchkdnahrfksnyhc dhkoch`dh yhrnkh omkursf dnahrfksnyhcdhkoch`n`nz,
(6.2
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48
^ 0 Q\
hntcnochrn yfrd`ycf (6.2
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41
^ 0 Q\
hntcnochrn yfrd`ycf (6.2
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jurfdfk
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4
Jurfdfk,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jurfdfk,
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85
Jurfdfk,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Jurfdfk,
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86
,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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8:
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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84
Jc` 6 Dnahrfksnyhc Dhkoch`chr
8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
165/199
88
Jc` 6 Dnahrfksnyhc Dhkoch`chr
8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
166/199
81
Jc` 6 Dnahrfksnyhc Dhkoch`chr
8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar
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87
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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8
Dmrucfrk pfrfchc mc`fs duru`u.
dnahrfksnyhc dhkoch`nkdh pfyvh pfydfdfon dmrucfrk
hn`chrn hnt yfkn dmrucfrpfrfchc nsh dhkoch`nk z` nnk ffdfonfd`cfr uygucfkr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Duru` NN.
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15
Dmrucfrk pfrfchc mc`fs duru`u.
dnahrfksnyhc dhkoch`nkdh pfyvh pfydfdfon dmrucfrk
hn`chrn hnt yfkn dmrucfrpfrfchc nsh dhkoch`nk z` nnk ffdfonfd`cfr uygucfkr.
dhrsho
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
n cnn d cd dncn
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12
hntcnnkdhk hcdh hdncnr.
hntcnn (6.2>)df yhrnkh omkursf
Jc` 6 Dnahrfksnyhc Dhkoch`chr
ntcnn d cd dncn
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14
hntcnnkdhk hcdh hdncnr.
hntcnn (6.2>)df yhrnkh omkursf
(6.45)
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
vh ju hntcnnk dzhkchk`hsn smkubukdf
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11
(6.46)
jucukur. Grcd gnjn vhrnchk dnahrfksnyhc
dhkoch` dhnohkchrnkh fyrc` mcduukdfk
hntcnnknk nkthgrfcn fckfrfo smkubf gndncnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.:5.
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17
dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
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Jc` 6 Dnahrfksnyhc Dhkoch`chr
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75
mcduukdfk dmrucfr pfrfchcdnr.
z 0 x :y dhrsho
Dmcfysycf,
mcur.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkh
omkursf
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76
omkursf
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkh
omkursf
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7:
omkursf
vh ju hntcnnk dzhkchk`hsn smkubukdf
(6.4:)
hcdh hdncnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
(6.4:) kmcu hntcno dhnohkchrnkh fyrcfjncnr duru`f
dzhkchkhjncnr
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72
dzhkchkhjncnr.
Ju hntcnnk lhr non yfkkk nkthgrfcn fckrsf,
nafdhsnkdhk
Jc` 6 Dnahrfksnyhc Dhkoch`chr
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74
hntcnn jucukur. Ju hntcnothz 0 x :yomkursf
(6.42)
ghkhc z` hcdh hdncnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
rkho 6.:6.
dna n c d oc n n
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78
dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Hntcnothk dh grcd gnjn
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71
g g
mcduukdfk dmrucfr pfrfchcdnr.
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Hntcnothk dh grcd gnjn
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77
g g
mcduukdfk dmrucfr pfrfchcdnr. Qhrnchk dhkoch`dh,
Jc` 6 Dnahrfksnyhc Dhkoch`chr
Hntcnothk dh grcd gnjn
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7