Upload
eem-ders-notlari
View
399
Download
2
Embed Size (px)
Citation preview
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
1/74
IGHjG
yAsGA
PGALdMdKe
pxsrers
GMG APXdLer
vrBGL@e
sre
e
lrvr
@ECGXJL^EUGM
@GLAMGOMGX
@GX^
LDPMJXE
Goge
Igglmgr;
Bjajl
zlvrl
Gmgatrdlea
Obgl`ese
Ojmzgog
Hemeoe
vg
Obgl`ese
Jyg PDOHJA
Gmgatrdlea
Obgl`ese
Es
tga
v g
Dlgrem grelez
Eel
; jhemgreluhese. balFiouem.
n
do
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
2/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
3/74
-M
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
4/74
-.K.
k;
e,e
_
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
5/74
K|.
EK)
_
m
M
e-,
m
__
_m
ee
rm
]_
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
6/74
-k_:5T41MRe
E
r e et:
|-_--,
-e
em_:
'--_
e_e
m
'-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
7/74
tq)
,
_me
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
8/74
E
e
.,.,. -
ee
m
TTTTT_
eme`
--;
t
m
t
e
j
e
e
t
m
t
m
m
E
E
m
E
E
m
E
m
E
E
E
(
m
E
(
(
E
(
e(
(
(
{
m
m
(
m
m
m
(
:
e
m
-.
M
ee
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
9/74
eT_
r m/1
|
_
:---
k.
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
10/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
11/74
_
T-
---m
K_e
pQx)e
-m.-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
12/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
13/74
e-
u/
k
+ _gkc ,
r
:-:-T-,
_
E
_
e_
TT _Te
r
e_
E
e
:mm
e
me'm
_m_
m--.*e-
_,
,)
E
:
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
14/74
LG{Q*
tg...|e
'
'nmgT
k----e-r-
m
-*-|--:.eT1
--:-'-,
-,
-,
.-";-.--T.---.T-T-.--'-m-m-----|-T._-;T
TTt
-
+
T
r,T-.k
:-.
-jrgVr.
#tr+
PC-,,
C--;;
,-e-e-e
e
-_--_
m--
_-
--
_ TeTT-,-_ t
Tr
_
,-e-*- -T-m-M-e---k--e-'|-|-m--_---_--_.
|e--c.-cce-e-{-m-
M
e
m],_a]
e]\'.e
_
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
15/74
ec}
-
,-T-_
TTT>TT
_
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
16/74
.k
m
ctq]
-----].-K
:.,;e
u..Q-7;;;
.-mT
V\Tyl`_r TM:
M-
-|-Tm
M t :.:
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
17/74
tTe
m
T?.
\,e7en*_i0jT [EeT.T,,
,--
be|
--
{_rd
_.-T
V
e
-r, |
J0m
m
1
m,.
1]k
n,
{
m_
ee-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
18/74
m
T_-m
e
_M-m
_
M
emee
e
k
e
ml
,em
m,te
e
_
mm
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
19/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
20/74
emMT;
r.-_ _.,m :,: m1.-m'
,,r
S|
e _; _-
:k.
rm
'mm
|--e-eTT--
mm_e
mE
/
/
tNd
E
_mt
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
21/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
22/74
m
,.|
e
e
j
E
{
m
1m
,j
m
m
,(
mm
m
e
m
m
eTT--
M-
|-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
23/74
_
m
.)
e-?
---|--
rT,
_
_mm
t
e__m
___
eT'm----Tk-|
mm_e
_-_-_
1
kmmek
Me,K
_TTTr+1
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
24/74
ek
Em
m_
--?r
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
25/74
T_
m
E
m
_
e
-TT:
m
--m
m
--
_1
m
TTm
m
E
T_
_
Tm
m
_
e
m
_
:
--_
_
m
M
m
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
26/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
27/74
e
m
m
K];
+.]--
M_
meme
-,-1-K
m-
T
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
28/74
mmmm
MM
mxm
{r)
,.
9
K
T-.-*
ct
e_+
q4Tdm
cK
Ctr
1?E
,Qm,
mT
.-|K
Ee
_
ee
+
0r
Kr_,
m
m
Mp,
m;m"
m.
Em
_
E
m
t
_
-|mm
+e+
.:..:,
b
m
,7C
T|M;
M,
||r;1i
crct
:Qcmr,
G+
_kcc
,.'_j; ,*?
-*mm
*
:
_
.
:W]?7.tr}
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
29/74
M.,,.,,,,T *
tn"t,
m
{;-,,,,,,
},
{',rr,,:
Q
;
K"'
-
>;K"*
eK'*
m,(x}
,}0c,s,
"
,
K
,.,
0':":m
N
tG
.0
tr
G
G
-:
*
:
G
-,0
t
z
G
.:
G
.>
G
J
tr
tr
g
-
tr
0
M
-
m
j
]
t
;
,-;"
/"
mmg,
t
m
R
.m
m
--,
}
.-e
eu'
n:
(x]
mt
,{x
"c
.
"mm
"--t e
({Tm
m
7]
ee{"(T_
+( m
E
K
)..
E
g'
x
{)i
t
],: n
'
(,
n"]"];
,r
((
,
, r,"{
,:
*,"
,.;
g{.,'.,'}
}
"r
(T|
n
{.
s
c,mr,..Kg*
Kn,mr
m*.,,,
m
,:
K
\.
1
e,n.
m
(,,.
:---.
>p(xm
(t{.
*,t,
]
+
`.
-mU
r
jU
:T{
(Km(k
9
nr (
*
Z
r,r.|(v
t
,m
t
x
0
(x
}
-9
-)p
(x
}
{d
)
-
K*m
j9\;
,
,,.
(J-T,m,,*+t);,
D
jy
r
;
(
(
+
(: m.(sr
T|;
-e
),,...
c'+
>e,e
t;
K
8t
\0
::
M:
;,
m
,}1
4,]
. }
N
/
(
U
,-
t.,
m
.}
;
}
(-tuKT
K
-
(,}(
R)q,(
e
.,.j'
:,4
{ye
.
n"
e
Km
(,'(u
)
y,
'(
m
+
(
,'
("
}
.y.,'{
}
+
n|e(()
e_,"{
r}-+r,m:x)
yr"(t
,, r,r,.
""
,:d
.{,
-,"
K;
{'
KT
'K
(.
(
x
)
y
Tm
(y
}'-
-K
K
>
. ,
m|
k
'(,
|.U,},
,r)
;.{
e,
{;
.-
eB.
t]
|
U
K
t}
{.:
e
m
(
evm
-c
:,'(vm.
N:
(x).
m
'
g
,.9
r
mt'xm
T
'{
x
]
r
+
y
(;'
{
x
(+
E
,;
E-
]
K
Qe{}
b(reKr
9m'
,(Sm
-,
m(,
-
l
M
MN
}`M
9x
(T
,K(*),
(*
B
e'-r
B.
E
qr)
K
,,.
m,|q
M
g
t
;m
.
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
30/74
-c
-m
T_T.
6
e
-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
31/74
't"
"
'
m-r'
m: t
T
M,
-
-
?
ed
dm
K,
.."
,.m,
:
-
t
"'k',,,,,
-e dre,
+
,.*
,e
M-e
t
.
V,-m
) -+
M
*' e
',r
e
E
k,r
k,
{
,...`;.
,rme,(r.r,.
71,
,.,t
-m,"''-
Mt
ut
t"4
};.;rr,
:, t.M,,e
1" M,"
-d\M:,,:
""
r,"*jM
e-tr;rg,
,-;
{
Klj
"
e1n
*,c
/]
:,/1
:r
`,/,'-
,
m,
M
.,
-M
r
MlT
`
,'
'e
':..r}.'.1
rm
*,,,,
'e
.-..,.,.
0
*,
.e
1
e'
:
1
..
{
'*.
'u
.,t.,.
M
q
{.
K8,
j t,,
.e
e.'
et
r/
m.t
/
/-"]
m
{ jr.
._
eke.',
k
(M
;
-,/
t9
"
e
rx
'
e; e
K
:
._
-m|
/
""K
*,
m,
/
m.
1"
.;
,t-
Kj\
e
,e;
k1
e
,"
,,,m
e
,
.{
'*
Mt
{,,
emt
}m,t
..
,
].. e]mt(.,K.
T.c*
K
t
.',k',,.
|
.,
)l,
=,M.l)
. +c
,,e
e
.
t\*
,:
.
v( m j.
K
m.
m
M
T .,e
t"
:-,ju=-
.,(1
.{
e
M-,;.
@t-
e
,
` ;
;.,
{
._
,.
mt
-,
(m-m
{
,,e
}],
].T...],..,--m
M:
{,.:T
(jr,
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
32/74
-m
j'1e1
_
j1
E
e(k
_Et
t
tcc
e
-+
| r
(
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
33/74
md.
*.
{
,,,r.r.
g
e
:|j
".,
;
m,
l,
(M.Mm
;M,
t}7,
d
(Ml.M.,-{-4t-rk;.}r,
Mqt
.-Mr.}
).]x)
/",|,e,.1,,-,e,.,,"
,,,,{.,".,lM
,e,
,m.;
{.-,Jt
"c,mt"\
*>,r-,,",,,M1,/",
',)
()
*(n.,Q{,..,.,.-,";Kne,
P
M.,e
,.,,.,.,1
,
(')
dcr,tncc
.*
,e.,K,.*,:,e`:
r.'.,,,,,,r
y(-,"e
,1KK{.c,r-"
1K."eer;,;*T
.,,
,"m,1='
:e:
{jK
r,
m,,,'"
r,
,
-e
{:
:T
{,
"
K
e1;
.mn.ehmt1
.
{,,/nT
l..m,
k,r
,
k*K{1,,,.
-mKn
g
K,c*.
y
{R
{e
T,rcuM
|,,-m,M
e;
d.,kr,
K*T"..{:,"\\..,Tt"g
Mmc
,1.(,
y.,,
:mxe.,.
|-
11e,.-,,..e,,K
.4
K.,(1mmr;.Tr
*:-,kt,|,Kt;
Murmu.r,rr.
e.
)
5{Tx
),leTe>u
g Kr
l,.m;mm."
3
te*1,q
mm
tm
,c
0:u,q,-
tr
>.(x)
'g
,m,.;-k, ,m\'{
,m"{n,*
Kj
-_
M.
,e
)m
M.
ME
Me
M>
M}
Ml.
M>
b
{T)
}
M
e"T
m(Sm
Q;
-M\
M
-
Mt.
Mrm
}
"}]
?
t|
,
t{x}
TM,e,Tm
T/
(
_),K,
(x]
,-,,e;r,."eT,/""
.
m
,,
pk.
,'.,,.
e
'
-m
.
|`
g.{*tr,",.
m
M-j
mK;"
.T{,
y-,
(u
)
t
:" g
mKg t,e,",t
M
",|,-,
|.d(g\",],
r
)
{
1m vt
{'m
}
+mg,,
m
ec
*tr,"
,1j
"c
n{
,Q-,.-,g,
c-,,,
K,.,M.
(rr,Qm .r
d:r,n*
;,
r
e
}]
e
'
,,{.,"eM..*,eq.,;
,T
m ..1
t1,}{,
K,
mx)
.
"K{r,e
:"-1*"mTm
,,,",
Ql,t,'d
,r.,,r.MT
e e....e. ,
M t,u."
:,
{
r}
t\g-.
,.Mue
m
.(]
Krte**eTd:0
.: 11.,,
M1r1.-*m
|c,{.,/\|M;
,.,e., :;
kjt,{
q",T
t/1]
,(1Q;
,t-d,K
(.rM.r.e**..,,{.,d{,*.
t
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
34/74
_T_e
+
0-c
t
.
_
.*,-_rTM
-
_Tm
*
E_
M
j
j
m
j
E
j
j
t
t
;d
t
j
j
j
,
j
,m;,1k{em?mk*E
Mm.r_
9m
,e4e;k
m;1
|e-:
*-Em-rrk-,
::
cce#1T';.,
Q,|._e.-meee_+
. ,
mTjcdr*.]
.T
-
:
rt|1|.|_
e|_-
,
,
|.:mmm'e,mG
4"11, |*,t:"
#+Mtrk
jmm_*m-e*+e
,
eeT_
m
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
35/74
/,,
m
l\'
M. i. t-:
1
-TkU
e
,,|
-k
;,
{.r
e.
,{e.
_m
. ,'k
",k, \
,,
'e
)e
,
r,
r
,:e
Mje;.v:,.
]|
k
e :,,
-.
-
m v
,e
t
,
M
jeKK
*
|K,/,
,m
,-
-",,K
,,
-e-Kx
,
{m
*_
,
..K r
(x
e
,
,e,
e
t'
-e
M._
*t
'
_:
e,
.
m{
,.m
-.m
].,:
m',mRmt
,
-
W,
|
T;
,*
.;-
k
e
|
}r'
|m
t
('
(
|//
M
:
{-
M
+
.,e
1
--m
:
_:
x m
.1;*--
K
_+,
t
m
,. m
-
:
KTc,
M;.,
+T|
,
m,,
,,
,
'r.''}
.e
)
,,
ee
''
.m(
-*'
,''-'
.;']
(]
K"
]'
,
(,,
*
e ,
}
n
,e,
M:
-t
e M
T
}
'k(
-
QS:;
t)
t ] ,
..
t :
-K\
(
k(,
*(-c
/
("
g
{Kr.
r1{,
]M?
crr-
-,
(,
z
m(,
jT
)
K
t
t
]
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
36/74
mM
{EE
m4dmM
hm{*-.jlm17gm
$g\r
tg*r{gr
h
e{g,-**
7-xg-
bg+Mmg*
K{g.*
tg
>1d
d..s,.
cu
hgKg,
sj}ec,
Mdt^j"mt
t
/,gjmr esg
m
3G
rtgr*e.
1W
me,gzr
Ke{.
}gdMcu"-Kgj
jm"}j0,
jr
h,,
>Kt.:
*
ZT-}j|{G}*Z>
y,'(r)
+
m,
:r'(x:
+
M_
U:
(x)
?
b,
(r}
.()
;
b.
(x}
S:
mxK
c
0,,
U,(x
}
+
+
Hj7{r
hc-
^e},-,rm
7j
2.ej
mj
\,t
,
ld-"d.j.Q
'{*.**.
dK,
"'gr1|ml,
tMy]
"
n
,>
()
+
,,S(x}
+
b,m.}
t{x}
,
Zr,
5:
(R)
+
j::
>z
(r)
+
b.
tx}
hk*,1\
`
k*md
M|
-.
3gjg]
bjt
K
,
>,
{rm
,
::{r},
{dlM}>d\-
'mgrl
l
Qjgg,
KK{,
K,gT""a"Qg\, {-,{gd"t,"",
5t
St
"z"
7.'(x:
,
m'l
tlm
,
Z,,(x}
c,Mr) * Zt 9mx)+--,
+
Z,\
>d(,n}0a,t,v}
qdl
(t}
,
{x}
-t
j,.,
).
mx}*
*
J\,,,5r,{y),{
a"(x}
c
gtm,e\&0cc,
PgtT
je.
..:tx)
em;\*,gmgj
{h\t:,d*\j
c{1t""
(,j-m
+
..-'
}
't
jgr-*=MsKt
l}
j
`
$gMmt&
W
|
-
t
.T.*q>
-+
`,m^=r,".
*:gt
j,.K
?
b
(R)
*M|gr-;,
Kl\g
*'
C
,er{.&.,
.{g.j.d:19t
-c
}l|,mg,"l
Km"azjr-r-
b(r},br|}D
erg
K; c*gT
m..,*j1Vl `,c
r
,,
i(r}
>:t1
l
.t1rq`m
$"-.\
K0
e|.,..-*9*"
k{*Kgs'.
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
37/74
*r
n"Kjl
,
e1
Heml.*gmn,.
m1
m-hr7gj
7,'(Sm
.
Z, y,(xm
kr'M:
?
q8.
y|(y)
^u
KeK0n\\1:
Oem
m,1
m*;m
(}
K'?
l,7
K-
5n
nj
mr-
>*mtxm
1:'(xm
1
K,,
(x)
yu'el
?
,:"v),
)?exk
iT,,
")
?
:..,t)
v_
(x1
,
5'l),)?,
b(x)
-
Z\T
>l-
m-
c
m
8rm:*
Z
r
tet.T-
"jD}{j"-
K
U,(e)?
5:(t);
}r
m"x
}
,
q
y .?
U0tv),
y
(x}
,j:(
x
)
>"
trm
,(\"u
,
>'"
-')
(
*
*l
}mm1..
*g5gl'djmm'e:*lmu
Mem'cug.\
",Mmg,""
.. |-g.T*r
-:
E
E
m
K
>?
Q
5,'
.r)
m
M
),'(*}K
)eT{g,l
lm1":gm
,.T
,K,
tc
Zj,
},
{v}
,mn-"1
.*
e,tx)
.-
(x
)
j
/
]gjMmr*l
K,1
a
-*1
(jm,n,MZM,
J,-
m
""
l,"m
Mlr,
',r,
K
Mj-t
7d>,
*\{.1s',
(m
u.gT
*d*.t";
)
(
m)
',+..`mg
s"l
",\
xR
i?
Z
g
g
Mmel
`=-
.:-*gm
Z
:1*,1
(z]
|]e\
/
{.l
M
e
:
dz.
n.,l
:e
Kq0 h1.
(7T\_\
rd.r
g,,t:
Z
-7j,
r\
Kg
*-
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
38/74
$ll*M
,
(lrdMT0Tr_t,tM
`jnmtg",t,
mb,Jjm
(e.''}',
t*(rtgt*
e(l,
(>}
Mu-}
e,*"*
>:1''}**
l
-:,
kgKr0g-
kjrKnc
(
_}
g
tK-
g
K_m;;,
ere.j
ij*nec
,
T
h,rjtg.0
}'e
7ut-j_M
Kg0
a|
j
tt}
1g ,,.
`g`ert
,
,
(.
cu.e&
mj3/}
>',
J,)
1
,Mmmmm4JM
0Tt-tg-,jgngQ
Tj+
(
)
'K-+
*&
gJ
-d
1
me4
,,d.g`j
&T
M.
h,
Kdy*r.
(t}
ej.jar1"
.]x
m &g"
?-
{l-
xr)g
j? Q
,:
dm
Ed,
|
sjh,*
vgm.e
,
b
e7g
Kg.b. djr,g
E
l-
:
}
mJ
-r]m
r,
*
ZG
cm*
r"-,b
(q
e{nn{d
r1
J
,.ljr,&cr
c.
0j
m
e,t
d
tgj.,".*u.r
G
0{
a-
Kg,.t-tg'\&r-'
b0*:j,t
j"d}e
t
{g
r,.Z-
*"}'r-;d
m
HuM-+-*M*""-]{M`gjmt-,.,';jyx{Kq.Kr1m.J.n{re,e]$.`ggrtg,"&.,}r\
j
u"Qg'e0,
*1*,jjc,
}{*{
7j
K,*
t8{
.
,*
$jjr.7Qj
>jg(
=r1
]Jc,d,7g,t",
,j
J'
rutj,mec,
j
e
l,
,tr
K
,],,,
eJ{
,,d
7,
"Kjcg"t
=*mej 2,
Mi".s
t
{"\
t-
s'.}
\gT
T'*}
dms.r\
"
Q*e
dt*,.u\,j
*-q'
N')
`,z*1p"mg",
i
Q*,-'
gre
b-,.-mg.r:.'
el
-
(e}
}tR
q?
N
6 gT
+
S
.j.
,&*''*]
*
>:.
(x}
5r{)
*ktl}
*
&
>1t{.t}
*
'[
^,
(}
-
7,(x)
-t
Q,l"-)
m?r
;
,;KQ;.,;;K
m,
"}r,
K
'--
-"
*|:s"
;)
tr,,K
(->m,-(
,
b
}
d
}.
tt
Kgkr
e
q1,,
,9j,,d(r,l.
}
">
/
}
*}
M
-k
m,
9z
-
J.
}
q
c')
?d
jm
`ghg.*m,.1
(m)
t
("-l
)
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
39/74
J,
_
mt,,
>
q,"'-
luM'
M:,('|-8:r"]
*
u,)
,
c
)
T
mm
-
mMmm
{4d::
+
,
-tM
\,q
Mumulud
:t,
(')*
j
j4)
,.\ :|
;
h.t
\
u
e-
M
(,}
&;
,/;\
]1r..r.,
km
j,
P".,}m e mm
Q
1,,,K
?
|
,
K
*t-r
}
>z
(3-J)
D
r
r.e,
1}g.g
{
n,,7jK,-
y?
yq|{M -K
e
e\,
5,''M
::.(,'?d,
M
e,
(')
,
>
:."'
}"
k
(m
&'
t
m
T
\)
M
,
q.'u)
:'"E
"
'j
t
>,t
K
e
-
e
-')
?M:_
nt
tm
_gK\+,4
M
kK"
u*
d
$t{x)-
n{
5.(x},
Nt
>x
N :K
K(
g+M
Zx
NrZ
q(
N-
,
jK"7*"
E
gTr
ml
g{m
mng,e,
mtg,{m
SdrTd-7*(,\
j
J
"ld-}ree,t-1,
g1m dme,.
.
H,
Kg'gr]
\
?
\
'
}r
h.jr}
7g{n\
jrtM{." q.nl
Mer1
q G81
*,e
K"er.
-,..,dKd
&
k,,,gTM
mG,K
,
(J
-
mE
},
D*
?
d
K,l
emu\,",
1{g
m
,
vq
,m: nc
nkg
r {
-,"
e
rggm
M,g 7e,
hr,r
l
j
,mg
.1Kr}g-r{;\-.
&1rg\
h..l
`"KUg.g
.m
-jM
,>dr*,
i
(l}
?
s
*\*
(
l,
vga{.
,
`1+
KgT\,(g\,,
Kemn+.-el
er
hgm1r
mler
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
40/74
(s*)
h*c,*t,
M
v=.MTm,jll
}
^.
`eQ,t,",'jn
M*},
tg,t
j,g{*#r
gt^v,"
M,
K-
e;
,'K
m1K
p,*J7
-t
J?
Q
-{
_
M
t-
_
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
41/74
",(
Q
^Pm/
7.
?
D
dmT*r
-
Q
;K
}-'\
j,7\,
7
7:?
m
'**
MM
me1
',M-;lN
mmj=gr
hc
.*K
*
n.uez,,
,
U"t. u"(g+}
]rZT
VT;&
(JT),K
tm_
,Q
K_
k
>"
hgmrKe
Kg,rMtg",\d1"
he"
jgm
7?
Q,K*
r?Q:,.'1em?.,
M
m
_
.
l>n
Kx
:8
/m/
zdmz
,,
K,
mK.k ulK,",
y"},
K:
-
).
,
E
Ki\
|m
v\
je
rE
e;K),
},
y,
(t}
,
M
.
>.Q)
,
Mt,g
n..(x+1"g>R
>x
n.R
>,x
+
>x
+
t`g>g,t*,
tl
mmd*pm j|:
6-s
h}j{,
(d*-rd.K,md.r
vmjmr-em
ml,0
GTKg,
mg,,
adomgM
=;{01r
mg,,
1 cg
}.l
m
d.-
mj"
k,"g
,]?
0+1C
r]r,
(-
1
:>
H.-\
uQr**Dt
d.
t
gtT,m,-e
5:-jM
mjrg-
,
(
J-
Ej-
,gMt,,
,
-\
Md\
mg
|"-m
g
mgj.1
M*
(
C
MM
i|*u|-
:drg*j
gm
c
(],
Kgeg,e
tg
7e,j
Q,
s
m?c,\t,
mmu-;l)
],,-
z
ugaT{,
>
jm
'
7d,
,
Kd.
c
g]g-,m1`,tr.
:)
Z
?
d
ErmZ?
j
TKgk"rr,g
mm17M1
-
>,rz--Q>--e*9r-
*rj*q
(
..
)
IK
7K}g""
r
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
42/74
R.r"jimtt*m
K"e=rt',
&
&--Qm1;1m
cr
?
V-
m-"r,
dQ-
,
qGz?r-
>
((}
,
Z"
gJx
?
. } `
zgM-td'v
?QQK
",Q
;m
_
m
Ky
tre
/=+Cm
x'
Z.0M*d'U,,\
?
(r+1t-1
,,g`j\y
Eg
Kcu(};
"d*(
V,n,Vd
-
&,Ke,,
Vx}
+
d, j*
{g,"rc
r
r
c,"px)
1}gT,t".
Bd,,itg.T
Kgzt; qe*g,
.lnm
uq
T7*j*,{
M1"mq,
*>,]
*trt
gt*ti\.
QK*}*mj,r"*cjj
e
*
dz
+1C
z},gcgelg
M**
gtm-j
et-r
mtjg
rnnE
Kg1g.,t
g,
$* ,,gmKg
gmrl
s,(t}
-
Jg
QxmK
?
g{C
(
c n,s
Vx
-
l,*+
px}
7gtx}
*
m*
>t*mk
,
**
(
c-
*t
$x
+
c
,:C(,K
:
?.
-Rx
:.*
rgTm
vn
njjt
,MK,,,*mjr\Kjl ttdt*'m
,la-m:,:d'm.,,r
6 g
j-,.J
H
(
r=
}
j
,7-
(x
}
e
t *
*\,
n M;l,j"*g.
|
r-
c-
m,
*.m
'
|;
M0-\
,d
m
m
t0,0+m;
|
]
K
E
t1_
-(
0
+
rl"M
.
#d}md\*.{
glr
\,l*..m'*e
.
._
,q
D
K,'
(x)
?
tK-
>9T
4
.:
.'Qx}r
>>,+ry.
)
c+m.M
1r+g,t{,q
cNT
s.
`
g
(
=-rm3x
+
1 )10
4}
},$.
`t,Mtn.
}tr0n,\\r,"r:
,\d{
g,1*\1,r,r
G-6*0"tT,
m8-
m|
m
rt-
K-t
\Mmyd,r*,rl,m(mr
*.`**m*rj`
*.
&*"0j*
&?
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
43/74
jm
j-h?tK
j+mh;
@
)
q.KzK*
Eg,-mqKgr
hd 0jrgt.
-
7?.m"
tz9
m
/
z
gT'
,-,.
+e
re,
&
g-':
n,
(-2-:+
e
7,-
(-g)
?t5{&-1m.'.{^)
3-
>x
-':
7r x}
d
K(
&"
,}n\
ej
Qx)1
m
E
K
Q1_,EK_
t\jt-a
`;
t{,
&n,lMmg\
7;c
}=-\\.
\;
(,eue*-.-
x), FT
l}*,
?|,
m
*(u'k1}\
?
r,
Kg"*.
(
M1K
M1K
,:M]
-.*-
Q
(j1}x
-
E
t1l_v
+
y"gm
Zze*
+
N..7.Qxm
tjt
]8=
-
N
Ker,
m
71>x
+
(:(nm,K
rg
g,m
"}r:d,:;
etj>x
1
(5cmx
5:
{e?
y,
Q{) ;
::tx),
P
:8
(x)
"
**{l'-P
"dm1,".,,,1K
.
}
t-}Te
*j |-sr])
En",m,_
n: .y,
(x)
"*'Q-
N:
MM:
k,
mem;j\.g-mtl
;
:|"-,_gl
.\1-*
Jueg,
t
jr
{-j{1
KJ-
\
K
|
?
Mu,qMmg="}mmT
Kr"le*"
eg"
\1
Kg0
m
),:]z
*J
m 7g-
bj.,
7,r
,]
c
jmgcg"
reg
(l-
}P
)&'')
|gr,mt.-\\m":
tg
&,m ,
g,j
dm
S"lc,MM
>gm;r.
((
)
,]
R
5(x)?
N(.Z.
g
+
Mrjq:Tr
tpc
jc
j
}e{=-.j",
s1.:n m
a\M
m"-,
T
,Kg\
k
`(m)
e,g t,cc,
n
&"}
g\"
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
44/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
45/74
}j
:
n1
g
S-m:&
trr.
m
'{
:,")*
e.'"
*
3:r"
T
md:,
(')
.r.
:.'''T
::
:
-8:(d)
0
t.''' -
5 :
]
tm
,,
M.
`gl
,TQg
e,'"?
-e,'
jj'\,
ql"'
j
4e
]d\
}j,, ,
jcg*
em
&
{-r
)
}vgM.{--,_
1rl
TJ1mg,g
*
g,T,
zKyg,m\g
)\]
-.W]
,1*
,,
(
"]
'.' . .,K /
`d"M
m
gT\,\
&/,).
\,
,cm*-
m
hmum"{r
,
'mg\.
E
t,d')
/
t,"}
M
\
{:
m:
P
,
-]
Kr
M
;,
K
g
l;
?)
e.")
*
7g,q
g,
7c
a
m?)
K
TKg
)
-*M
-
4
q,{'
k
Mr
r.m
*,-,
}
k,
K
e
mg
m
)
l**+
''
e;}u"-,K
EeKT,-
:
{e
'K;
:
m5"n.e
{*"tj|
g
,0
,
'e
q
-,
M*j
tnTe
cc,
c{-,-
r;k},,{,g.tl,Tel
*
:,lr*,
M
(_}
m ;
-m
m
-r
M
-e
( *m
N
m
-,
M,K
Ad*t,
j)
eM
h)
,,,
bltmgT.e
d)
,r1
,
"t,u-
$q
(e
rrm"c*
ge,m,
Kgkg"
.K.
'r*e***0{,eK,r
&*0,,
E
J,m't
m
M,`
:..(g{,
(,(,{,
WM.-dem
':
,-a/
m
,]+}-,J;Sd-ndm+g1$,+et(KG.\m{g\\e.Me\Mmg-m.M)>tte\
,e,
hd
e}
Gd*mc
",,*g
M..,
,
1g{n,
7
_*
,
&"/
ez
,*
M..-{
'
w
,
j/\\,(
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
46/74
mz;
)
).,K(*)
5"}*}
*
,d 1,.KgKm..
\K
d}.m*.
,m,l
1,,'
5cj,{'&T
st
M**k,,n "l,:',
2,,
|h;
Mj+m &*
1m1,1\
et",.,
M"\1,:
tvga0,mj,-
('?
t.M*
C
S
W
,G,
S
]Kmr
M-
Mg
me)
r,.md.Tn
Mer
.K.
*rQTme,r.
r{qjjheeK
j, q,
em,,\n1
m,T\gtr
t
c,"*,
,]
nT,",eeK*
/m
:"
J
:('}(*e,
N"'g"'
3mn,rj|,,{",
(>:
..r
(:T:
+r),l'*
k
e-1\q-"
h,jcr-,c
/,1,K*
h-,nt"{Kg
u'"mtk;
]m
K
'Cm
-
>.0,\-\j|'
K'
m''m
.,
;
.
Ktl
3glu]
n"r*;'l">l
>t)
(G*_\|gr
],t
]/j*
7
vgM{K,;
Kg
t
-
)
Kd.,
mr
g\;
e
mg
|
t
-mqer
tn,
y,,
",m
,'T
,{17
{rr.\
me\gir
TT.,
r}
,'m=
v
e,
{K'e
e,
x+
Q
d.
^'')
+
ns
(
&'d',
(xm
,
K*
Cg,=
K
tj)
U
Q?
(j-
1t
:mn
1:
Ee-r
}tR
T
(8]
(Kr
3mu
(
)z
E)
K
;
&
,'{dKgr;
?
N,,
&('mg
Mem
*.:?
y''m*)
-
,
)K
R,x
')
G
Mj{
t,
,],
?
]"
',
b",
;-
H-
dm,-"*Kd
m)
,k
],",,:t,
,'m
.,
1
Kg
.,0,
G1MT
mmd(e
,]
e
,,
,,
m="..,
0Q
"r
um*
m
M,
ka".;"g,
,,m
t,-,\ md.
e. '
M9,*
,
,T
d^..
kt..,
K
Kmr,r
ZT+hl
7*4't
:,.,e;
,em"
|ell"T,
Q,
oe,
t-, '.t--J,
t*:
,*r,.
G_m
:mmx}; (*x+:
')
(v)
;.
.(']
*
\'
'
"a*e
J
*
le
q
1''
c
.\
gT.
k;,
{.e-'
*
*&
,3M.-j../.
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
47/74
{'m
`*-tr.{g*e,*Kqr.
M,
]-
M-
`dM.".
{}
(
z
)
MrM,&
+t.MZ
jeur,
(.'le.:
-h.
m-cdK*r' (j)
K
9gr ,
]*lup
r.
v0
M:
/..K\\
{*
m.
t
5'
:.-
u\m 3tjrd
a
h1q.*
Kg
M=|ermgl;
r
s,c
\>
,:
>")
(
*
,i
tKg g
Km
E
r
r.
|rjct"Kmj.r
m
g
ter
m
',
n .
e,
sd0r
_,m.,
Qgl(*g
t{
t
*
Me0
c"r|,t,,*&-
e*Tr-t,
MT,,
cm
ci*:;."*
i
.=
m,* m*
*..,d
T
K,yr&0-
M',{d,.,,*,
hee>t" j,
,*h"
m1
\
Kr
r;,$
j
G
glct
*
r
4-
vg
M,/]V
()
(q)
']}
(:)
-
E m
K
v4M+d,
mgc
.
T ]T
(0-
\K
)mn
?
tct*}}K;
.\.d,*edd
M..&
dQ"G,
7
y,m{,e,Kg
(l-]r)K"&
et(
t,jm,mg/]e,
A
,
u
vy
/t,]
vg-t-e
,-|J;,,l",
\e
,{e,
Q,z ,*,* a,
e
-,m'd
,
k(r)tl)
?
(
m(x:+
Mx
t-)
g
"
{d,i1,1
Sc"
-'""*\,
}x
(zrr+M
}/*
{:.Tm-|-'
,
\(Mxt-Mx+,\-)
`x
KgmKiQ1 $n.gue,+ k,, l1
K;
{*',cec(
*mr\.'ct*
'm''1"0'g'-l*'t'**i
r,c`1.(
:tr,,}*,(,dd,
,'*1mgm
a,e*mgr 1l
*,,{g/g},ge9
Q
]r',d
(
+
'zM}x
-,]l,+
M
m
;
J
(
"*j-+
Mx+&)
MK
h,,ltd
0u,
R
,el
KKK*-l
|*-,wm
m-|,:
el
{d{>*lr, ,l,, 4]r0
c{,n:
1+
t(l,,M-{.r,)tT
Kd
,
ee
)
\
M M.K\E
em1*tQ-.,
],T./g*T.g
k",
M
a1,
K
glS|g.\\T,,l
K"-*c
ud7\mgr".'
m
,{,):x),
D.lr"
>'"'{),
ecm-x+7}l\'
K'')(*),
{
Q
r4+
T.t'
+..
*}'
(J-};r)A*,
tc
(0-}P}M,Mmn
(J-\jKl\,
t
g
/K
,K1e
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
48/74
*M 0-0
' d.*.
"e
*,.
Mkj
MT.{
(
/
v
2*
K."
Kg
t
{
J*
}:}
[Kx+"rKg\*
`......
*
K*
,:g\g
{
tK
.(x)?
7
K(
*0
r.
j
m*
t
|g\\1
t,\rd
r,,m
cZ,
,
mn
M)
,Kr.,,
Nm
+
n.
(
&x+
^}
+
n,
(
Dxt+
|*e1. l-\.
"
.j-
}u
j
cj-m
:j-,
K
*
>.
e,
Me,=
7,}t\.
Eq,-
vg
k7+ge
0,(ne$resm=-
)
m(t)
?
J
y
(
x)
,
1",kQ
"{*,{
e"K
KJ}}{,
m1"g,-
jb,,M|d\
)1
e,0
\,
mq-,e,:1"d
M$._umg.-,,
Kd.e*j,
y,.Y
(y
},
g
()
t1
",*,,ln
k;M* c""
31t'"
_'\,
d
Kdj
w(t]/
7MmT}...,|0r:
Utkr?
*,1)*\kR
r.*,-,-,e.-.
,,rt m"mr.'\*.j
:>er.'* 'tm.-.
M1'
arr.,
*d."mTr.1*
jt*{,
,y*
r*j*eg
:e:-:*,*.1,r
k;k"T;;;,;;';
#g,*1',
G*
31:,}.\`TgT
d,/jn,\j4T
,
>,?
>
tMme,,,Kg:
m
{.t,,T
,tul|,],\e,
rx
*.,Tme.,
Kg
MTe
n,e*e*Kel.,mj0
gc1
,jmglgr}mm*
K'?
J
'jmr3,E
u- g
,N
mm
K-
Kg\
]*]g-1
\
T
her
=,
7
-m
\\
,tn,
t
e
m
l\g
t--m,t
ej,g]+1r
.
_].K,...,"...rm..d*\\*`re,,(),KjM1w(x)}C.s|\"''r.1e.,-1l
edm.-0,-
d>:
dl,dM
mjm'r',
3er Ml\pmgm Kj.5l,l
m{P
{ ,{"-'
{'
me'ml'*
/
+
-:
g'?
Mm
t
k
g
T+
9
Jt
m-
-t
i-
J
+
c
*
T
-t
9m
.0
t-c
K.
lE
J,
+
lm
*,md,
,m*,\:*tdl,,.*
Q1*"
*,\\
TK"
t*,j
M
J
r\',0r,e'
'K]KJ
\y.
dKv,m-
{-.
t,e- mnT*"
\-rn,|rec
:]g
e
,e-m{gm
t--'0r{
g
\y
\
-,
s,: ,{M* M1,r
*\|rrt ,r
]gKe
s,
,tn
{d\
r*t
r
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
49/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
50/74
M
e.
J
l\.\
e.
e
.
l,'0
ju
c
vg
h
,l=j
|'e
7]i""
`
er- "l
J--
V.h
V-m
}g TrejKg
e,{\Kg
gK,'egK{;r
H."
gM1Q- nMe
7rtg-,
m""\
g*''
dj|
r,c|jK,
tr,l3 l
M1re*:
mj1"|rc
(;']
:Kg
-t
cc?
V@}' >",m"
Jx
\
V(
}R)
V"'
T
M-
Jy*
g
",}*
7y1MT
K
c,*Kg,
4
(h
"m
t
T-:
K
-|
8T
K-"
A)
.j
,m
)
;h
x)
m
K
}
(
h
4
(
r
]l)
,
c
(
c
m
(
,,
1,
.
m
8
,m
4,
Vedx)p-'
Q
*
(hx)
tr
R
gW
,x
Dn
Ku
((:
:+
E+
V
m
M
h{
T{
V(
hxm
C
x)r
+.,
V-|
d
(*,M
':g'c
g
m
.d'?
Jx
,.-: m
mE
m
K,g
KemgMter
K"g.tM(n-
merQnT\\.Z
1m1emz","
gJU
mg'-,nm
l-,.d.}r,c,.\;
]"{
7170g,,",
y=lg
t
-*t;*e*K
-(*nP
4
m
"r,t
Q.
1
(
m
,'",",
,j
M
{,
tg,
Mt
K
}
8Ut
>K,':;-
>';
J
>
C
U,
:m}
m
7?
m m
M
."}
,k
m eq mm
Q;,n,)m
k'M
>
-,KM
5,,)K
M
K
-m
t
|
-
(-t
e/*m
-K|*
.d{
x}
J? Q
q
m
M
c
,
(.*dM
g(&
_.)
mJ-):{--'-)
K
m.
,.
-4-
SJ,
}:,7
r1nK
g,c-tT,,m
j,r
]-Mj
um
K.'m"1 r.
'"),
m
;
_
em
&
t'm,
{
-m
I
zTvtM.m
.
0-\*..mret1
,
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
51/74
Jx
?
H,t-..
.m,."Tl
*tq
ud*
-T
r J,
c-m
1'y
**
,
d,md,.j
gmmTg .t.e|r
.''m
d
m
M
d
,"K
-Mu
l
Em
n
-8g-e"*
."
-j*
3c
jqM,.m..c
.
p
{Kj
g
K1m1
r"
r m
-z
:
m>
K
Kj
"T3x
0
J
+n"
e- : z m m-
g
m /td
|
-;
)
,K
Tzv
c
x
t
K'
g
3g
:
M
-Kg,M**
>"'u
}M1_
.|.
e'0
0
\*1*
K,m-
9M
cr
'M
.
7,R.
j
1, r,
,
{
",
k"
e,,e,\,_
}
-4*,
\cR
TMu''l
i +
h+
,Mx
"\k,
3n,"
->g.
+
K
?
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
52/74
Mmmm
/
'M"z
@p
tr.
]- S (,q
u0
.,'
---
J
0
u/
T|
T|
-
(
'-
,]d,+
^E
-
'{
)
mn.
emc
,,
.
t,,r
+.
.
(mn-u,
.?9,.:
d,d.
]
?
M.
}.?
]:
?
M
m d{{)
c
{'
k
{
m
,J
(5
;
E
]?
5?m
d-'.m
M,1K
.
.d
{ ,m
m
t
t
m
K
ml0
K
e
4
eK
E
,r(
MKK
mj-
E:m
>
-}
]-
..(
K-
rm
K,v
U,
;
--k
K,g
(-
,:-
J,
>{
,.>d
jTvg
mn
*{
,
c=z
n{-\c
:,:
s-.
-
,*
Kcl,
-
/) e
m ,/
M
-|
-|
-:*E
,K.
Kc
g",.g
tg,.\.,
,
t
a
t)
e_
&"
?
()
mT
t-
m
E
}:
.,
K
M
*
-
,0
(
)
4:,
*
.. tt
}
]
'+
\
Mm)
Kr
-}
z
u,g "-|rKlKm,
mt'
c-t
m
|,
e
m -.. {
:,:
M
"
m
}
-m1
K
t.,
"'
-
:rm'm
-? j
jqru'm
-
er")
,d
l
\
(t}
T ) d
/,)
T
D
KKr
''K
e]
\
m|}
i
;
":T
q..
m'r. :0
(')
,K - ,
|{
{m m
-
,{'m.*
}m
t
K
K
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
53/74
,,
Ee
}+
c,
M1
_gM'+
"
Q"
4_
."
|,
a
,>
-:
:
M111
;_
,,
},
Q>
m:
."
,,
K
M,KK
{
+
-,
R
(
Q
..,-e
e+
.-,
e
|,m
:g
>',
M"k"e|_
)'
Kert..
>gmu
J-}:m
.
-
(
:-:
}r.8
1]
eij:
,m{
^
tk_g
mrd.>drjM
.K
K'?
J
K
*
3:
+
"
,r--c
)
tl}
>?
N{
_
+
7,
J?
}J
-z]::?
,lK*''
*
'r,r''}
,k
-(
,g-r
|0?
J:
(K
cm?
':m
-
''
m8
(0
T
-
H"
>
Kgk
grg
td
|
MJ*.)
?
(,)
j)
Mx
5."m
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
54/74
m
Mel
n
(:}
m,',.gg
r
Mk,*r,
M*
?
(Mr+K).g-
tM}
nT,1zK"=
.u
gM,K,
i-nrk*r*
":
m
1
,.,
..
J,{
.}-{}
z(M")
+
tT
J
(Mx+*c)
/
lT*
J
e4T)
]-
MM+
M)
(J-4i
)mM16
$,,^.
MJ,j;mK-
jm
.. :n
-
-
M K-
K|
-
M
"*.d.*m*\
Mrycr,s{dlj
-
MK
mM
?
1JK
EM
3e,+mMKz-j]_e-j
-Kk
-
(K.
+
>
?
m
-^r
4Kr?
Kt,
Kt
,
:g
j" etnTr
g
E
1
t
K
mG:tg
n
- M')
-
)-
[.
:*
m"
e'q
(,,--m
]m-MK
:|
m
em
e
]'
M
K
x+
e
E
K
m
4
.'*
e
))
8.*
,
',*
,
(k)
T
(
K-]
.]4K 0
)-{.
, T
4/
(8G-x+
MKt
rygT
MR+\l
)
g
:y
/
y
v4
+
M
y
*
mu}
g
-e
`"
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
55/74
i'
?
m]
>
M,mt
\\
g,
j T
9T
(rr:".uMU+\0)
c
M\*MPK
a.i
,-mP] M ?Mt
-
(J\tmml"?M
{J*
jt)
l
?
t-
(zrx,tu)
,;,
J
(Mym+M-xO)
M?
&.
M;[K
(
,.,,
+
::*
g
,
Mu>
->o,
-
3d,
,`
t-9
-
tq|
-8
*\
-,
4-
(t}
.g)
,:
(>m
}--m
}k*
{
t
_
,;-
|
(>)
>")l"
"
M:
K?
U'?
`,7
c
dr..
r.tg-;
:
{
"M
:q,
-m
t
2q
t()
B
Nj
>
-m
Tq
-4
|M
,(
pt
e\
d-
mJ-}.]
?
9
K
ri: m
m,,E
'K
)
+
(
m
'm
."
tmK:
E
d
m"
Q
-
K
-?|-k
tJ-8-j
}
M}
-T,
8+>'mr-{:>;
>(T
+
:.(
,
D
3:
*4*t?;;
m
q\
+
M_
::;6
mM
q,(')
-
m
u
(d)-
d
m-
t
-,
:'m,
K
*'*}- m
mz)
t-
jv
E
d
"m
t
''
:0m
M-K
K
I)
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
56/74
M-T
a,
s.,,
*
M.
DrT
jx\Mx
+Ug
(Dv
+-7)
V,'
^-Tl
mK
?m -
K K m m-
^,
m
--
m
-z
m
m
4
m
1
Kt
1k
K
m
-
:--E
M
q-
G
(l
E
|;1m
-.r,-_t.
-K
Kgt
,
mT|
Mz
*:M.
-
K
|-r
tJ
(M
*:
an\{,
(-
mrdm,
,"m
Kc
"
,cK\c"
,,,,,\K
1\,
\ )
Me
vO
]*
m
'M
S-
d
|um,
mj.m,.-
4,:m,.,l
]dm,.
> d.r,.
-4-
?M,
]Q
"
-8M-_M,,T
S:;,
;m
rqcu)
(-*--***
'-. ,-
t
t*T*.,*T
(.tgK
)
8T:
+
K7T,
+ ;
):1
}:;
]K
m.
"
K
7*
e"
d
c
,"'?
t
M
rK
}?
.,--4`",
.
E
..
:U
jy
*
K
K
mm
.rm
T
Q
:
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
57/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
58/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
59/74
zgmm1 *
].
.{-
e
Mt.ggdm,W-
U(t)
vg
):({)
tgm1,,
Sm'
^
Kg71g,;
1:10
ma.'
|"\MT,
{:
v{
KgT
wg,m"\{
emn";
lM1}
Mnmp
mj
=j
KK\11
1*.1
d.{
d+
e7-.d.
H,,
K-ru\/q
(
y(+)K
?
tQ.,
y,
(
t
)
+
(
ME
Q
r,{"}K
tN.
_
()K
E,,
1Tgm1m,-
Z.t
,
e"gv,-
(.t
)K+
ev
A*
vrg{
cd.a
,5
8\
'
t,{
yt+)
?
t
H,l
{\,
M me
)n]K\
*-,_r,=
v
ej
k'j
Q(
K0Mtr
*rd
Kdj
*K*
7
Ut)
-
-:Md)
M
jp
|
,,
ng
]t'"
,"-e
,
\
",,
t
l9
M
M*ri{nli
nm,"e|
-\-rK
(.
(d)
T
im(d,)
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
62/74
M
e
,"K;
}g\
..
rT"
et,
"+r7*"t,d
^., k
(t)
\
m:::
:lTt )
-
7
)'(d)
T
s'
M
erk
Tsy(.)
-:'(Td)
'
{*mr}
-
rm>Q"
*
:'(d}
-K"(d)
j.Q-;
m
Kg
,,,K
?,
m(*r')')'K
j
3glg{m.zm1r,r
}a,
Mm
r'"'>
?
,
.t
7,
tr
c
y(r)K(
EK
W
1.:
*sr
=,,
m,'
e
e
K
D
,r
|,:
E
`.-
{
EK"r(r)K(K
D
t
--
(,K(j)KP"
)
d
nrt.m"1}..r,zr:
7,*'e?
MK
)
-
Ks
Mn:*
tj
c.n
'h,7\e
1/,*e;
-
K
-lSeK
n,"M,c,]
em,"e
M-
Gt*
rgnc1-,mz
Mymj,rg
dm\""1,1
g
,rr,y
m*
*
d
Q*
,"
,,
',,
,)(9
)
K"
t
lm
':{).?
9({,)
yjmt'.t'*
tm
j( g
,G(
)
?
t
Mm"'K
E
"m-
*c
Q"
?)
,
e
Q,_
M
c,>
((n1
,
d
Sg>"
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
63/74
G.lgtT,,
r,'moe
C
7''
M
j
`T
SX.
`>
(
dzt
9 T
(
m-)g ]-
]
(KK
-K
r
gqm
-,t.0
-n
_
;q
.:
7't
d
m
de_,
dt-n,
tT
-{s-d=)
0
g
-
"(
,
M
n,
mn\ t-
jT,.
nm
K"
t -'
-"
1
,
7
},mT
7*j
Kj*T-G
m
|;
E
.)
E
M,;\\:
K
?
-t
,(+
g
g
N-t
;,
r
-^0t
Eg ,,n.t
K+
)
('}
"
KS
d=)
r
-]:
(
*-'.'s
e\j0
0:
K
d
e
ct
T{{
)
(p
-
:rd|K,
/)
j
&/
r
-^{
] .
-
K1l,jtKt
K
-7t
-m
*
d(
T
-K
D(
g,K
\
\|
\,
(
je
m
-
7T
Q.-"r,t
d,
tK-{
,
1)
j
|
M1
,,q1
"m-Kt
K*+
1
K
1'-,,"
-"'
m
*
st
(1.
T
edm
1M-
m
q.
K?)
,
e)d
TQ,,\"l?
+
Ke=
dtr,
dM
K,,
i\m..ee,,gm ,M
7
m.
d,
,-{
,,
M)
,=
(\
K
P>,
*-'-'KK?*
MQ-'"r*
m
4
j\
r*"
.m
-)
t
m m
3
r)-m*
-"1
M
-,|j-r
]e
jjr
j
1.t
c-
-M;-
,
'j0
cp
]*
K
ec.-1']0(
mm)
tQ
.,,*0K-
M{
?E
{K\
]+K e*E-
]
t
]
kT1*}
(s,1d)
/"r..K,,m
j.,*
r
,.:." *mq,,,
:0
+
M
m".,r"
*
K
:
r)d'
:"'*dM
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
64/74
K(te
(
.t"
g
mlj{
1,,
"",
",r
}n)
t
e
jE
j-m
m
^
)-
c
-
("
.e
)"
7)D
)d
c
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
65/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
66/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
67/74
-l"M
"
":
7dM
8-,U.nM
mM;m?
Q(
T.,+
c
(s}
"
{
mt
)
?
7,\{
`""\m
,}1-:"-,u\
t",t
-,t
,M-
e,Tr
,cd"*
1m
e,'
Mjp
m
n
nZ
K#,,1
m
,_\, ,
l
,-
b g
jjmrr,:
,j
,
,
/t
,m,
0tm
y
:jr,K*".
qdm.k
",d
,
M.r"
4(,
{T
:
:"c
(
K
d
-|:
K,r_
7elt?
Eg.
8',
M,
(j(;
T| .{
*
"}
g
g
Ml
m*t
m1
d
(d7t
.t.
Q:lt
nd
{
*
e,,t
*,
j
(
)
c
-s4
]
s,,
[,
g
KP
K
T7"
* K
imlP-
g
7
1''
Kz
-7P
-M9.
|*g
'*
s4+m
T* ****`
\,"rK
M,g
'P189(
1,10
$()
G,
T
m
74*
|
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
68/74
rd
mt-
M.
"
3
Mdlvdmu^>d,
(rd+(*\-.}
jgd,g-\;
kEM+)
vgT
>d(,+}
,'eg >Krtr"1{r
vg
y,
t-t)
*
U:
{-*,}
*
M
-\v
d:,>K*:
d\-
>,
(+}
,r
U"
Q{-)
mery
-[
m
`
M
7
(
+
-zm
cr."
md,
ee*
/ejm
/t
0
{-
]
y,Nr)
ym
(,0
-K
K,MM[}K[
r,
j
t
(
y,{P),e-,(t.-rm,
K
,
[)d
e|
K:(+)
",{"nKg,'",;l
Mj(en,,(q
jD
.-
-]m
y,*ydm
?
Q
Q
d,
*,,K
g
"Km
)KW
W
q
(:
t\
]
K,
{+)
tn
-{
b
*-K
d
/i.TTT,
?')
tr,e" c
d'ju
8vt "
E
-70,,
KP
g
n{t-
9
td
1
,e
9]
>(,)
_-
KzK,]
-K,(\+zK
(z)
K(\)
g
K9K)
Ul
(K
)
'm:
)m
KrK+
-K
TQc
.
:*
[
?]
K+?
KJ
.K7c
tg
( (
r''r
(z}k(+-r)
)K
t,
E
-7(rlJK
r,-
c m m/
\j
*7
eT
:
+,.e
r
+
(9)
>(\9
K-g
))
'm;
m
/
)
?
Kc
,,
68
{ z
hz
m
/
]#
*em
m
,t*t}
:;}K?
>-'
E
r.
;
-)K
g
]-9
r] M
Q
y,(
m)
t
;,
(sK
+
:,
(
:}
y.
({}
Ur
(.)
yt
e
K
M,'
r
5(t)t
y,T(t)
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
69/74
-.*Q
*|
(
M}
t
m
(,
Me)t
-.
K1.
"(
M*,
U{,m
?
{,.
M
:e
-,
d
lu\
uc:
|um
c
di
t
*
n
mj,:9,*Kd--
(:+
rt1r
Sl
v
jm
KK
:
d
,:
d-
mm
K
m,'"
Mumm
"l
d-d
M
M
{
I*r,k,
K
>>'},j,|
:T,
.tE
-m
9
;-.
)
Trt
c4TMj
]
E
y.
rr
,:,
K.)
(4
:
{+
h'K
\'r
)
"+
M'
_
t,t}
1
).t+)?
5m};
h
{;;*
M?
,k
Qt
}
,
t
(,
m
l-
4..{c
}
?
N"M
MMM,t)KM
+
dr\d1,1,,n, g
>0gmgr,\g,
F
(t-j)
|
?
(
"''l
y
(.0
-
d")
K
))
q
l
u
({-0)
,m
E
)
+
*+-
_
-
T*
-T -".*
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
70/74
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
71/74
>"sm0
mM(m),
Q*d0
.r,,
j
e1
-
/, M
:
V9s
j
,)
K
7>+ :
kt*
,tr4K
+
>>
D(r)
Z(re
-
(-*()
(rtx)
(-">)
V(
s
}
c--,
E
J:
)
Kt
k*d
)
i(em.
Jt
+
7-
L
y''M*)+>m(+)+0KWm.?m
hecjj1,
]
J|>r,
p
"dM
|g",te"e
K,
y{r)*
7-sy(}+
0
U("r}"
U(m}.
{T.-T
G
(r4+c7+qm
0,0,uu
Z|},1
m,reg\,,
eg
>0,{.\.Qm(;/.
J,t
,
JM,
J9
m'g7ndc
mjjc,
,
>(d)
?
7'td}
,
Nre'r,-
m-
T-e
tK,c*T
,
",
.]
3,,:,.;,{
Grl
E*,
m
-c
-
K(s+q)(s*.)
JHN
--*-
-r
"r
7
7++
K+m
t.T"*"-T-*.
*""-"-."--
* u
kT.,k_,e-*T:.:tkm,k..
-.
-.,-
kk,e
*
ct(+)
7?d e)rj
^
?-K-
_";j
K
-T
T]
ee0
0J*
M
?)
T_,
3,
m
-]
m|e]
m
;]
m/"
7+5
U
nrmgr1el
exr,
m".
n
q
(*,,)ts
t}+,{
'('+t_}
?|
,(?
K
t
m
/
.
-(
K
2?
K
M
Em
.
y*
].
r,4
e,
g.r.c
0
-K
"
cd
c,
.:.
r',
]
U(r)
?
(t)?
mI)
?
mmq
K
Tm
-V'(
"(-
(
+K
mM
M"'
E
K+
c
y
e ?;
c,-'
+)
q
-aM
-t
g
- g
K
(T
9
-
k
s+qK
K
Tk"
M:
:m
M
K--m( m
:"
7m )
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
72/74
..gM
t->:
i, {d)
,
E
+
3>,*>d
*8>-
>
,
4rtd}
-t
?Rg
-{
-
b
t )K
m
1
e.,.
ed-c
,,.t
E
M
K
jt'
KM
c
;{trr\*
t1,-c
m
K*,&cn\,
}"e,^(",ter
}r,r,
,{*.r.d
,
{{,
l*N,
q
k't,,'*,e
t,
MT*,.,
mjng.
7
t(7)-{
^m
j{klut,**
{-
3y,es)
TjU*U
+
>'MM
(c)
?
]T
jm
K-
t
?M
s\m
(s-g
)'0
-
[m
t
T+
(-m):T'T
--;TT
KT|
U,{r);
-T 8t
t7-
},}
T
^+0
(sT|)( -,q)
mr-,
(;->}
(K-qm
-^+P
T
J
+
(
:T)
(
-u)
c-t
]p8gr
7gtjm(*mi
":
St()
?
*T;T.-..*
(
,
t)
(s
*,K
c( T
-t
T1T
;
-
)--
K-q
7-
m
^-q
K""'E
yesek
;;
M'E
(
*J
1P
sM:
m9
"K
]
m
;*, ^*Mt
)
.K-m
*m
,
0 ]
tmK
/
`T
:
*T.
m
M
-er
K
*
,ke
(
d)
+
T
-}g
'+
MqT
">_
qt
//
= r,|
e
?
7.:
y(Tr)-
U(dm
`]/(')
M
Er'
,?
7
U(em-
K(")
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
73/74
@e
{*r..0
K"mj,|
}'"
M
m
gi
mN,
M,t$
*'g,*,d
m
Kx
*r
0
]-
}
-
yu':1"x
m,.
m
}
M
`x
`'>
11
d
+
PtP5
?
71,
P
R
Kb
+l:,9
?
Z
\ *ZrR
g
-,
"/j
x
-
j
E
r
j.
..R
t
l'N
TTT
-
'jj
r',,,
e
tcr
-1/M
R
5l-T
`.
+
M,Tg
n:
(nrsP(y-me9eS}
+
N: (*n-ex
{
NT
+()
*js
ee,
+,
";|
s1"r,x
]T,\."T-/
(.
K-"
^r*
?
N,>
g7
ce'
*
(,
",
71
-c
y
>4,
{7 {x)
1-Gx
,{*
{{t}
71mc,R
\M1E
(-
r.,,ry
+
7_\x
- ''
)
"'
n-.1., re, *--K
]n*K,
r,
mh\"*1g
nmg,q
Mmg,\"
pm jK.-
lm
-4
}":
{
d
,
*
kK..
d
Kx
a
(,,
{,.
*,
, *;
t
},.j
-
M:
1
jm|j,R
Kle
r
y
-m
?
ys/,]R
T
+{*
m
M
t
K
,r., Td'*r.
t
-
Nx,\xky-N
wM
)
\/
*
Kr'
rr
-
e
G-n
..+
e"eQx)
F
8/9/2019 Diferansiyel Denklemler - Gebze Teknik niversitesi Defter Notlar
74/74
c.1{y}
"
(3(
}
*
,e'
(
*
},
(]]
`x
1
m
'
n|y
}r
c Jgt
w(
]
"---
-*
)
Kg{
w
(
R
{'-
Ku0
*,
(,
|
-\\"*,-
Kg
w(l,}
:p,
?
(,e0'{
' ,rz
*,
e,',,te"
0
e
'Kr,,
Krt
w1;
m
M :r,'
:{rr
*{}{w)?/t
]u,
k
E
D
'l',:
M
7,"l]
:pz
K
Kn{Q\r,).
,K1mctx"
,,
8g,
'8
:
-'Q '11,,
,,""K
Kg+
Q"y),T
^ejEEU
(,,
m
K
,
Q jmlmlxKx
)
EtT
K
d
m..cxv
+(0{3'ee'
".
*.,,
(
--
Smms
'Kg
(.Q*p,
7"*,
,.
{n..,
lT:,
,
)\
>
e
m l,
d\,.
m
,;d