Xususiy hosilali differensial tenglamalar umumiy yechimi - Algebra - Kurs ishlari

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Telzor Uchun

Ro'yxatga olish sanasi 21 Aprel 2025

9 Sotish

Xususiy hosilali differensial tenglamalar umumiy yechimi

Sotib olish

URGANCH DAVLAT UNIVERSITETI
FIZIKA-MATEMATIKA FAKULTETI
AMALIY MATEMATIKA VA INFORMATIKA
YO’NALISHI 203 -GURUH TALABASI AVEZOV DIYORBEK
MATEMATIK FIZIKA TENGLAMALARI FANIDAN
MAVZU: XUSUSIY HOSILALI DIFFERENSIAL
TENGLAMALAR UMUMIY YECHIMI
Topshirdi: ___________________
Qabul qildi: Yakubov Hamdam
Urganch 2021.
Ur g a n c h D a v lat Un i v er s iteti Fi z i ka -
1KURS ISHI

Matemati k a fa ku lteti M atemati k a y o ’ na lis h i
20 3
- gu r uh tala b a s i Avezov Diyorbekn i ng
“
M atemati k fizi k a te ng lamala ri” fa n i dan
“
Xususiy hosilali differensial tenglamalar
umumiy yechimi ”
ma v z us i d a g i ku rs is h i ga
TA
Q R I Z
Avezov Diyorning k
ur s i sh i “Xususiy hosilali differensial tenglamalar umumiy
yechimi
” m avzu si ni o‘rg a ni s hg a b a g‘i s h la ng a n.
T
a l a b a Avezov Diyorbekn i ng ku r s i sh i reja a sos i d a a d a b i y otl a r dan
foyda
l an i lg a n xo l d a t a h l il q ili n i b c h i q il g a n v a g ra m mat i k x at o lar s iz , yuq o ri
s
av i yada yo z i l g a n. I s h y a kun i d a m u allif t o m on i d a n a n i q xu l os alar k e ltiril gan
bo‘
li b, ul a r olib bo r ilg a n a niq ta h lillar n atija s i d ir .
M
az kur k u r s i s hid a y ana ma v z ug a do ir mi so llar k eltiril g a n bo ’li b , k u rs
is
h i n i b a ja r i s hd a t a la b a Avezov Diyorbekn i ng n aza ri y ta dq i qo tlar o li b
b o
ris hda qob ili y a t g a e g a e k a n l igini ko‘ r s at d i v a shun i ng u c hun un i ng i sh i n i “ ”
bahoga
l oy i q d e b o ‘ yl a ym a
I
l m iy r a hb a r: Yakubov Hamdam
2

Ur g a n c h D a v lat Un i v er s iteti
F
i zi ka - M atemati k a fa ku ltet M ate ma ti ka
yo
’ n ali sh i 20 3 - g u r uh tala b a s i Avezov
Diyorbek
“ M atemat i k fi z i ka te ng lama lari”
fa
n i d a n “Xususiy hosilali differensial
tenglamalar umumiy yechimi
” ma v z us i d a g i
ku
r s i sh i g a
T
AQR IZ
Ta
q riz g a ta qd im e t il gan tala b a
Avezov Diyorbek n
i n g k ur s is h i 34 b e t, k i rish q i sm i, reja as o si d a
yo
zil g an, x u losa va 10 ta a d a b i y ot va i n t erne t sa yt la r idan fo y dala n il g a n
a
d a b i yo tlar ro‘ yx at i d a n ta r ki b to p ga n b o‘l i b , “Xususiy hosilali differensial
tenglamalar umumiy yechimi
” m avz u si n i o‘ rga n is h ga b ag‘is h la n ga n .
Avezov Diyorbek
k ur s is h i “Xususiy hosilali differensial tenglamalar umumiy
yechimi ”
a d a bi yo tlar as o si d a ta h lil q ili b c h i q ilga n .
Ta
q ri z g a ta qd im q ili ng a n Avezov Diyorbek n i n g k u r s is h i n i y ak un la n ga n i sh
d
e b , m u all i f n i n g is h i n i “_ _ __ _ ” b a h oga lo y i q deb o‘ y la y ma n.
Ta
q ri z ch i : __ ___ _ __ _ _ __ _ _ ___ _ __
3

MAVZU: ORTONORMAN SISTEMALAR VA UMUMIY FURE
QATORI .
REJA:
I. KIRISH.
II. ASOSIY QISM:
1. Xususiy hosilali differensial tenglamaning xarakteristikasi
haqida tushuncha
2. Xususiy hosilali differensial tenglamalar sistemasining
tipini aniqlash
3. Ikki o`zgaruvchili ikkinchi tartibli xususiy hosilali
differensial tenglamalarning umumiy yechimini topishga
doir misollar
III. XULOSA.
IV. FOYDALANILGAN ADABIYOTLAR.
4

Kirish
Ma`lumki, tabiiy fanlarning har xil sohalarida uchraydigan
ko`pgina jarayonlarni o`rganishda xususiy hosilali differensial
tenglama yoki xususiy hosilali differensial tenglamalar
sistemasini
oldindan berilgan boshlang`ich va chegaraviy shartlarda yechish
masalalarini o`rganishga duch kelinadi. Bunday jarayonni
ifodalavchi
matematik masalalar ko`pgina umumiylikka ega bo`lib,
matematik
fizika tenglamalarining predmetini tashkil etadi. Matematik
fizika
tenglamalari matematikaning asosiy fundamental va tadbiqiy
bo`limlaridan bo`lib, u bakalavryatning matematika, mexanika,
amaliy
matematika va informatika kabi yo`nalishlari o`quv rejasidagi
umumkasbiy fanlardan biri hisoblanadi. Xususiy hosilali
differensial
tenglama yoki xususiy hosilali differensial tenglamalar
sistemasini
5

o`rganish uchun ularni sinflarga ajratish maqsadga muavfiqdir.
Uslubiy qo’llanmada xususiy hosilali differensial
tenglamalarning va differensial tenglamalar sistemasining
klassifikatsiyasi masalasi, hamda ikkinchi tartibli xususiy
hosilali
differensial tenglamalarning umumiy yechimini topish usullari
bayon
etilgan.
Ushbu uslubiy qo’llanma universitetlarning “Amaliy matematika
va informatika” yo’nalishlari bo’yicha bakalavrlar
tayyorlaydigan
fakultet talabalari uchun mo’ljallangan bo’lib, shu yo’nalishning
namunaviy dasturida kiritilgan “Matematik fizika tenglamalari”
fani
rejasiga asosan yozilgan
6

1
n
 ...

 
n


u1 - § . X U SUSIY HO S ILALI D I FF E R E N SIAL
TEN G L AM A NING X A R AKTE R IS T IK A SI H A QI D A
T U SH U NC HA
 – o r q a l i x , x
2 , . . . , x
n , n  2 , ortog o na l d e kart
k o o r di nata l i x n u q ta l a r n i ng n – o’ l c ha m l i R n
e v k l i d f a z o s id a n
o l i n g a n s o h a n i
b elg i l a y mi z .
 – so ha d a n ol i n g a n
x n u q t a v a 
1 , 
2 , .. . , 
n ,


j  k , k
 0 , . .. , m , m
 1 , m a n fi y mas
b u t u n i n d e ks l i
j  1
p
12
. . . n h a qi q iy o ’ zg ar u vc hi l i
F x , .. . , p , .. .
1 2 n 
h a qiqi y q i y mat l i f u n ks i y a b er i lg an b o’ l i b , h e c h b o’ l m a g a n d a
 F
 p
12
. . .
n , b u n da


j  m , h o si l a l a r d an bi r o rta s i n o l d a n
j  1
m
7

1
1 

u

1
n
nf a r q l i b o’ l s i n . B u y e r d a
p12
. . . n

 x 1  x
2 2
  x
n n d e b o l a m i z .
k
F
 x , . .. ,
 x
1 .. .
 x
n n , . . .
  0 ( 1 )
s h a klda g i t e n g l i k u ( x )  u ( x , x
2 , . . . , x
n ), x  
n o ma ’l um f un gs i y a g a n i s b a t a n
m  ta r t i b l i x u s u s i y h o s i l a l i d i f f e r e n s i a l
t e n gl ama
d e y i l ad i v a b u t e ng l ik n i ng c h a p to m o n i e sa,
m  ta r ti b l i
x u s u s i y h o si l a l i d i ff e r e n s ia l o p er ator d e y i lad i .
 so had a a niql a ngan u ( x ) h aq iqi y qi y mat l i
f unk s i y a v a u n in g ( 1 ) t e n g l ama d a q at na s h g a n b arc h a x u s u s iy
h o si l a l a r i u zl u k s i z b o ’ l i b , b u t e n g l ama n i a y ni y atga a y l a n t i r s a , u
h o l d a shu f un k s i y a g a r e g ul y ar y e c hi m d e y i l a di .
A g a r F f unk s i y a p
12
. . . n
, b u n da
 


j
 k ,
j  1
k  0 , . .. , m b a r c ha o’ z g aruvc h i l ar g a ni s b a t a n c hi zi q l i
f u nk s i y a b o ’ l sa , u h o l d a ( 1 ) t e n g l ama g a c hi ziq l i t e ng l ama d e b
ata l a d i . A g a r F
f u n k s i y a p
12
. . . n
, bu n d a
 


j  m
o’ z g ar u v c hi g a g i n a
j  1
n i s b a t a n ch i z i q l i b o’ l sa, u h ol d a ( 1 ) t e n g l ama g a k v a z i c h i z i q l i
t e ng l ama d e b a ta l a d i .
8


l
 


pLu  f ( x ) c hi z i q l i t е ng l ama u n ing o’ ng t o m o ni d a g i f
( x ) f u n k s i y a n in g b a r c ha
x 
 u ch u n n olg a t е n g y o ki a y nan
n ol d a n f a r q l i b o’ l i s h ligig a q a r a b bir j i n s l i y ok i bi r j i ns l i
b o ’ l ma g a n t е n gl a m a d е b a t a l a d i .
Os o n g i na k o ’ r s at i s h m u m k in k i , a g a r u ( x ) v a v ( x )
f u nk s i y a l ar b i r j i n s l i b o’ l m a g an Lu  f ( x ) c h i z i q l i
t е ng l ama n i n g y е c h i m l a r i b o ’ l sa , u h ol d a u l a r n i n g a y i rmasi w ( x )
 u ( x )  v ( x ) esa
L w  0 b i r j i ns l i t е ng l am a n i n g y е c hi mi
bo’ l ad i . B u n d a n t as h q a r i , a g a r u
k ( x ), k  1 , .. . , l
fu nk s i y a l ar bi r j i n s l i t е ng l am a ni n g y е c hi m l a r i
b o ’ l sa , u h o l d a
u


 c
k u k ( x ), b u n da
c
k – ha qi qi y
o’ zg a r mas l a r ,
k  1
h am s h u t е ng l am a n i ng y е c h i mi b o’ l a d i .
( 1 ) t е n g l ama n in g t ip i
K
 
1 ,..., n   
 F

 m 1 . . . n
 1 1... n n
x a r a kt е r i s t i k f o rma o r qa l i ani q l a n ad i.

a ( x )D  u
 f ( x ) ( 2)

 m
t е ng l ama
m – t ar t i b l i x u s u s iy h o sila l i c h i z i q li d i ff е r е ns i al
9

u
1
1 
  
1
1t е ng l ama ni ng u m u m i y s h a k l i , b u n da
  ( 
1 , 
2 , . . . , 
n )
– m u l t i in d е k s ,

j  0 , j  1 , n , b u n d a n t a s h q a ri 
j – b u t u n
s o n l ar,

 
1  
2  .. .  
n m u l ti in d е ks m o du l i , D  u 

x

...

x
n n b o ’ l s i n . D

   a l mas h t iri s h
or q a l i ( 2 ) t е ng l ama n i n g

a ( x ) 
s i m vol i n i h o s il q i l ami z , b u n da  
 
1 1 2 2
...
n n .

 m

a ( x )  f o rma g a ( 2 ) t е ng l ama n i n g b o s h s i m vol i y o k i

 m
x a r a kt е r i s t i k k o ’ p h a d i d е b a ta la di .
x 
 t a y i n l a n ga n n u qta b o ’ l sin. No l d a n fa r q l i

 ( 
1 , 
2 , . . . , 
n )
 0 v е k t o r u c h u n
 a ( x ) 

0
b o’ l s a , 
 m
u h o l da b u v е k t o r x a r a k t е r i s t i k y o ’ na l ish d е b a ta l a di.
F ( x , x
2 , . .. , x
n )  0 for m u l a b i l an b е r i lg an g i p е r s i r t
u ch un h ar b i r n u q t a x ar a kt е r is t ik y o’ n a l i s h g a e g a b o ’ l s a , y a' n i
 F ( x , x
2 , ... , x
n )
 0
10

1 
 F  F


 


 
1
 F  F  F

n n
 x  x
1
n n
 x  x

m a

( x ) 

x 
1
.. .
 
x
n  n

0 , g r a d F 
0 ( 3)
b o ’ l sa , u h ol d a x ar a kt е r i s t i k s i rt d е b a t a l a d i . B u x ara k t е r is t i k a
t е ng l amas i d i r , b u n da
grad
F 

 x
,
 x
2
,...,
 x
n 
.
I kki n c h i t a r t i b l i k va z i c hi z i q l i ( b ar c h a y u q o ri t ar t i b l i
h o s i l a l a rg a n i s b a t a n ch i z i q l i) u zl u k s i z
a
i j (x)
ko e f fits i е n t l i t е ng l ama n i q ara y m i z :
  a
i j ( x )  2
u
 Ф ( x , u , gra d u )
 0 .
i j
x  ( x , x
2 , .. . , x
n ), n  2 o ’ z g ar u vc h i l i F ( x )
f u n k s i y a C 1
s in f d a n o l i n g a n b o ’ l i b , F ( x )  0 s i r t d a gr ad
F ( x )  0 v a
  a
i j ( x )  F
  F
 0 ( 5 )
i
 1 j  1 i j
11

2
2
n


 F  F

 t е k i sl i k l ar o i l as i d a n i borat b o’ l a d i.
 u  f
b o ’ l s i n . U h o l d a F ( x )  0 e sa, ( 4 ) k va z ich i zi q l i d i f f е r е n s i a l
t е ng l ama ni ng x a r a k t е r i s t i k s ir ti d е b , ( 5 ) t е ng l ama e sa x ar a kt е r is t ik
t е n g l amasi d е b a y ti l a d i .
n  2 u c h u n x a r a kt е r i s t i k s i r t i
x ar a kt е r is ti k c h i z i q d е b a t a l a d i.

t
u

a 2
 u
t o’ l qi n t a r q a l i s h t е n gl ama s i u c h un x ara k t е r i s t i k t е ng l ama
F 
0
d a
s h a k lg a eg a d i r . 2 2

 t
 
a 2
i 
1 
 x
i
 
0
Uc h i ( x
0 , t
0 ) n uq t a d a b o’ l g an x ara kt е r is t i k k o n us d е b
a t a l u v c h i
a2(
t  t 0)2 
x  x0

2


0
s i r t x a r a k t е r i s t i k s i rt b o’ l a d i.
12


u
 xt 
a 2
 u  f
i ss i qlik o ’t k a z uv c h a n lik t е ng l ama s i u ch un x ar a kt е r is t i k t е ng l ama
F 
0 d a  a 2 n

 F
 2

0
i 
1  i 
s h a k lg a eg a d i r . U ni n g
x ar a kt е r is t i k a l a r i os o n g i na
k o ’ r i n a d i ki , t  C
13


 F

P u a sson t е n g l a m asi u ch u n x ar a k t е r i s t i k t е n gl ama
n 2
F  0 д а
i 
1 
 x
i  
0
s h a k lg a eg a di r. Bu n dan F 
0 да g r a d F  0 e k a n l i g i k е l ib
c h i q a d i , b u e sa m u m k i n e m as, y a' n i x a r a k t е r i s t i k s i r t g a e g a e m as.
E n d i x u sus iy h os i l a l i d i ff er e n s ia l t e n g l ama n i n g ta r ti bini v a u n in g
c h i z i q l i di f f e r e n s i a l t e ng l ama e k a n l i g i n i a ni q l a s hga d o i r miso ll ar
k el t i r a m i z :
1 – m i s o l.
s i n(u
x u y )s i n u
x cosu y  s i n u
y cosu x  0
t e ng l ama x u s u s i y h o s i l a l i diff e re ns ia l t e ng l ama b o’ l a di m i?
Y ec h i s h:

F 
 s i n ( u
x  u
y )
 s i
n u
x c o s u
y  si n u
y c o s u
x 
 u
x  u
x

cos(u
x u y )cosu x cosu y s i n u
y s i n u
x 

cosu
x cosu y  s i n u
y s i n u
x cosu x cosu y s i n u
y s i n u
x  0 .
X u d d i sh u n da y ,  F
 u
y  0 e kan l i gi n i k o’rsati s h m u m k i n. F
f u n k s i y a d a n x u s u s i y h os i l a l ar b o ’ y i c h a ol i n g a n h os i l a l ar n o l g a t e ng
14

3

3
2

u2 2
2 2 2 2 22
3e ka n . D e m a k , b e r i lg a n t e ng l ama x u s u s iy h o s i l a l i di ff e r e n s i al t e n gl ama
b o ` l mas e k a n .
2– m is o l.
u
x x  u y y   u
x x u yy 
2

0 t e ng l ama x us us iy
h o si l a l i d i ff e r e n s ia l t e n gl a m a b o’ l ad i mi?
Y ec h i s h: A vv al b er i lg a n t e n g l am a ni s o dda l a s h ti r i b ol a y l i k .
F

u
x x  u yy   u
xx u y y 
2

u
x x u yy u xx 

2u
x x u yy  u
yy  2u
x x u y y . En di e s a , x u s u s i y h os i l a l ar b o’ y ic h a
h o si l a l a r n i h i s o b l a y mi z :

F
u
x x 
2u
yy 
0
,  F
 u
yy  2 u
xx  0 .
D e ma k , b e r i l g a n t e ng l ama 2 – t a r t i b l i x u s u s iy h os i l a l i diff e re ns ia l
t e ng l ama ek a n .
3– m is o l. c
os 2 u
xx  s i n 2 u
x x  4u
x  2u
y  u  0
t e ng l ama ni ng t a r tib i n i a n i q l a n g.
Y ec h i s h:

 c os 2 u
x x  s i n 2 u
xx  4u
x  2u
y  u

u
x x
  2 c o s u
xx s i n u
x x  2 si n u
x x c o s u
x x  0
15


u

u
 c os 2 u
xx  s i n 2 u
x x  4u
x  2u
y  u


12u
x  0
x
D e ma k , b e r i l g a n t e ng l ama 1 – t a r t i b l i x u s u s iy h o sila l i d i ff e r e n s ia l
t e ng l ama ek a n .
4– m is o l.
x 2 u
x x  xu
x y  sin y  u
x  y 2 u
 0
t e ng l ama x u s u s i y h o s i l a l i c h i zi q l i d i f f e r e n s i al t e ng l ama b o’ l a di m i?
Y ec h i s h:

 x 2 u
x x  x u
xy  s i n y  u
x  y 2 u


x 2

0
xx
b o ` l g a n i u c h un b er i lg an t e n g l ama 2 –ta r t i b l i d ir. Ha m d a t e n g l ama
Lu  f k o ` r i ni s h i d a b o ` l i b , b u e r d a
Lu
 x 2 u
x x  x u
xy  s i n y  u
x  y 2 u
o p e r ator n o ma ’ l u m f u n k s i y a v a un in g h o si l a l a r i
u
x x , u
x y , u
x , u l a rg a nisba t an c hi zi q l i d i r, c hu n ki
L ( u
  v )  x 2 ( u
  v )
xx  x ( u   v )
xy 
 sin
y  ( u   v)x
 y2( u   v)
  x2uxx
  xuxy   sin y  ux
 y2u   x2vxx    vxy
  sin y  vx
 y2v   L u   L v
16

2
2v a t e n gl am a n i n g o` n g tom o n if ( x )  0 d i r . S h u n in g u c h un 2 –
ta r t i b l i x u s us i y h o s i l a l i bi r j i n s l i c h i z i q l i di ff er e n sia l t e n g l a m a b o’ l a di .
5– m is o l. x
u
x x  ( x  y )u
x y  5 y  u
x  y 2 u
 s i n( x  y )

0 t e ng l ama x u s u s i y h o s i l a l i c h i zi q l i d i f f e r e n s i al t e ng l ama
b o’ l a di m i?
Y ec h i s h: B er i l g a n t e ng l a m a c hi z i q l i d i f f e r e n s i a l t e ng l ama b o ’ l a
ol m a y di , c h u n k i
Lu  x u
x x  ( x  y )u
xy  5 y  u
x  y 2 u
o p e r at o r u
x x g a n i s b a t a n b i r inc h i ta r ti b l i ( c hi zi q l i) e mas.
6– m is o l.
 sin ( x  y )  x 2
 0
x
u
x x y  y 2 u
yy y  5 y  u
x y  l n y  u
x  y 2 u

t e ng l ama x us u s i y h os i l a l i c h i z i q l i d i f f e r e n s i a l
t e ng l ama b o’ l a di m i?
Y ec h i s h: B e r i lg an t e ng l ama b i r j i n s l i b o’ l m a g an c h i z i q l i
d i ff e r e ns i al t e ng l a m a b o ’ l a d i, c h u n k i
Lu
 x u
x x y  y 2 u
y y y  5 y  u
x y  l n y  u
x  y 2 u
o p e r ator
u xxy,
u yy y, u xy, u x, u l ar g a ni sbat a n b i r in c h i
d ar a ja l i d i ff e r e n s i al i f o d a v a f ( x )   sin ( x  y )  x 2
 0 .
17

7– m is o l. u
x xy  t gy  u
yy  7 y  u x y u y y  y  u
x  yu  xy  0
t e ng l ama x u s u s i y h o s i l a l i c h i zi q l i d i f f e r e n s i al t e ng l ama b o’ l a di m i?
Y ec h i s h: B e r i lg a n t e n g l ama c h i zi q l i di f f e r e n s i al t e n gl ama b o’ l a
ol m a y di , ch u n ki
Lu
 u
x xy  t gy  u
yy  7 y  u x y u y y  y  u
x  yu
o p e r ator
u xy, u yy l ar g a n i s b a t an bir i n ch i d ara j a l i di ff e r e n s ia l
i f o d a e m a s . A m mo u sh b u t e ng l ama kv a z ic h i z i q l i di f f e r e n s i a l t e ng l a m a
b o ’ l a d i , ch u n k i
Lu
 u
x xy  t gy  u
yy  7 y  u x y u y y  y  u
x  yu
o p e r ator y u q o ri t a r t i b l i h o s i l a
u xxy g a nis b a t an b i r i n c h i d a r a ja l i
d i ff e r e ns i al if o d a b o’ l a d i .
2. X U SUSIY HO S ILALI D I FF E R E N SIAL
TEN G L A M A LA R SI S TEMASIN I N G TI P INI AN IQL A SH
Bi zg a u 1
, u
2 , . . ., u
N n o ma ` l u m f un k s i y a l ar q a t n a s h g a n h a r
b i ri m – t a r t i b l i q u y i d a gi N ta x u s u s iy h os i l a l a l i d i ff e r e ns ia l
t e n g l ama l ar s i s t e m asi b e r i l g a n b o ’ l sin:
19

 1
211 ii
iiii
2 n 2 n1 N
11 ii
iiii
2 n 2 n1 N
ii
12 n 12 n1 N




1 iii
2 nj
2
1 nii
i
1
1
1
1
1 1
1 1
1
1
1 1
2
2
1 2
1
2 1
N
N
N1A 



 




 

 n
F
 x , u
1 , u 2, , ... , u N , , ..., p x1x2
...xn , .. . p x1x2
...xn
  0
 F x , u
1 , u 2, , ... , u N , , ..., p x1x2
...xn , .. . p x1x2
...xn

0

............. . ......................... . ....... . ......................... . ...
 F
N
 x , u 1 , u 2, , ... , u N , , ..., p x1xi2...xin
, .. . p x1xi2...xin

  0 ,
b u y e r d a p
x
1 x 2
.. . x
n  i
u
j
 x
1  x
2 . . .  x
n , 0  i  m , 0  j 
N .
Us h bu t e n g l a m a l ar s i s t e m a s i n i n g t i p i n i a n i q l a s h u c hun u n i n g
x a r a kt e r i s t i k f o rmas i n i t u z a m i z . B u n i n g u c h u n b i z g a q u y i d a gi
k v a d r a t ik m a trit s a l ar z a r ur b o ’ l a d i :

F

p
i i 2 .. . i n

 F
i i
2 . . . i n  
p
i i 2 . . . i n
.
  F
N

p
i i 2 ..
. i n
 F p
i
i
2 . . . i n

F
2 p i
i
2 . . . i n
.
 F
N
p
i
i
2 . . . i n




 F

p
i i 2 . . . i n


F
2
p
i i 2 . . . i n


,
.


F
N
p
i i 2 . . . i n
  i
k 
m
.
k  1
20

1
1
1 2 n i




1A n
2
1AB u m a trit s a l ar d an fo y d a l a n ib,
1 , 
2 , .. ., 
n h aq i q i y s k a l y a r
p aram e tr l ar g a n i s b a t a n u s h bu N
m – t a r t i b l i x a r a s t e r i s t i k for ma n i
t u z a m i z :
K (

1 , 
2 , 
, 
n )
 d e t

 A i
i . . . i 
i


n n
 .
 | i |
 m

Y uq or i da g i s is t e m a n i n g t i p in i a n iq l a sh u s h bu x ar a k t r i s t ik f or ma ni ng
s h a k l i g a qar a b,
m – t a r t i b l i b i t ta t e n gl ama q a r a l g a n i s i ngari t i p l a r g a
b o ` l i n a d i .
1– m is o l.  2 u
x  4 v
x  3 u
y  8 v
y  u  0
 3 u
x  2 v
x  6 u
y  3 v
y  2 u
 0 t e n g l ama l ar
s i s t e m a s ini n g t ip i n i an i q l a n g.
Y ec h i s h : A vv a l am b or , b i z
ii2
. . . in
,  i
k  m

k  1
matr i ts a l a r n i t u z a m i z . B i zni ng mis old a N = 2 , n = 2,
 i
k 
1 ,
u
1 
u ,
k  1
u
2  v b o’ l g ani u ch un
ii2
. . . in
ma t r it s a l ar q u y i d a g i c h a b o ’ l a d i :
22

1
2
2A 
1

u
 

 

 ,
1
1
2A 
1

u
 

 

 1
1
1
1 2 n i
2 
 
22
 



 

 8



 4 
 9 
 1 2 
48

3 

6 
 2 
 3 




 F
1 0 
  F x

 u
x  F
 v
x 
 2 
4

 F 
 3  2

 v
x  b u e r d a i  1 , i
2  0 ,
  F
0 1 

 F y
  u
y  F
 v
y 
 3
8
  F
2 
 6
3
  v
y
 , b u e r da i  0 , i
2  1 .
E n d i e sa
K (

1 , 
2 , 
, 
n )
 d e t

 A i
i . . . i 
i


n n

 |
i |
 m

x ara k t e r i s t i k k o’p h a dni t u z a m i z :
K (
1,2) 
det
3
4 
3
2 

1
6
3 
2




det
2 1

 32
 1 2
 8
1
392 .
4 1
 82  2 2 2 2 1 2 1 2
1 2
B 2
 AC  0 2
 8  (  3 9)  3 1 2  0 . D e ma k , b er i lg a n
t e n g l am a l ar s i s t e m a s i t e k i s li k n i ng h am m a n u q t a l a r id a gi p er b o l ik t i p d a
b o` l a d i .
2– m is o l.  u
x  u
y  v
y  v
z  x y u 
0
 v
x  u
y  v
y  u
z  2 u 
0 t e n g l ama l ar
s i s t e m a s ini n g t ip i n i an i q l a n g.
23

Y ec h i s h : A v v a l am b or , bi z
24

1A n
3
1A
1
2
2A 
1

u
 

 0



 1
1
2A 
1

u
 

 1



 1
1
2 2
A 
1

u
 

 1


  1
1
1
1 2 n i
ii2. . . in
,  i
k  m

k  1
matr i ts a l a r n i t u z a m i z . B i zni ng mis old a N = 2 , n = 3,
 i
k 
1 ,
u
1 
u ,
k  1
u
2  v b o’ l g ani u ch un
ii2
. . . in
ma t r it s a l ar q u y i d a g i c h a b o ’ l a d i :

 F
1 0 0 
  F x
  u
x  F
 v
x 
 1  F 
 0  v
x  1
 , b u er da
i
 1 , i
2  0 , i
3  0 ,
  F
0 10 

 F y
  u
y

 F
0 01 
  F z

 u
z  F
 v
y 

1  F
2 
  1  v
y

 F
 v
z 
 0
 F 
 1
 v
z  
1
 , b u e r d a i  0 , i
2  1 , i
3  0 ,
0
 , b u e r da i  0 , i
2  0 , i
3  1 .
E n d i e sa
K (

1 , 
2 , 
, 
n )
 d e t

 A i
i . . . i 
i


n n

 |
i |
 m

x ara k t e r i s t i k k o’p h a dni t u z a m i z :
25

1 1
1 1
1 1
2 2 

 
 

 
2
2
F 22 


  






 



.

     
   


 

 

 

     


   


22




1A n
2
1AK ( 1
,

2 , 
3 ) 


 F 
d e t 
 
u x
   u
x 
F
   F

v
x
 u
y  F  1

 F
 v
x
 
 u
y  F

 
F  v
y


u
z  F 
 F
2  v
y

  u
z  F
 
 v
z

 F  3
 v
z 


 1 0

 1 1

 0
1

   0 1
 1

 1
 1
 2
 1
0
 3


de t 
1 

2
 2 3  2  
3 
2 2 2 2 2 2
1 2 2 3 1 3
1 2
X a r a k t e r i s t i k f or m a ning K (
1 ,

2 , 
3 ) 
1  
3 k a n o n i k s h a k l i d a
i k k i n c h i k o e ff it s i e nt n o l g a t e n g di r . S h un ga k o`ra, b er i lg an t e n gl am a l ar
s i s t e m a s i fa z o n i n g ham m a n u q ta l a r i d a par a bo l ik t ip d a b o ` l a d i .
3 - m is ol .  u
x  u
y  v
y  u  0
 v
x  2 u
y  v
y  x u 
0 t e n g l ama l ar s i s t e m as i n i n g
t i pini a n i q l a n g .
Y ec h i s h : A vv a l am b or , b i z
ii2
. . . in
,  i
k  m

k  1
matr i ts a l a r n i t u z a m i z . B i zni ng mis old a N = 2 , n = 2,
 i
k 
1 ,
u
1 
u ,
k  1
u
2  v b o’ l g ani u ch un
ii2
. . . in
ma t r it s a l ar q u y i d a g i c h a b o’ l a d i:
26

1
2
2A 
1

u
 

 0



 1
1
2A 
1

u
 

 

 1
1
1
1 2 n i
0
.

1
    
 
  

 




  

2  

 


22

 F
1 0 
  F x

 u
x  F
 v
x 
 1  F 
 0  v
x  1
 , bu e r d a
i
 1 , i
2  0 ,

 F
0 1 

 F y
  u
y  F
 v
y 
 1 1

 F
2 
  2  1

 v
y
 , b u e r da i  0 , i
2  1 .
E n d i e sa
K (

1 , 
2 , 
, 
n )
 d e t

 A i
i . . . i 
i


n n

 |
i |
 m

x ara k t e r i s t i k k o’p h a dni t u z a m i z :
K
(
1,2) 
det


1
0   1
1

 1 

1
2
1

2


det


22
2
2  2 2 2 2 2 1 2 2 1 2
1 2
Xar a k t e r is t i k f o rma ni ng K (
1 ,

2 ) 
1  
2 k a n o n i k s h a kl i d a
i k k i a l a k o e ff i t s i e n t h a m bi rga t e ng d i r . Sh un g a k o`ra, b er i lg an
t e n g l ama l ar s i s t e masi t e ki s l i k n i ng h amma n u q t a l a r i d a e ll i p t i k t i p d a
b o ` l a d i .
M u s t a q i l y ec h ish u c hun m i sol l a r
28

Q uy i d a be r i lg a n t en g l ama l ar s is t e ma sin i ng t i p in i a n i ql an g :
29



























8 . 1 .
8 . 2 .
8 . 3 .
8 . 4 .
8 . 5 .
8 . 6 .
8 . 7 .
8 . 8 .
8 . 9 .  u
x  v
x  u
y  v
y  u  0
 2 u
x  v
x  4 u
y  2 v
y  u 
0 .
  u
x  u
y  v
x  xy  0
 2 u
x  3 u
y  3 v
y  u  0
.
 u
x  2 u
y  3 v
x  v
y  x  0
 3 u
x  3 u
y  3 v
y  3 v
x  u  0
.
 4 u
x  u
y  3 v
x  xu  0
 u
x  5 u
y  2 v
x  u si n x  0 .

4 u
x  5 u
y  3 v
x  x 2
u  0
 2 u
x  5 u
y  v
y  u c o s x  0
.
 5 u
x  3 v
x  v
y  x 2
y  0
 u
x  2 u
y  v
x  u sin y 
0 .

1 0 u
x  2 v
x  u
y  x 2
c o s y  0
 4 u
x  3 u
y  5 v
x  v
y  4 x sin y  0 .
 2 v
x  u
y  6 v
y  x 3
y  0
 u
x  2 u
y  2 v
x  3 v
y  3 x y 2
 0 .
 6 u
x  2 u
y  3 v
x  4 v
y  x 4
 y  0
30

 3
u
x  2 u
y  8 v
y  u  y 2
 0 .
31

x y x y



x x y


























 u  6 u  5 v  3 v  x  s i n u  0
8 . 1 0 .
 3 u
y  5 v
x  v
y  u  c o s x  0 .

2 u  2 v  12 u  2 u  0
8 . 1 1 .
 v
x  4 u
y  v
y  x y  0 .
8 . 1 2 .
8 . 1 3 .
8 . 1 4 .
8 . 1 5 .
8 . 1 6 .
8 . 1 7 .
8 . 1 8 .
8 . 1 9 .  2 u
x  7 u
y  v
x  2 u  0
 3 u
x  3 1 u
y  v
y  3 v
x  e y
sin x  0 .
 2 u
x  3 v
x  v
y  5 u  0
 3 u
x  3 v
x  3 u
y  4 v
y 
0 .
 u
x  u
z  3 v
x  v
z  x  0
 u
x  v
x  2 v
y  2 v
z  u  0 .
 v
x  u
z  2 v
z  5 u  0
 u
x  v
x  v
y  v
z  x u  0 .
 u
x  v
y  2 u
z  3 v
z  u  0
 u
y  2 v
x  2 u
z  v
y  2 u 
0 .
 u
x  u
y  2 v
y  3 v
z  5 u 
0
 u
x  v
x  2 u
z  v
z  4 u
 0 .
 u
x  2 u
y  v
y  v
z  6 xu  0
 2 u
x  4 v
x  2 u
z  v
z  4 yu 
0 .
 u
x  v
y  v
z  2 xyu  0
 2 u
x  2 u
z  v
z  4 x
 u
 0 .
32


















8 . 2 0 .
8 . 2 1 .
8 . 2 2 .
8 . 2 3 .
8 . 2 4 .
8 . 2 5 .  2 u
x  u
y  v
z  u  0
 u
x  2 v
x  2 u
z  4 u  6 y  0 .
 u
y  2 v
y  v
z  x y 
0
 u
x  v
x  2 u
z  v
z 
0 .
 u
x  u
y  v
x  2 v
z  5 u e x
 y
 0
 u
x  v
y  2 u
z  v
z  3 u  0 .
 2 u
x  u
y  v
x  u  0
 v
y  2 u
z  v
z  2 u e x
 0 .
 u
x  u
y  v
x  2 u
z  u  e y
 0
 u
x  v
y  2 u
y  u
z  3 u  0 .u
x  v
x  u
z  u  e x y
 0
u
x  v
x  2u
y  v
z  u e x y
 0 .
3 . IKKI O ` Z G A R U V C HILI I K K I N CHI T AR T I BLI
X US U SIY H OS I L A L I D I FFE R E N SI A L T E N GL A M A L A R N I N G
U M U MIY Y E C HI M I N I TOPISH G A D OIR MI S O L L A R
B u p ara g raf d a i k ki o ` z g ar u v c h i l i i k k i n c h i ta r t i b l i x u s u s iy h o si l a l i
d i ff e r e ns i al t e ng l a m a l a r n i n g u m um i y y e c hi m i n i t o p i s h ga d oir
m i s o l l a r n i qa ra y mi z .
1– m is o l.
u
x y  0 t e n g l a m a n i n g u m u m i y y ec h im i ni t o p i ng.
34

2
1 1
C
Y ec h i s h: u
x  v d e b b elg ilash k i r i t am i z . U h o l d av
y  0
t e n g l ama g a e g a b o’ l ami z . U s h bu t e ng l ama n i y e c hi sh u c hu n
u n i int e g r a ll a y miz v a v  C ( x ) t e n g l i k k a e g a b o’ l am i z . T o pi lg an
i fo d a n i
k i r i ti lg a n b elg i l as h g a o l ib, b or i b qu y i b , u
x  C ( x ) t e n g l ama g a
eg a b o ’ l am i z . Bu t e n g l am a ni int egr a ll ab u ( x , y )  f
( x )  g ( y )
u m u m i y y e c h i m g a eg a b o’ l am i z , b u y e r d a f ( x ) v a g ( x )
f u n k s i y a l ar i x t i y or i y di ff e r e sia l l an u vc h i f u nk s i y ala r di r.
2– m i so l .
u
x x 2u xy 3u y y  0 t e ng l a m a n i n g u m u m i y
y e c h i m i n i t o p i ng.
Y ec hi s h :
A  1 , B   1 , C   3 ,   B 2
 AC  1  1  (  3 )  4 
0 y uq or id a gi t e n g l ama g i p e r b o l i k t i p d agi t e n g l ama ek a n . En d i
e sa x a r a kt e r i s t i k t e m g l amas i ni t u z am i z v a u n i y e c ha m i z :
y
 2

2 y


3 
0 ,
y
  
2  4  12
 
1
 2
,
y 
 
3 ,
y


1
y   3 x  C , y  x  C
2 , C  3 x  y , C
2  x  y ,
b u er d a
1 , C
2 o ’ z g ar masl arni m o s ra v i sh da
 v a  l ar
b i l a n a l mas h tir a m i z , y a` ni

  3 x  y
   x  y .
U h ol d a
u f u n k s i y a n i m u r a k k a b f u n k s i y a d e b q a r a b  v a 
o ` z g ar u v c h i l ar
x v a y o` z g ar u vc h i l arn i ng chi z i q li f un k s i y a l a r i
35


        


e kan l i gi n i h i so b g a o l i b b i r inc h i v a ik k i n c h i ta r t i b l i x u s u siy h os i l a l a r ni
h i s o b l a y mi z :
u
x  3u  u , u
y  u u ,
u
x x  9u
 6u  u , u
x y  3u
2u
u 
, u
y y  u 2u  u .
To pi l g an i f od a l a r n i
u
x x 2u xy 3u y y  0 t e n g l ama g a o l i b b o r ib
qu y a m i z . N a t i j a d a
9 u

 6 u 
 u 
 2

 3
u 
 2 u 
 u 
  3

 u 
 2 u 
 u

 
0 ,
yo k i 16u 
 0 ,
u

 0
t e n g l ama g a eg a b o’ l ami z . U s h b u t e n g l ama n i n g u m u miy y e c hi mi
y uq or id a gi 1 - m is olg a a s o s a n q u y ida g i c h a b o’ l a d i :
u (
 ,  )  f (  )  g ( ) .
B u e r da
 v a  o ` z g a r u v c h i l ar o`r ni ga u l a rn i n g x v a y
o ` z g ar u v c h i l ar or qa l i i f od a l a r i n i q u y i b ,
u ( x , y )  f ( 3 x  y )  g ( x  y ) u m u miy
y e c h i m g a e g a bo’ l ami z .
3 - m is ol .
u
x y 2u x  0 t e ng l am an in g u m u miy y e c h i m i n i
t o p i n g .
36

Y ec h i s h: u
x  v d e b b e l g i l ash k i r i t a m i z . U h o l d a v
y 2v  0
t e n g l ama g a eg a b o’ l am i z . Us hb u c hi z i q l i t e n gl am a n i y e c ha m i z v a
v  C ( x ) e 2 y
t e n g l i kk a e g a b o’ l am i z . To pi l g an i f o da n i k i r i t i l g a n
b elg i l as h g a o l ib b o r i b q u y ib , u
x  C ( x ) e 2 y
t e n g l ama g a e g a
b o ’ l am i z . B u c hi zi q l i b i r j i n s l i t e n g l am a n i y e c h s a k ,
u ( x , y )  f ( x ) e 2 y
 g ( y ) u m u m iy y e c hi m n i h o s i l q i l ami z ,
bu y e r d a f ( x ) v a g ( x ) f u nk s i y a l ar i x t i y o r i y
d i ff e r e s i a ll a nu vc hi
f u n k s i y a l a r di r.
4 - m is ol .
u
x y  2u
x  3u
y  6u  2 e x
 y
t e ng l am a n i ng
u m u m i y y e c h i m i n i t o p i n g .
Y ec h i s h:
U s h bu t e n g l am a ni y e c hi sh u c h un a v v al t e n g l a m a d a gi
b i r in c h i t a r t i b l i x u s u s i y xo s i l a l ar n i y o` q o ta m i z . Bu n i ng
u c h un
u ( x , y )  v ( x , y )  e
 x
  y
a l ma s h t iri s h b a jara m i z , b u y e r da gi

v a  o ` z g armas l a r n i k e y in c ha li k t a n l a y mi z . Bi r i n c h i v a
i k k i n ch i ta r t i b l i x u s u siy h os i l a l ar n i h i s o b l a y m i z :
u
x  v
x  e
 x
  y
 v   e  x
  y u
y  v
y

e  x
  y
 v   e  x
  y
u
x y  v
xy  e  x
  y
 v
x   e  x
  y
 v
y   e  x
  y
 v
   e  x
  y
To pi l g an i fo d a l ar n i u
x y  2u
x

3u
y  6u  2 e x
 y
t e n g l ama g a
37

ol i b b or i b q u y a mi z . N a t i j a d a
38

v
xy  e  x
  y
 v
x   e  x
  y
 v
y   e  x
  y
 v    e  x
  y

 2
  v
x  e
 x
  y
 v
  e  x
  y
  3

 v
y  e  x
  y
 v

 e  x
  y
 
 6 v  e  x
  y
 2 e x
 y
,
yo k i
v
xy  e
 x
  y
 (   2 ) v
x  e  x
  y
 (   3 ) v
y  e  x
  y

 (
  2   3   6 ) v  e  x
  y
 2 e x
 y
t e n g l ama g a e g a b o’ l ami z . U s h bu t engl i kd a
 v a  o ` z g armas l a r n i
s h u n day t a n l a y mi z ki, o x i r g i t e n gl i k d a bir i n c h i ta r t i b l i x u s u s iy
h o sila l ar q a t n as h m as i n . Bu n i n g u c hun
  3 ,   2 d e b
t a n l a y m i z v a q u y ida g i t e n gl ama g a e g a b o’ l am i z :
e 3 x
 2 y

v
x y  2 e x
 y
,
y a` ni
B u e r da v
x

 v
xy 
2 e  2 x
 y
.
d e b b elg i l ash ki r i ta m i z . U h ol d a u sh b u

y  2 e  2 x
 y
c h i z i q l i t e ng l ama g a e g a b o’ l amiz v a u n i y e c h a m i z . N a t i j a d a

  2 e  2 x
 y
 C ( x ) ,
y a' n i
v
x   2 e  2 x
 y
 C ( x )
c h i z i q l i t e ng l ama g a e g a b o’ l ami z . Bu t e ng l ama n i i n t egr a l l a b ,
39

v  e  2 x
 y
 f ( x )  g ( y ) e kan l i gi n i h o s i l
q i l ami z .
u ( x , y )  v ( x , y )  e 3 x
 2 y
a l mas h ti r i s h g a
asos a n,u(x,y) e
xy


 f (x) g(y)
  e
3x2y
u m u m i y y e c h i m n i h os i l q i l ami z , b u y er da f ( x ) v a g ( x )
f u n k s i y a l ar i x t i y or i y di ff e r e sia l l an u vc h i f u nk s i y ala r di r.
M u s t a q i l y ec h ish u c hun m i sol l a r
Q uy i d a be r i l g a n t en g l amalarn i ng u m u m i y y e c hi m ini t o p in g : 9 . 1 .
u
x y 4u x  3u y 12u  0 .
9 . 2 .
u
x x 4u xy 5u y y  0 .
9 . 3 .
u
x y s i n y  u
x  0 .
9 . 4 .
u
x y  2u
x  5u
y  10u  2 e 3 x
 2 y
.
9 . 5 .
3u
xx 5u xy 2u yy  3u x  u y  2 .
9 . 6 .
u
x y  au
x  bu y  abu  0 .
9 . 7 .
u
x x  a 2 u
yy  0 .
9 . 8 .
u
x x  2au
x y  a 2 u
yy  u
x  au
y 
0
. 9 . 9 . u
x x 4u x y  4u
yy  u x 2u y  0
.
40

2x yx9 . 1 0. u
x x 9u y y  6u x  0 .
9 . 1 1.
u
x y  3u
x  4u
y  2 e 3 x
 2 y
.
9 . 1 2. u
x x  x  u
x  0 .
9 . 1 3.
u
x x  u
x y  6u
y y  2u
x  3u
y  e 2 x
 y
. 9 . 1 4.
u
x x  2u
x y  8u
yy  2u
x  e 3 x
 2 y
.
9 . 1 5.
u
x y  y  u
x  0 .
9 . 1 6.
9 . 1 7.
9 . 1 8.
u x x 16u y y  6u y  0 .
u
x x u yy  6u x  4u y  0 .
u  1
 u  0 .
c o s y
42

2n Xulosa
Ma ' l u m k i , c hi z i q l i a l mas h t i r i sh n i m o s t a n l a s h y o ’ l i bi l an
k v a d r a t ik f or m a n i nga
i j ( x 0 )
ma t r i t s a s i n i d i a g o n a l s h a k lg a , y a' n i


i   i k a n o n i k s h a k l g a k е l ti r ish m u m k i n b o ’ l i b , bu n d a
i  1
 i
, i  1 , n k o e ff i t s i е n t l ar 1 ,  1 , 0 qi y ma t l a r n i q a b u l q i l a d i ,
b un d a n t a s h qari in е r t s i y a q o n uniga ko’r a , m u s b a t, m a n f i y v a n o l g a
t е n g k o e ff i t s i е n t l ar so n i k v a d r a t i k for ma n i k a n o n i k shak lg a
k е l t i ri s h da g i c h i z i q l i a lm a s h t iri s h g a n i s b a t an i n v a r i an t d i r.
A g a r b a rc h a
n t a 
i k o e ff i t s i е nt l ar b i r x i l i s h or a l i
b o ’ l sa , u h ol d a t е ng l ama x 0
nu q t a d a e l l i p t i k t i p da gi t е n g l ama
d е b, a g a r
n  1 t a 
i k o e ff i t s i е n t l ar b i r x i l i s h or a l i v a b i t t a
k oeff i t s i е nt un g a q a r a m a – q ars h i i s h ora l i b o ’ l s a , u h ol d a t е n gl a m a
x 0
n u q t a d a
g ip е r b o l ik t i p da g i ( y o k i n or m al gip е rbo l ik t i p da gi ) t е n g l ama d е b,
a g a r

i k o e ff i t s i е n t l ar n i n g m t a si bi r x i l i s h or a l i v a n m t a si
un g a qa r am a – qars h i i s h o ra l i
 m
 1 , n
 m
 1

bo’lsa, u holda
t е nglama x0 nuqtada ultragip е rbolik tipdagi t е nglama d е b, agar i

koeffitsi е ntlarning h е ch bo’lmaganda bittasi nolga t е ng bo’lsa, u
holda t е nglama x0 nuqtada parabolik tipdagi t е nglama d е b ataladi
43

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44

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