Alisher navoiy nomidagi samarqand davlat universiteti differentsial tenglamalar kafedrasi
Download 8.22 Mb. Pdf ko'rish
|
|
t
U U a U t xx t ; 379
) ( ) 0 , ( ), ( ) 0 , ( , 0 ) , ( ) , 0 ( ), ( 2 x x U x x U t l U t U x f U a U t xx t ;
R r y x U 0 , 0 ) , ( , R r y x f y x U ), , ( ) , (
, 2
2 2
r y x .
40) Qaysi tenglama gaz tarqalish va diffuziya tenglamasi bo’ladi
r y x U 0 , 0 ) , ( , R r y x f y x U ), , ( ) , ( , 2 2 2 2
r y x ; ) ( ) 0 , ( ), ( ) 0 , ( , 0 ) , ( ) , 0 ( , 2 x x U x x U t l U t U U a U t xx tt ;
x U x x U t l U t U U a U t xx tt ) 0 , ( , ) 0 , ( , 0 ) , ( ) , 0 ( , 2 2 ;
) ( ) 0 , ( , ) 0 , ( , 0 ) , ( ) , 0 ( , 2 x x U o x U t l U t U U a U t xx t
.41) Qaysi tenglama issiqlik tarqalish tenglamasi bo’ladi ) ( ) 0 , ( , ) 0 , ( , 0 ) , ( ) , 0 ( , 2
x U o x U t l U t U U a U t xx t ;
r y x U 0 , 0 ) , ( , R r y x f y x U ), , ( ) , ( , 2 2 2 2
r y x ;
) ( ) 0 , ( ), ( ) 0 , ( , 0 ) , ( ) , 0 ( , 2
x U x x U t l U t U U a U t xx tt ;
x U x x U t l U t U U a U t xx tt ) 0 , ( , ) 0 , ( , 0 ) , ( ) , 0 ( , 2 2
42) y x U xy tenglamaning umumiy yechimini toping.
) ( ) ( 2 2 1 2 2
x xy y x U ; ) ( ) ( 2 1 2 y x y x U ;
) ( ) ( 2 2 1 2 2 y x xy y x U ; ) ( ) ( 2 1 2 y x xy U .
43) 0 yy xy U U tenglamaning umumiy yechimini toping.
)
) (
y y x U
; ) 2 ( ) ( x y y x U
;
) ( ) ( x y e y e x U
; ) 2 ( ) 2 (
x y x U
.
44) 0 3 10 3
xy xy U U U tenglamaning umumiy yechimini toping.
)
) 3 ( x y x y U
; ) ( ) ( y x y x U
;
) 3 ( ) 3 ( x y x y U
; ) 3 3 1 ( ) 3 1 (
y x y U
.
45) 0 5 6 yy xy xx U U U tenglamaning umumiy yechimini toping.
)
) 5 ( x y x y U
; ) ( ) 5 ( x y y x U
;
) ( ) 5 ( x y x y U
; ) 4 ( ) 4 5 ( x y x y U
.
46) Koshi masalasini yeching: 2 2 2 ) 0 , , , ( , ) 0 , , , ( z y x z y x U xyz z y x U U U U U t zz yy xx tt
7 5 2 2 2 3 2 2 2 2 2 2 2 105
1 ) ( 15 1 ) ( 3 1 ) (
t z y x t z y z x y x t xyz xyz U ;
U .
xyz xyz U 3 ) ( ; 105
7 t xyz U .
380
47) Ushbu xy z y x U z y x z y x U U U U U t zz yy xx tt ) 0 , , , ( , ) 0 , , , ( 2 2 2 Koshi masalasi yechimini ko’rsating
t z y x U 2 2 2 2 3 ; 2 2 2 z y x U ; 2 2 2 2 3t z y x U ; 2 2 2 2 5t z y x U . 48) Ushbu
1 ) 0 , , , ( , ) 0 , , , ( 2 2 z y x U y x z y x U U U U U t zz yy xx tt Koshi masalasi yechimini ko’rsating
2
2 2t t y x U ; 2 2 2 2t t z x U ; 2 2 2 3t t z x U ; 2 2 2 2t t z y U . 49) Berilgan Koshi masalasi yechimini aniqlang:
1 ) 0 , , , ( , ) 0 , , , (
y x U y x z y x U U U U U t zz yy xx tt
t y x U ;
y x U ; 2 t y x U ;
t y x U .
50) Berilgan Koshi masalasini yechimini aniqlang:
z y x U z y x z y x U U U U U t zz yy xx tt ) 0 , , , ( , ) 0 , , , (
tz z y x U ; tz t z y x U 2 ) ( ; tx z y x U ; ty z y x U . 51) 0 y x U U tenglamani yechimini ko’rsating.
.
.
. 2 2 y x U . 52)
0 y x yU xU tenglamani yechimini ko’rsating.
.
y x U . xy U . . 2 2
x U
53) 0 2 2
x U y U x tenglamani yechimini ko’rsating. y x U 1 1 .
x U . y x U . . 2 2
x U 54)
0 y x xU yU tenglamani yechimini ko’rsating. 2 2
x U .
y x U . xy U y x U y x U 381
x U . . x y U
55) 0 y x U U tenglamani umumiy yechimini ko’rsating. ) (
x U .
) ( y x U .
) (xy U .
) ( 2 2 y x U . 56) 0 y x yU xU tenglamani umumiy yechimini ko’rsating.
)
y x U ..
) (
x U .
) (xy U .
) ( 2
x U . 57)
0 2 2
x U y U x tenglamani umumiy yechimini ko’rsating.
)
1 (
x U ..
) ( y x U .
) ( 2 2 y x U .
) ( 2 2 y x U . 58) 0 y x xU yU tenglamani umumiy yechimini ko’rsating.
)
2 2
x U ..
) ( y x U .
) (
x U .
) (xy U . 59)
0 y x yU xU tenglamaning birinchi integralini aniqlang.
..
y x . c y x .
y x .
60)
x U x x U t x U U t xx tt , 0 ) 0 , ( , ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
x t x U ) , ( .
x t x U ) , ( .
t x U ) , ( . t x t x U ) , ( .
382
61)
x U x x U t x U U t xx tt , 0 ) 0 , ( , sin ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
x t x U cos
sin ) , ( .
x t x U cos
sin ) , ( .
x t x U cos
sin ) , ( . t x t x U sin
sin ) , ( .
62)
x x U x U t x U U t xx tt , ) 0 , ( , 0 ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
xt t x U ) , ( .
x t x U ) , ( . 2 2 ) , ( t x t x U .
x t x U ) , ( .
63)
x U x x U t x U U t xx tt , 1 ) 0 , ( , ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
t x t x U ) , ( .
) sin(
) , ( t x t x U .
x t x U 2 ) , ( .
xt t x U 2 ) , ( .
64)
x U x x U t x U U t xx tt , 0 ) 0 , ( , cos ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
x t x U cos
cos ) , ( .
x t x U cos
cos ) , ( .
x t x U sin
cos ) , ( . t x t x U sin
sin ) , ( . 65)
x U x x U t x U U t xx tt , 0 ) 0 , ( , ) 0 , ( 0 , , 2 Koshi masalasi yechimini aniqalng
2 ) , ( t x t x U .
xt t x U ) , ( . t x t x U ) , ( . 2 2 ) , ( t x t x U . 383
66) x x x U x U t x U U t xx tt , sin ) 0 , ( , 0 ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
x t x U sin
sin ) , ( .
2 sin
sin ) , ( t x t x U . t x t x U sin
sin ) , ( .
x t x U cos
sin ) , ( .
67)
x x U x U t x U U t xx tt , cos ) 0 , ( , 0 ) 0 , ( 0 , , Koshi masalasi yechimini aniqalng
x t x U sin
cos ) , ( .
x t x U cos
sin ) , ( . t x t x U sin
cos ) , ( .
x t x U sin
cos ) , ( . 68) Quyidagi tenglamalardan qaysi birini 0
V xy kurinishga keltirish mumkin.
0
x xy U U U
0 2 2 2 y x xy U y U x U
0 1 1 y x xy U y U x U
0 cos
sin y x xy yU xU U
69) U U U U y x xy 3 2 tenglama qanday turdagi tenglamaning sodda kurinishidan iborat. Giperbolik turdagi. Elliptik turdagi. Parabolik turdagi. Elleptiko-Parabolik turdagi.
70) 0 yy xx U U tenglama qanday turdagi tenglamaning sodda ko’rinishidan iborat Elliptik turdagi. Giperbolikik turdagi. Parabolik turdagi. Giberbolo-Parabolik turdagi.
71)
U U U U y x xx tenglama qanday turdagi tenglamaning sodda ko’rinishidan iborat Parabolik turdagi. Giperbolikik turdagi. Elliptik turdagi. Elliptiko-Giberbolik turdagi.
72) Nyuton potensialini aniqlang
3 2 1 | | 1 dy dy dy y y x x u D
384
y y x x u | | 1
2 1 1 ln dy dy y y x x u D
2 1 ln dy dy y x x y x u D
73) Logarifmik potensialini aniqlang
2 1 1 ln
dy y y x x u D
y y x x u | | 1
3 2 1 | | 1 dy dy dy y y x x u D
2 1 ln dy dy y x x y x u D
74) Quyidagi funksiyaning qaysi biri
2 1 2 2 2 1 | | 1 ln dy dy y x y y x u D logarifmik potensialning zichligi bo’ladi.
2 2 2 1 y y y
2 2 2 1 1 y y y
2 2 2 1 1 y y y
1 y
75)
3 2 1 2 1 1 | | 1 dy dy dy y x y y y x u D Nyuton potensiali zichligini aniqlang
2 1 1 y y y y
2 1 1 y y y
1 2 1 y y y y
1 y
76) ) ( ) 0 , ( ), ( ) 0 , ( , 2 x x u x x u u a u t xx tt masala yechimi uchun Dalamber formulasini kursating.
d a at x at x t x u at x at x ) ( 2 1 2 ) ( ) ( ) , (
2 ) ( ) ( ) , ( at x at x t x u
385
d a t x u at x at x ) ( 2 1 ) , (
d a at x at x t x u at x at x ) ( 2 1 2 ) ( ) ( ) , (
77) 0 5 2 2 zz yy xy xx U U U U kurinishdagi elliptic tipli tenglamaning sodda ko’rinishini aniqlang.
0 V V V .
0
V V V
0 V V V .
0
V V V V .
78) Quyidagi giperbolik tipli tenglamaning sodda kurinishini aniqlang: 0 2 2 3 U U U U yz xz xy
0
V V V .
0
V V V
0
V V V V .
0 V V V V V .
79) Ikkinchi tartibli ko’p o’zgaruvchili parabolic tipli tenglamalarning sodda kurinishini ko’rsating 0 2
2 4 4 U U U U U U U yz xz xy zz yy xx .
0 2
V V .
0
V V V
0 V V V .
0 V V V .
80) Fazoda Laplas tenglamasining fundamental yechimini aniqlang.
x U 1 .
y x U 1 ln . y x U .
x U .
81) Tekislikda Laplas tenglamasining fundamental yechimini aniqlang.
x U 1 ln .
x U . 386
y x U .
x U 1 .
82) 0 yy xx U U tenglama uchun
R r y x g y x U R r r y x ), , ( ) , ( 0 , 2 2 2 Dirixle masalasi yechimini kursating.
2 0 2 2 2 2 ) cos( 2 2 1
d r ar a r a U 2 0 2 2 2 2 ) cos( 2
d r r r U 2 0 2 2 ) cos( 2 1 1
d r ar a r a U 0 2 2 2 2 ) cos( 2 2 1
83) 0 ) , ( , 0 ) , 0 ( , ) ( ) 0 , ( , ) ( ) 0 , ( , 2 t l U t U x x U x x U U a U t t xx tt
masalaga mos keluvchi Shturm-Liuvill masalasini kursating
0 ) ( ) 0 ( , 0 2 '' l X X X X
0 ) ( ) 0 ( , 0 2 2 '' l X X X a X
0 ) ( ) 0 ( , 0 2 ''
X X X a X
0 ) ( ) 0 ( , 0 2 ''
X X X X
84) 0 ) , ( , 0 ) , 0 ( , ) ( ) 0 , ( , ) ( ) 0 , ( , 2
l U t U x x U x x U U a U x t xx tt
masalaga mos keluvchi Shturm-Liuvill masalasini kursating
0 ) ( ) 0 ( , 0 ' 2 ''
X X X X
0 ) ( ) 0 ( , 0 2 2 '' l X X X a X
0 ) ( ) 0 ( , 0 2 ''
X X X a X
0 ) ( ) 0 ( , 0 2 ''
X X X X .
85) 0 ) , ( , 0 ) , 0 ( , ) ( ) 0 , ( , ) ( ) 0 , ( ), , ( 2
l U t U x x U x x U t x f U a U x x t xx tt
masalaga mos keluvchi Shturm-Liuvill masalasini kursating
0 ) ( ) 0 ( , 0 ' ' 2 '' l X X X X
0 ) ( ' ) 0 ( , 0 2 2 '' l X X X a X
0 ) ( ) 0 ( ' , 0 2 '' l X X X a X
0 ) ( ) 0 ( ' , 0 2 '' l X X X X
86) Quyidagi aralash masalaga mos keluvchi Shturm-Liuvill masalasini kursating 0 ) , ( , 0 ) , 0 ( ) , 0 ( , ) ( ) 0 , ( , ) ( ) 0 , ( ), , ( 2 t l U t HU t U x x U x x U t x f U a U x t xx tt
0 ) ( , 0 ) 0 ( ) 0 ( , 0 ' 2 '' l X HX X X X
0 ) ( ' ) 0 ( , 0 2 '' l X X X X
387
0 ) ( , 0 ) 0 ( ' ) 0 ( , 0 2 '' l X HX X X X
0 ) ( ' ) 0 ( ' , 0 2 '' l X X X X
87) Quyidagi aralash masalaga mos keluvchi Shturm-Liuvill masalasini kursating 0 ) , ( ) , ( , 0 ) , 0 ( , 1 ) 0 , ( , 0 ) 0 , ( ), , ( 2 t l HU t l U t U x U x U t x f U a U x t xx tt
0 ) ( ) ( , 0 ) 0 ( , 0 ' 2 '' l HX l X X X X .
0 ) ( ' ) 0 ( , 0 2 '' l X X X X
0 ) ( , 0 ) 0 ( ' ) 0 ( , 0 2 '' l X HX X X X
0 ) ( ' ) 0 ( ' , 0 2 '' l X X X X
88) Chegaraviy sharti uchunchi tur bulgan quyidagi aralash masalaga mos Shturm-Liuvill masalasini kursating: 0 )
( ) , ( , 0 ) , 0 ( ) , 0 ( , ) ( ) 0 , ( , ) ( ) 0 , ( , 2 t l HU t l U t HU t U x x U x x U U a U x x t xx tt
0 ) ( ) ( , 0 ) 0 ( ) 0 ( , 0 ' ' 2 ''
HX l X HX X X X .
0 ) ( ) 0 ( , 0 2 '' l X X X X
0 ) ( ' , 0 ) 0 ( ' , 0 2 '' l X X X X
0 ) ( ) 0 ( ' , 0 2 '' l X X X X
89) 0 ) , ( , 0 ) , 0 ( , ) 0 , ( , 0 ) 0 , ( , 2
l U t U x x U x U U a U t xx tt masalaning xos qiymatlarini kursating.
. ,... 3 , 2 , 1 , n l n n
. ,...
3 , 2 , 1 , 2 n l n n
. ,...
3 , 2 , 1 , 2 ) 1 2 (
l n n
. ,...
3 , 2 , 1 ,
n l n
90) Quyidagi chegaraviy sharti ikkinchi turdan iborat bo’lgan aralash masalaning xos qiymatlarini kursating: 0 )
( , 0 ) , 0 ( , 1 ) 0 , ( , ) 0 , ( , 2
l U t U x x U x x U U a U x x t xx tt
. ,...
3 , 2 , 1 ,
l n n
. ,...
3 , 2 , 1 , ) 1 2 ( n l n n
. ,...
3 , 2 , 1 ,
n l n
. ,...
3 , 2 , 1 , 1 2
n l n
91) Quyidagi aralash masalaning xos qiymatlarini kursating: 0 ) , ( , 0 ) , 0 ( , cos ) 0 , ( , sin ) 0 , ( , 2 t l U t U x x U x x U U a U x t xx tt
388
. ,...
3 , 2 , 1 , 2 ) 1 2 (
l n n
. ,...
3 , 2 , 1 , 0 , n l n n
. ,...
3 , 2 , 1 ,
n l n
. ,...
3 , 2 , 1 , 0 , 1 2 n l n n
92) xx tt U U tenglama uchun Gursa masalasi shartlarini ko’rsating.
. ' 0 ), ( ) , ( , ' 0 ), ( ) , (
bo x t agar x t x U lsa bo x t agar x t x U
) ( ) 0 , ( ), ( ) 0 , ( x x U x x U t .
) ( ) 0 , ( ), ( ) 0 , (
x U x x U t x .
) ( ) 0 , ( ), ( ) 0 , (
x U x x U x x .
93) Bir ulchovli to’lqin tenglamasi uchun qo’yilga Koshi masalasi yechimi qanday formula orqali beriladi? Dalamber formulasi. Grin formulasi. Puasson integrali. Krixgorf formulasi.
94) Ikki ulchovli to’lqin tenglamasi uchun qo’yilga Koshi masalasi yechimi qanday formula orqali beriladi? Puasson formulasi. Grin formulasi. Dalamber formulasi. Krixgorf formulasi.
95) Uch ulchovli to’lqin tenglamasi uchun qo’yilga Koshi masalasi yechimi qanday formula orqali beriladi? Krixgorf formulasi. Grin formulasi. Puasson integrali. Dalamber formulasi.
96) Fazoda Laplas tenglamasi uchun Grin funksiyasining kurinishi qanday buladi? ) , ( 1 ) , ( y x g y x y x G .
) , ( 1 ln ) , (
x g y x y x G .
y x y x G 1 ) , ( .
y x y x G 1 ln ) , ( . 389
97) Tekislikda Laplas tenglamasi uchun Grin funksiyasining kurinishi qanday buladi? funksiya garmonik y x g y x g y x y x G ) , ( ), , ( 1 ln ) , ( .
) , ( 1 ) , ( y x g y x y x G .
y x y x G 1 ) , ( .
y x y x G 1 ln ) , ( . 98) Fazoda Laplas tenglamasi uchun qo’yilgan Dirixle masalasi yechimi qanday ko’rinishda? Puasson integrali ko’rinishida. Kirxgorf formulasi ko’rinishida. Grin formulasi ko’rinishida. Dalamber formulasi ko’rinishida.
99) Tekislikda Laplas tenglamasi uchun qo’yilgan Dirixle masalasi yechimi qanday ko’rinishda? Puasson integrali ko’rinishida. Kirxgorf formulasi ko’rinishida. Grin formulasi ko’rinishida. Dalamber formulasi ko’rinishida.
100) Gel’mgol’ts tenglamasini ko’rsating. 0 2 U k U U U zz yy xx .
0 zz yy xx U U U .
0 yy xx U U .
0
yy xx tt U U U U .
390
MATEMATIK FIZIKA TENGLAMALARI
TAQDIMOT SLAYDLARI 391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
|
