UCH O‘LCHOVI DIFFERENSIAL TENGLAMALAR SISTEMASINING MUVOZANAT NUQTALARINING TURLARI

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

304

UCH O‘LCHOVI DIFFERENSIAL TENGLAMALAR SISTEMASINING MUVOZANAT

NUQTALARINING TURLARI

Xusanov Bazar

Samarqand davlat arxitektura-qurilish universiteti, dotsent

E-mail:

bozorboyxusanov98@gmail.com.

Shodiyev Kamoliddin

Iqtisodiyot fanlari bo’yicha falsafa doktori, PhD

E-mail:

shodiyevkamoliddin91@gmail.com

Vahobov Mehroj

O‘qituvchi

E-mail:

vahobovmehroj62@gmail.com

https://doi.org/10.5281/zenodo.15089083

Annatasiya.

Uch o’lchovli fazoda o’zgarmas koeffsentli bir jinsli differensial tenlamalar

sistemasini trayektoriyalarini muvozanat nuqta

𝑂(0,0,0)

atrofida joylashishini o’rganiladi.

Kalit so’zi

: Muvozanat, xaraktrestik tenglama, asimtotik, tugun, turg’unmas, trayektoriya,

turg’un tugun, Turvis sharti, turg’un fokus, turg’unmas fokus, focus markaz, karrali haqiqiy va
har xil, qo’shma kompleks turg’unmas egar, turg’un tug’ma tugun.

Bu maqolada o’zgarmas koeffesintli bir jinsli tenglamalar sistemasi uch o’lchovli fazoda

berilgan bo’lsin

{

𝑑𝑥

1

𝑑𝑡

= 𝑎

11

𝑥

1

+ 𝑎

12

𝑥

2

+ 𝑎

13

𝑥

3

𝑑𝑥

2

𝑑𝑡

= 𝑎

21

𝑥

1

+ 𝑎

22

𝑥

2

+ 𝑎

23

𝑥

3

𝑑𝑥

3

𝑑𝑡

= 𝑎

31

𝑥

1

+ 𝑎

32

𝑥

2

+ 𝑎

33

𝑥

3

𝑑𝑒𝑡𝐴 ≠ 0,

𝐴 = (𝑎

𝑖𝑗

) (1)

Sistema traektoriyalarning

𝑂(0,0,0)

muvazanat (yoki maxsus) nuqta atrofida joylashini

o’rganamiz (1) Sistema yechimini

𝑥

𝑘

= 𝛼𝑒

2𝑡

(𝑘 = 1,3

̅̅̅̅)

ko’rinishida izlaymiz

𝜆

ni topish uchun

xarakteristik tenglama tuzamiz

∆(𝜆) = |

𝑎

11

− 𝜆

𝑎

12

𝑎

13

𝑎

21

𝑎

22

− 𝜆

𝑎

23

𝑎

31

𝑎

32

𝑎

33

− 𝜆

| = 0 (2)

Yoki

𝜆

3

+ 𝑎

1

𝜆

2

+ 𝑎

2

𝜆 + 𝑎

3

= 0 𝛼

1

, 𝛼

2

, 𝛼

3

- lar aniq ko’paytuvchilarga ikkitasi

uchinchisi orqali quydagi sistemadan topiladi.

{

(𝑎

11

− 𝜆)𝛼

1

+ 𝑎

12

𝛼

2

+ 𝑎

13

𝛼

3

= 0

𝑎

21

𝛼

1

+ (𝑎

22

− 𝜆)𝛼

2

+ 𝑎

23

𝛼

3

= 0

𝑎

31

𝛼

1

+ 𝑎

32

𝛼

2

+ (𝑎

33

− 𝜆)𝛼

3

= 0

(3)

Quydagi hollarda bo’lishi mumkin ularni ko’rib chiqamiz alohida-alohida
a) Xarakteristik tenglamani ildizlari

𝜆

𝑖

(𝑖 = 1,3

̅̅̅̅)

haqiqiy va har xil, u holda uning yechimi

𝑥

𝑘

= 𝑐

1

𝛽

𝑘

𝑒

𝜆

1

𝑡

+ 𝑐

2

𝛾

𝑘

𝑒

𝜆

2

𝑡

+ 𝑐

3

𝛿

𝑘

𝑒

𝜆

3

𝑡

𝑘 = 1,3

̅̅̅̅ (4)

ko’rinishda yoziladi.

𝛽

𝑘

, 𝛾

𝑘

, 𝛿

𝑘

lar (3) sistemaga mos ravishda

𝜆 = 𝜆

1

, 𝜆 = 𝜆

2

, 𝜆 = 𝜆

3

larni quydagilar orqali

topiladi

𝑐

1

, 𝑐

2

, 𝑐

3

lar erkli o’zgarmas. O’z navbatida bu yerda quydagi holler bo’lishi mumkin.

1) barcha

𝑖

lar uchun

𝜆

𝑖

< 0.

U vaqtda muvozanat nuqta

𝑥

1

= 0, 𝑥

2

= 0, 𝑥

3

= 0

asimtotik

to’rg’un, chunki (3) formuladagi ko’paytuvchi

𝑒

𝜆

𝑖

𝑡

𝑙𝑎𝑟 𝑡 = 𝑡

0

moment koordinatalari boshini

𝛿

atrofida bo’lgan barcha nuqtalar

𝑡

ning istalga kata qiymatlari uchun koordinatalar boshining

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

305

istalgan kichik

𝛿

atrofiga tushib qoladi va

𝑡 → ∞

da koordinatalar boshiga intiladi. 1-chizmada

muvozanat nuqta atrofidagi traektoriyalar joylashishi ko’rsatilgan. Muvozanat turg’un, tugun
turg’un deyladi. Strelkalar

𝑡

o’sishi bilan traektoriyalar bo’yicha harakat yo’nalishini bildiradi. Bu

hol (1) Sistema quydagi koefsentli tengsizliklar bajarilishi o’rinli bo’ladi.

1- chizma

𝑎

1

> 0,

∆

2

= |

𝑎

1

1

𝑎

3

𝑎

2

| > 0,

𝑎

3

> 0, 𝐷 < 0.

Bu yerda

𝐷 = 𝑞

2

+ 𝑝

2

xarakteristik tenglamaning diskriminanti

2𝑞 =

2𝑎

1

3

27

−

𝑎

1

𝑎

2

3

+ 𝑎

3

, 3𝑝 =

3𝑎

2

− 𝑎

1

2

3

2)

𝜆

1

> 0, 𝜆

2

> 0, 𝜆

3

> 0

bo’lsin. Bu holda

𝑡

ni

𝑡

ga olmashtersak, yuqorida ko’rilgan

strelkalarini teskari quyish kerak) muvozanat nuqta turg’unmas tugun deyladi bu hol koefsentli

𝑎

1

< 0, ∆

2

< 0, 𝑎

2

> 0, 𝐷 < 0

tengsizliklar bajarilganda o’rinli bo’ladi.

2- chizma
3) Agar

𝜆

1

> 0, 𝜆

2

< 0, 𝜆

3

< 0

bo’lsa, muvozanat nuqta turg’unmas chunki

𝑥

𝑘

= 𝑐

1

𝛽

𝑘

𝑒

𝜆

𝑖

𝑡

Traektoriyalar bo’yicha harakat qiluvchi nuqtalar (

𝑐

1

ni qanchalik kichik olmaylik)

𝑡

o’sishi bilan koordinatalar boshidagi

𝜀

atrofidan chiqadi

𝑡 → ∞

koordinatalar boshiga intiluvchi

harakatlar

𝑥

𝑘

= 𝑐

2

𝛾

𝑘

𝑒

𝜆

2

𝑡

+ 𝑐

3

𝛿

𝑘

𝑒

𝜆

3

𝑡

ko’rinishda bo’lib, butun bir tekslikni tashkil qiladi (2-

chizma).

(1) sistema

𝐷 < 0

gurbis sharti bajarilgan holda uchraydi

b) Xarakteristik tenglamani ildizlari bajarilmagan holda uchraydi ikkitasi qo’shma

kompleks, ya’ni

𝜆 = 𝜆

1

, 𝜆

2,3

= 𝑝 ± 𝑞𝑖 , 𝑞 ≠ 0

bu holda Sistema uchun umumiy yechim

𝑥

𝑘

= 𝑐

1

𝛽

𝑘

𝑒

𝜆

1

𝑡

+ 𝑒

𝑝𝑡

(𝑐

1

𝑐𝑜𝑠𝑞𝑡+𝑐

2

𝑠𝑖𝑛𝑞𝑡) 𝑘 = 1,3

̅̅̅̅ (5)

ko’rinishda bo’ladi

𝑐

2

𝑖

> 𝑐

3

𝑖

𝑙𝑎𝑟 (𝑖 = 2,3) 𝑐

2

𝑖

, 𝑐

3

𝑖

larning birorta chiziqli konbinatsiyasidan

iborat bo’ladi. Shu sabali bu yerda quydagi holler uchraydi:

1)

𝜆

1

< 0, 𝜆

2,3

= 𝑝 ± 𝑞𝑖 , 𝑝 < 0 𝑞 ≠ 0

ko’paytuvchi

𝑒

𝑝𝑡

, 𝑝 < 0 𝑡 𝑛𝑖𝑛𝑔

o’sishi bilan

nolga intiladi, ikkinchi davriy ko’paytuvchi (5) formulada har doim chekli bo’ladi.

𝑡 = 𝑡

0

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

306

momentda koordinatalar momentda koordinatalar boshininh

𝛿

atrofida bo’lgan nuqta traektoriya

buylab harakat qilib

𝑡

ning o’sishi bilan koordinatalar boshiga yaqinlashib boradi va

𝑡 → ∞

da

nolga intiladi. Nol yechim- asimtotik turg;un muvozanat nuqtaga ega turg’un focus deyladi (3-
chizma ).

Bu holda

𝐷 > 0, 𝑎

1

> 0, ∆

2

> 0, 𝑎

3

> 0

bo’ladi.

3-chizma

2)

𝜆

1

> 0, 𝜆

2,3

= 𝑝 ± 𝑞𝑖 , 𝑝 > 0 𝑞 ≠ 0

bo’lsin bu yerda

𝑡 𝑛𝑖 𝑡 𝑔𝑎

almashtersak qolgan

hollarga kelamiz. Traektoriyalar joylashishi 3-chizmadagi kabi bo’ladi, lekin stelkalarteskari
quyiladi. Muvozanat nuqta turg’unmas focus deyladi.

Bu holda

𝐷 > 0, 𝑎

1

< 0, ∆

2

< 0, 𝑎

3

> 0

bo’ladi

3)

Agar

𝜆 = 𝜆

1

, 𝜆

2,3

= 𝑝 ± 𝑞𝑖 , 𝑝 < 0, 𝜆, 𝑞 ≠ 0

bo’lsa

(5)

formuladagi

qushuluvchilardan bittasi o’suvchi, hisoblanadi

𝑡 = 𝑡

0

𝑚𝑜𝑚𝑒𝑛𝑡𝑎 𝛿

atrofida bo’lgan

𝑡

o’sishi

bilan traektoriya bo’ylab koordinata boshidan uzoqlasha boshlaydi, demak nol yechim
turg’unmas. Muvozanat nuqtaga egar-fokus deyladi.

𝐷 > 0

bo’lib, Turvis sharti bajarilmasa

muvozanat nuqta egar-fokus bo’ladi.

4)

𝜆

1

, 𝜆

2,3

= ±𝑞𝑖 , 𝑝 = 0

bo’lsa

𝜆

1

ning ishorasiga qarab nol yechim turg’un yoki

turg’unmas bo’lishi mumkin.

Muvozanat nuqta focus markaz (4-chizma) egar-fokus bo’lishi mumkin.

4-chizma
b) Xarakteristik ildizlari karrali

𝜆

1

= 𝜆

2

, 𝜆

3

, 𝜆

1

≠ 𝜆

3

ya’ni xarakteristik tenglamani

diskeriminanti

𝐷 = 0.

Bu holda (1) sistemaning umumiy yechimi

𝑥

𝑘

(𝑡) = (𝑐

1

𝛽

𝑘

+ 𝑐

2

𝛾

𝑘

𝑡)𝑒

𝜆

1

𝑡

+ 𝑐

3

𝛿

𝑘

𝑒

𝜆

3

𝑡

, 𝑘 = 1,3

̅̅̅̅ (6)

Kurinishda bo’lib kupaytuvchi

𝑒

𝜆

1

𝑘

, 𝑡 → ∞

da nolga intiladi. Shu bilan birga koordinatalar

boshining

𝛿

- atrofidagi istalgan nuqta

𝑡

o’sishi bilan

𝜀

- atrofiga tushadi va nolga intiladi. Nol

yechim asimtotik turgun bo’ladi. Muvozanat nuqta turg’un deyladi bu holda muvozanat a), 1) bilan
turg’un focus b), 1) orasidagi chigaraviy dolda mos keladi, chunki

𝑎

𝑖𝑗

parametrlari ozroq

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

307

o’zgartirilganda muvozanat turg’un tugun, a), 1) holda yoki turg’un fokudga o’tishi mumkin, ya’ni

𝑎

𝑖𝑗

parametriga o’zgarish berilganda xaraktrestik tenglamaning karrali ildizlari oddiy haqiqiy yoki

qushma kompleks ildizlari ajralishi mumkin.

2) Agar

𝜆

1

= 𝜆

2

> 0, 𝜆

3

> 0 𝑏𝑜

′

𝑙𝑠𝑎 𝑡 𝑛𝑖 𝑡𝑔𝑎

almashterilsin 1) holdagi natijaga

kelamiz. Faqat traektoriyalar buyicha harakat teskari yo’nalishda bo’ladi. Muvozanat nuqta
turg’unmas tugun deyladi.

3)

𝜆

1

= 𝜆

2

> 0, 𝜆

1

∙ 𝜆

3

< 0

bo’lsin. Nol yechim tugunmas bo’ladi,chunki (6) formulada

𝑡 → ∞ 𝑑𝑎 ‖𝑥(𝑡)‖ → ∞.

Muvozanat nuqta egar deyladi, lekin bu egar

a), 3) holdagi egardan farqi shundaki

𝑎

𝑖𝑗

parametrlarni o’zgarterganda a), 3) holdagi egar

yoki egar fokus b), 3) dagi tugunga o’tish mumkin, chunki

𝑎

𝑖𝑗

paramertlarning o’zgarishi

natijasida ajratilishi mumkin.

a)

𝜆

1

= 𝜆

2

= 𝜆

3

bo’lsin. Xaraktrestik tenglamaning diskerminantida

𝐷 = 0, (1)

Sistema

uchun

𝑥

𝑘

(𝑡) = (𝑐

1

𝛽

𝑘

+ 𝑐

2

𝛾

𝑘

𝑡 + 𝑐

3

𝛿

𝑘

𝑡

2

)𝑒

𝜆

1

𝑡

, 𝑘 = 1,3

̅̅̅̅ (7)

ko’rinishda yoziladi. Quydagi hollar bo’lishi mumkin .
1)

𝜆

1

< 0

bo’lsin

𝑒

𝜆

1

𝑡

ko’paytuvchi

𝑡 → ∞

da tezroq nolga va shu sababli

(𝑐

1

𝛽

𝑘

+ 𝑐

2

𝛾

𝑘

𝑡 + 𝑐

3

𝛿

𝑘

𝑡

2

)𝑒

𝜆

1

𝑡

ifoda

𝑡 → ∞

da nolga intiladi. Nol yechim asimtotik

turg’un. Muvozanat nuqta turg’un tugun boladi.

2)

𝜆

2

> 0

bo’lsin

𝑡

ni

𝑡

ga almashtersak, 1) holdagi natijaga kelamiz. Traektoriyalar

buyicha harakat teskari yunalishda bo’ladi. Muvozanat nuqta turg’unmas tugun bo’ladi.
Xaraktristik sonlar noldan farqli bo’lgan barcha hollarni qarab chiqdik, chunki shartga ko’ra

𝑑𝑒𝑡𝐴 ≠ 0

. Demak yo’qoridagilarga asoslanib tekslikda muvozanat nuqtani quydagicha sinflash

mumkin ekan.

a)

𝜆

1

𝑣𝑎 𝜆

3

haqiqiy va

𝜆

1

≠ 𝜆

3

. Bu holda muvozanat nuqta;

1)

𝜆

1

< 0 , 𝜆

3

< 0

turg’un tugun;

2)

𝜆

2

> 0, 𝜆

3

> 0

turg’unmas tugun;

3)

𝜆

2

< 0, 𝜆

3

> 0

turg’unmas egar;

b)

𝜆

2,3

= 𝑞 ± 𝑞𝑖

. Bu holda;

1)

𝑝 < 0

turg’un fokus;

2)

𝑝 > 0

turg’unmas focus;

3)

𝑝 ≠ 0

markaz tugun;

d)

𝜆

1

= 𝜆

3

. Bu holda;

1)

𝜆

2

= 𝜆

3

< 0

turg’un tug’ma tugun;

2)

𝜆

2

= 𝜆

3

> 0

turg’unmas tug’ma tugun;

Agar

𝑑𝑒𝑡𝐴 = 0

bo’lgan holda (1) tenglamalar sistemasining muvozanat nuqtalarini

kelgusi ishlarimizda sinflaymiz.

Xulosa

Xulosa qilib aytganda bir jinsli uch o’lchovli differensial tenglamalar sistemasini

yechimlarinin muvozanat nuqta

𝑂(0,0,0)

atrofida sinflash uchun tenglamalar sistemasini

xaraktrestik tenglamasini yechimlari

𝜆

1

, 𝜆

2

, 𝜆

3

larning qiymatlariga qarab

𝑑𝑒𝑡𝐴 ≠ 0

da bturg’un tugun, egar focus va turg’unmas tugun , egar focus, markaz

ekanligini teksheriladi.

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

308

Adabiyotlar:

1.

Хусанов, Б., & Кулмирзаева, Г. А. (2022). О РАСПРЕДЕЛЕНИЕ
ИЗОЛИРОВАННЫХ ОСОБИХ ТОЧЕК ОДНОЙ СИСТЕМЫ n-МЕРНОМ
ПРОСТРАНСТВЕ. In

" ONLINECONFERENCES" PLATFORM

(pp. 319-324).

2.

Husanov, B., & Mahfuza, T. (2022). GEODESICAL VIEWS IN THE MATHEMATICAL
WORKS OF ABU RAYHAN BERUNI.

Central Asian Journal of Theoretical and Applied

Science

,

3

(6), 123-127. Retrieved from

3.

https://www.cajotas.centralasianstudies.org/index.php/CAJOTAS/article/view/568

4.

B., Khusanov, and Fatkhullayev F. "Existence of the Isolated Special Points Three-
dimensional Differential Systems of a Special Look."

JournalNX

, 2020, pp. 239-242.

5.

Bazar, Khusanov, and Kulmirzaeva G. Abduganievna. "Singular Points Classification of
First Order Differential Equations System Not Solved for Derivatives."

International

Journal

on

Integrated

Education

,

vol.

4,

no.

3,

2021,

pp.

448-450,

doi:10.31149/ijie.v4i3.1533.

6.

Matyokubov, B. P., & Saidmuradova, S. M. (2022). METHODS FOR INVESTIGATION
OF THERMOPHYSICAL CHARACTERISTICS OF UNDERGROUND EXTERNAL
BARRIER STRUCTURES OF BUILDINGS. RESEARCH AND EDUCATION, 1(5), 49-
58.

7.

Bolikulovich, K. M., & Pulatovich, M. B. (2022). HEAT-SHIELDING QUALITIES AND
METHODS FOR ASSESSING THE HEAT-SHIELDING QUALITIES OF WINDOW
BLOCKS AND THEIR JUNCTION NODE WITH WALLS. Web of Scientist:
International Scientific Research Journal, 3(11), 829-840.

8.

Egamova, M., & Matyokubov, B. (2023). WAYS TO INCREASE THE ENERGY
EFFICIENCY OF BUILDINGS AND THEIR EXTERNAL BARRIER STRUCTURES.
Eurasian Journal of Academic Research, 3(1 Part 1), 186-191.

9.

Nosirova, S., & Matyokubov, B. (2023). WAYS TO INCREASE THE ENERGY
EFFICIENCY OF EXTERNAL BARRIER CONSTRUCTIONS OF BUILDINGS.
Евразийский журнал академических исследований, 3(3), 145-149.

10.

Husanov, B., Shodiyev, K., & Mehroj, V. (2024). FUNKSIYA EKSTRUMLARINI
IQTISODIY VA QURULISH MASALALARINI YECHISHGA TADBIQI.

Gospodarka

i Innowacje.

,

44

, 11-16

11.

Husanov, B., Shodiyev, K., & Mehroj, V. (2024). TEKISLIKDA TO’G’RI CHIZIQ
TENGLAMALARINI IQTISODIY MASALARNI YECHISHGA TADBIQI.

TA'LIM VA

RIVOJLANISH TAHLILI ONLAYN ILMIY JURNALI

,

4

(1), 11-14

12.

Shodiyev, K., & Mehroj, V. (2024). Chiziqli tenglamalar sistemalarini yechish usullari.

Gospodarka i Innowacje.

,

43

, 49-56.

13.

Khusainov ShamshidinYalgashevic, Shodiyev Kamoliddin Shamsiddin o’g’li, &
KimDinaraVladislavovna. (2021). HEALTH OF CHILDREN OF PRESCHOOL AGE
ANDOPPORTUNITIES OF RECOVERY UNDER THE INFLUENCE OF PHYSICAL
STRESS OFCHILDREN’S PRESCHOOL INSTITUTIONS OF SAMARKAND CITY.

World

2025-YIL

28-29-MART

“YANGI O‘ZBEKISTONDA MUHANDIS KADRLAR TAYORLASHNING ISTIQBOLLARI VA

YOSHLARNING IJTIMOIY - SIYOSIY FAOLLIGINI OSHIRISHNING DOLZARB MASALALARI”

Respublika ilmiy-texnik konferensiyasi

309

14.

Mardonov, B., & Zikiryayev, S. (2024). BA’ZI GEOMETRIYA MASALALARINI
YECHISH USULLARI.

Theoretical aspects in the formation of pedagogical

sciences

,

3

(7), 183-186.

15.

Axmadovich, M. B. (2020). Sfera sirtida joylashgan uchburchaklarni yechishning ba'zi
usullari.

Science and Education

,

1

(2), 23-27.

16.

Usarov, S., Zikiryaev, S., Mardonov, B., & Namazov, G. (2024, May). Numerical analysis
of the process of heat transfer in inhomogeneous media. In

AIP Conference

Proceedings

(Vol. 3147, No. 1). AIP Publishing.

17.

Aхмадович М. Б. . (2024). Интерактивные Веб-Технологии Для Развития
Логического Мышления Инженеров Будущего В Условиях Цифровой
Трансформации Образования.

Miasto Przyszłości

,

52

, 755–761. Retrieved from

https://miastoprzyszlosci.com.pl/index.php/mp/article/view/4713

18.

Mardonov Baxodir Axmadovich. (2024). KELAJAKDAGI MUHANDISLARNI
RAQAMLI TA’LIM ASOSIDA O‘QITISH, SAMARALI VEB-KONTENT YARATISH
METODOLOGIYASI.

IJTIMOIY

FANLARDA

INNOVATSIYA

ONLAYN

ILMIY

JURNALI

,

4

(9),

42–45.

Retrieved

from

https://sciencebox.uz/index.php/jis/article/view/11916

19.

Khusanov, B., Shodiev, K., & Vahobov, M. (2024, November). On exceptional directions
of a homogeneous polynomial system of the second degree. In

American Institute of

Physics Conference Series

(Vol. 3244, No. 1, p. 020039)

20.

Shodiev , K., & Jumanazarov, R. (2025). MATHEMATICS AND SCIENCEAT A HIGH
LEVEL FEATURES OF THE PROBLEM THE DEVELOPMENT OF THINKING
ABILITIES. Modern

Science

and

Research, 4(2),

316–322.

Retrieved

from

https://inlibrary.uz/index.php/science-research/article/view/65793

21.

Shodiev, K., & Jumanazarov, R. (2025). EXCEPTIONAL DIRECTIONS OF A
HOMOGENEOUS POLYNOMIAL. Modern Science and Research, 4(2), 164–171.
Retrieved from

https://inlibrary.uz/index.php/science-research/article/view/65685

22.

Kamolidin Shodiev; Predicting prospects for providing sustainable development of tourism
in the innovative economy.

AIP Conf. Proc.

27 November 2024; 3244 (1):

020001.

https://doi.org/10.1063/5.0241472

23.

Bozorboy Khusanov, Kamoliddin Shodiev, Mehroj Vahobov; On exceptional directions of
a homogeneous polynomial system of the second degree.

AIP Conf. Proc.

27 November

2024; 3244 (1): 020039.

https://doi.org/10.1063/5.0241696

24.

INNOVATSION

IQTISODIYOTDA

TURIZM

SOHASINI

BARQAROR

RIVOJLANISHINI

TA'MINLASH

ISTIQBOLLARINI

BASHORATLASH.

(2024). Aktuar

moliya

va

buxgalteriya

hisobi

ilmiy

jurnali , 4 (02),

123-

135.

https://finance.tsue.uz/index.php/afa/article/view/100

25.

Shodiev , K. . (2024). Econometric Models of Forecasting the Sustainable Development of
the Tourism Network in the Innovation Economy.

Miasto Przyszłości

,

46

, 549–558.

Retrieved from

http://miastoprzyszlosci.com.pl/index.php/mp/article/view/2900

26.

Shodiyev, K., & Abduraxmonovich, Q. A. (2023). The Model of Optimization of
Enterprise Production and Increase the Profitability of the Enterprise in a Market Economy.