BERNULLI DIFFERENSIAL TENGLAMALARNI O`QITISH METODIKASI

T A D Q I Q O T L A R

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BERNULLI DIFFERENSIAL TENGLAMALARNI

O`QITISH METODIKASI

Xolturayeva Kamola Bahrom qizi

Termiz davlat universiteti “Axborot

texnologiyalari” fakulteti “Amaliy

matematika” ta’lim yo‘nalishi

bakalavr II bosqich talabasi

Email: kxolturayeva@gmail.com

Tel: (+998)919051602

Annotatsiya:

Hozirgi kunda matematikaning differensial tenglamalar bo’limi

juda rivojlanmoqda. Ta’lim sohasida esa alohida e’tibor qaratilmoqda shu bilan
birgalikda differensial tenglamalar orqali ko’pgina masalalar o`z yechimini topmoqda.
Differensial tenglamalarga oid masalalarni yechishda turli sohalarda keng
qo’llanilmoqda. Masalan: ta’lim, tibbiyot , qurulish va boshqalar. Differensial
tenglamaning Bernulli hamda chiziqli ko’rinishlari bilan differensial tenglamalar
faniga kirib boramiz va ularni yechishni o’rganamiz.

Kalit so`zlar:

Differensial tenglamalar, chiziqli differensial tenglamalar,

𝜇

-

kiritish usuli, Bernulli, oddiy differensial tenglamalar, bir jinsli differensial
tenglamalar, kvadraturalar.

Annotation:

Currently, the field of differential equations in mathematics is

rapidly developing. Special attention is being given to this area in education, and
simultaneously, many problems are being solved through differential equations.
Problems related to differential equations are widely applied in various fields such as
education, medicine, construction, and others. We will explore the Bernoulli and linear
forms of differential equations and learn how to solve them.

Keywords:

Differential equations, linear differential equations,

𝜇

-integrating

factor method, Bernoulli, ordinary differential equations, homogeneous differential
equations, quadratures.

Аннотация:

В настоящее время раздел математики — дифференциальные

уравнения — активно развивается. В сфере образования этому уделяется особое
внимание, и одновременно с этим многие задачи решаются с помощью
дифференциальных уравнений. Задачи, связанные с дифференциальными
уравнениями, широко применяются в различных областях, таких как
образование, медицина, строительство и другие. Мы рассмотрим уравнения
Бернулли и линейные дифференциальные уравнения и научимся их решать.

Ключевые

слова:

Дифференциальные

уравнения,

линейные

дифференциальные уравнения,

𝜇 −

метод интегрирующего множителя,

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Бернулли,

обыкновенные

дифференциальные

уравнения,

однородные

дифференциальные уравнения, квадратуры.

Kirish

Bernulli tenglamasi

1-Ta’rif.

Ushbu

𝑑𝑦
𝑑𝑥

= 𝑝(𝑥)𝑦 + 𝑞(𝑥)𝑦

𝛼

(1)

tenglama Bernulli tenglamasi deyiladi. Bu tenglamada berilgan

𝑝(𝑥)

va

𝑞(𝑥)

lar biror

𝐼

intervalda aniqlanga funksiyalar.

𝛼 −

biror haqiqiy son

(𝛼 ∈ 𝑅)

. Ravshanki, agar

𝛼 = 0

bo’lsa,

𝑑𝑦
𝑑𝑥

= 𝑝(𝑥)𝑦 + 𝑞(𝑥)

chiziqli differensial tenglamaga ega bo’lamiz, agar

𝛼 = 1

bo’lsa, o`zgaruvchilari

ajraladigan differensial tenglama hosil bo`ladi.

𝑑𝑦
𝑑𝑥

= (𝑝(𝑥) + 𝑞(𝑥))𝑦

Demak Bernulli tenglamasi

𝛼 = 0, 𝛼 = 1

bo`lganida bizga ma’lum differensial

tenglamalarga aylanadi Endi

𝑎 ≠ 0, 𝛼 ≠ 1

deb faraz qilamiz.

Asosiy qism
1-Teorema.

Agar

𝑝(𝑥), 𝑞(𝑥)

funksiyalar

𝐼

intervalda aniqlangan va uzluksiz

bo’lib,

𝛼 > 1

bo`lsa,

𝐺 = { (𝑥, 𝑦): 𝑥 ∈ 𝐼, −∞ < 𝑦 < +∞}

sohaning

∀(𝑥

0

, 𝑦

0

)

nuqtasidan

(1)

tenglamaning

𝐼

intervalda aniqlangan bitta integral chizig`i o’tadi.

U holda

(1)

tenglamani yechish uchun tenglikning har ikki tomonini

𝑦 ≠ 0 , 𝑦

𝛼

ifodaga bo’lamiz. Ya’ni

𝑦′

𝑦

𝛼

=

𝑝(𝑥)

𝑦

𝛼−1

+ 𝑞(𝑥) (2)

(2)

tenglamani hosil qilamiz , bu yerda

𝑦′ =

𝑑𝑦

𝑑𝑥

.

(2)

tenglamadan

1

𝑦

𝛼−1

= 𝑧(𝑥)

almashtirish bajaramiz:

𝑧′ = (1 − 𝛼)

𝑦′

𝑦

𝛼

1

1 − 𝛼

𝑧′ = 𝑝(𝑥)𝑧 + 𝑞(𝑥) (3)

Bu

(3)

tenglama.

𝑧

ga nisbatan chiziqli differensial tenglama. Uning umumiy

yechimi

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𝑧 = (𝐶 + ∫ 𝑒

− ∫(1−𝛼)𝑝(𝑥)𝑑𝑥

(1 − 𝛼)𝑞(𝑥)𝑑𝑥) 𝑒

∫(1−𝛼)𝑝(𝑥)𝑑𝑥

ko’rinishda yoziladi.

Misol.

Quydagi Bernulli tenglamasini yeching.

𝑥𝑦

′

+ 2𝑦 + 𝑥

5

𝑦

3

𝑒

𝑥

= 0.

Bunda

𝐶 = 1

.

Yechish:

𝑥𝑦

′

+ 2𝑦 + 𝑥

5

𝑦

3

𝑒

𝑥

= 0.

𝑦

′

+

2𝑦

𝑥

+ 𝑥

4

𝑦

3

𝑒

𝑥

= 0.

𝑦

′

𝑦

3

+

2

𝑥𝑦

2

= −𝑥

4

𝑒

𝑥

.

|

1

𝑦

2

= 𝑧 𝑧

′

= −

2𝑦′

𝑦

3

|

.

−

𝑡

′

2

+

2𝑡

𝑥

= −𝑥

4

𝑒

𝑥

.

𝑡

′

−

4𝑡

𝑥

= 2𝑥

4

𝑒

𝑥

.

𝜇(𝑥) = 𝑒

∫

−4

𝑥

𝑑𝑥

=

1

𝑥

4

.

∫ 𝑑 (

𝑡

𝑥

4

) = 2 ∫ 𝑒

𝑥

𝑑𝑥.

𝑡

𝑥

4

= 2(𝑒

𝑥

+ 𝐶).

1

𝑦

2

= 2𝑥

4

(𝑒

𝑥

+ 𝐶).

2𝑥

4

𝑦

2

(𝑒

𝑥

+ 𝐶) = 1.

𝐶 = 1 𝑏𝑢𝑛𝑑𝑎 𝑦

2

=

1

2𝑥

4

(𝑒

𝑥

+ 1)

.

Demak berilgan differensial tenglamaning umumiy yechimini topdik.

Endi berilgan misolni Maple dasturida ishlaymiz.:
>

Bernoulli_ode :=diff(y(x),x)+(2*y(x))/x+x^4*y(x)^3*exp(x);

>

with(DEtools,odeadvisor);

>

odeadvisor(Bernoulli_ode);

>

with(PDEtools,dchange);

>

ITR := {y(x)=u(t)^(1/(1-3)),x=t};

:=

Bernoulli_ode















d

d

x

( )

y

x

2 ( )

y

x

x

x

4

( )

y

x

3

e

x

[

]

odeadvisor

[

]

_Bernoulli

[

]

dchange

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>

new_ode := dchange(ITR,Bernoulli_ode,[u(t),t]):

new_ode2 := solve(new_ode,{diff(u(t),t)}):
op(factor(combine(expand(new_ode2),power)));

>

ans := dsolve(Bernoulli_ode);

Natijada

𝐶 = 1

deb olsak, funksiya grafigi quyigicha bo’ladi:

>

with(plots):

>

Y1:=plot(((-1)*sqrt(1/(x^2*(1+2*exp(x))))), x=-5..5, y=-4..4,

style=line,color=red):
Y2:=plot((sqrt(1/(x^2*(1+2*exp(x))))), x=-5..5, y=-4..4, style=line,color=blue):
display({Y1, Y2}, axes=boxed, scaling=constrained, title=`Funksiya grafigi`);

trained, title=`Funksiya grafigi`);

Xulosa:

Bernulli tenglamasini ishlash uchun ma’lum bir algoritm yo`q. Rikk

tenglamasiga yechim qiladiganda ketma- ket ishlash natijasida Bernulli , chiziqli,
oddiy differensial tenglamalarga duch kelamiz. Maple 9.5 dasturidan foydalanib,

:=

ITR

{

}

,



( )

y

x

1

( )

u

t



x

t



d

d

t

( )

u

t

2 (

)



2 ( )

u

t

t

5

e

t

t

:=

ans

,



( )

y

x

1



2

e

x

_C1 x

2



( )

y

x



1



2

e

x

_C1 x

2

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berilgan differensial tenglamaga yechim qildik va yechimlarni grafiklari bilan ham
tanishdik.

Foydalanilgan adabiyotlar ro`yxati:

1.

Филиппов А. Ф. Введение в теорию дифференциальных уравнений.
Москва, «КомКнига», 2007.

2.

Салоҳиддинов М. С., Насритдинов Ғ. Оддий дифференциал тенгламалар.

3.

Ergashev T. Differensial tenglamalar. Toshkent, 2023.

4.

Dilmurodov N. Oddiy differensial tenglamalar. Toshkent, “Sano-standart”,
2019.

5.

Abdullayev O. B. Differensial tenglamalar. M. Ulug’bek nomidagi
O‘zbekiston Milliy universiteti, 2009.