Academic Research in Educational Sciences
VOLUME 2 | CSPI CONFERENCE 3 | 2021
Zamonaviy ta'limda matematika, fizika va raqamli texnologiyalarning dolzarb muammolari va yutuqlari
Toshkent viloyati Chirchiq davlat pedagogika instituti
CHIZIQLI DIFFERENSIAL TENGLAMALAR YECHISHNI
O’RGANISHDA MATEMATIK DASTURLARDAN FOYDALANISH
F. B. Xurramova
15-ixtisoslashgan maktab, Chirchiq, O’zbekiston
X. S. Ubaydullayeva
Toshkent viloyati Chirchiq davlat pedagogika instituti, Chirchiq, O’zbekiston
Xalimaubaydullayeva718@gmail.com
Bizga ma’lumki chiziqli differensial tenglamalar, differensial tenglamalar kursining
asosiy bo’limlaridan biri hisoblanadi. CHiziqli differensial tenglamalarni yechishda
har –hil kompuyuter dasturlari yordamidan foydalanishimiz mumkin.Yani matematik
dasturlardan biri Maple dasturi yordamida chiziqli differensial tenglamalarni
yechishni ko’rsatamiz.
Ushbu 𝒚′ + 𝒑(𝒙)𝒚 = 𝒒(𝒙)
(1)
tenglamaga chiziqli differensial tenglama deyiladi, bu yerda p(x) va q(x) x ∈(a,b)
oraliqda uzluksiz funksiyalar. (1) tenglamaning ikkala tomoni x ∈ (𝑎, 𝑏) oralig’ida
integrallovchi ko’paytuvchi
𝝁(𝒙) = 𝐞𝐱𝐩(∫ 𝐩(𝐱)𝐝𝐱) ga ko’paytirsak
𝒅
𝒅𝒙
(𝒚𝒆∫ 𝐩(𝐱)𝐝𝐱 ∫ 𝐩(𝐱)𝐝𝐱 )=𝒒(𝒙)𝒆∫ 𝐩(𝐱)𝐝𝐱
ni hosil qilamiz. Hosil bo’lgan sodda differensial tenglamani integrallab,chiziqli
tenglamaning umumiy yechimini topish formulasini keltirib chiqaramiz:
𝑦 = 𝑒 − ∫ p(x)dx [𝑐 + ∫ 𝑞(𝑥)𝑒 ∫ p(x)dx 𝑑𝑥],
1.misol. (𝑥 + 𝑦 2 )𝑑𝑦 = 𝑦𝑑𝑥.
Bu tenglama 𝑥 = 𝑥(𝑦)ga nisbatan chiziqli tenglama bo’ladi.
𝑑𝑥 1
− 𝑥=𝑦
𝑑𝑦 𝑦
1
1
𝑦
𝑦
tenlamani 𝜇(𝑦) = exp (− ∫ 𝑑𝑦) =
ga ko’paytirsak
𝑑
𝑥
( ) = 1 oddiy tenglama
𝑑𝑥 𝑦
xosil qilamiz,
bu yerdan 𝑥 = 𝑐𝑦 + 𝑦 2
Tenglama yechimini Maple dasturi yordamida tekshiramiz.
>d1:=diff(y(x),x)=y(x)/(x+y(𝒙)^ 2);
d
y(x)
d1 ≔ y(x) =
dx
x + y(x)2
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Academic Research in Educational Sciences
VOLUME 2 | CSPI CONFERENCE 3 | 2021
Zamonaviy ta'limda matematika, fizika va raqamli texnologiyalarning dolzarb muammolari va yutuqlari
Toshkent viloyati Chirchiq davlat pedagogika instituti
dsolve(d1,y(x));
𝐶1 √𝐶12 + 4𝑥
𝐶1 √𝐶12 + 4𝑥
+
, 𝑦(𝑥) = −
−
2
2
2
2
tenglama 𝑥 = 𝑥(𝑦) ga nisbatan qarasak
>d1:=diff(x(y),y)=(x(y)+𝒚^ 2)/y;
𝑑
𝑥(𝑦) + 𝑦 2
𝑑1 ≔
𝑥(𝑦) =
𝑑𝑦
𝑦
>dsolve(d1,x(y));
𝑥(𝑦) = (𝑦 + 𝐶1)𝑦.
Xulosa: CHiziqli differensial tenglamalar yechishda matematik dasturlardan
yordamchi vosita sifatida foydalanish informatika ,matematika va barcha texnika
yo’nalishida taxsil olayotgan o’quvchilarga bu tenglamalarni yechishda yuqori
natijalar olib keladi.
𝑦(𝑥) = −
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Academic Research in Educational Sciences
VOLUME 2 | CSPI CONFERENCE 3 | 2021
Zamonaviy ta'limda matematika, fizika va raqamli texnologiyalarning dolzarb muammolari va yutuqlari
Toshkent viloyati Chirchiq davlat pedagogika instituti
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