birinchi tartibli chiziqli differensial tenglama

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5-§. birinchi tartibli chiziqli differensial tenglama. ushbu (1) ko’rinishdagi tenglamaga birinchi tartibli chiziqli differensial tenglama deyiladi. bu yerda va funksiyalar biror oraliqda aniqlangan va uzluksiz deb qaraladi. agar bo’lsa, (1) tenglamaga chiziqli bir jinsli bo’lmagan differensial tenglama deyiladi. agar bo’lsa, (1) tenglamaga chiziqli bir jinsli differensial tenglama deyiladi va ushbu (2) ko’rinishni oladi. bu esa o’zgaruvchilari ajraladigan differensial tenglamadir. ko’rinib turibdiki funksiya (2) differensial tenglamaning yechimidan iborat. agar bo’lsa, (2) differensial tenglamani quyidagicha yozish mumkin: . bu tenglikning ikkala tomonini integrallab quyidagi (3) tenglikni olamiz. ushbu belgilashdan foydanib (3) tenglikdan (4) formulani hosil qilamiz. bu yerda -ixtiyoriy o’zgarmas son. (4) formula (2) ko’rinishdagi bir jinsli differensial tenglamaning umumiy yechi- mini ifodalaydi. bir jinsli bo’lmagan (1) ko’rinishdagi differensial tenglamaning umumiy yechimini topishning bir qancha usullari bor. avvalo biz lagranj, ya’ni o’zgarmasni variyatsiyalash usuli bilan tanishamiz. shu maqsadda (1) differensial tenglamaning yechimini ushbu (5) ko’rinishda izlaymiz. bu yerda hozircha noma’lum funksiya. (5) …

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shuning uchun (8) formula (13) ko’rinishni oladi. bu esa bir jinsli bo’lmagan (1) differensial tenglamaning umumiy yechimi bir jinsli (2) differensial tenglamaning umumiy yechimi bilan bir jinsli bo’lmagan (1) differensial tenglamaning xususiy yechimining yig’indisidan iborat ekanligini ko’rsatadi. endi, (1) ko’rinishdagi chiziqli differensial tenglamaning umumiy yechimini topishning bernulli usuli bilan tanishamiz. shu maqsadda (1) differensial tenglamaning yechimini (14) ko’rinishda izlaymiz. natijada biz ushbu , ya’ni (15) ko’rinishidagi differensial tenglamaga ega bo’lamiz. bunda funksiyani shunday tanlaymizki, natijada shart bajarilsin. bu esa o’zgaruvchilari ajraladigan differensial tenglamadir. bu tenglamani yechib (16) funksiyani topamiz. shuning uchun (15) differensial tenglama ushbu ko’rinishni oladi. bu differensial tenglamani integrallab (17) munosabatni hosil qilamiz. topilgan va funksiyaning (16) va (17) ifodalarni (14) tenglikka qo’yib (1) differensial tenglamaning umumiy yechimini olamiz. endi, bir jinsli bo’lmagan (1) differensial tenglamaning xususiy yechimini topishning koshi usuli bilan tanishamiz. shu maqsadda, biror nuqtani olib quyidagi bir jinsli differensial tenglamaga qo’yilgan (18) koshi masalasining yechimini …

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6-§. noma’lum koeffitsiyentlar usuli. ushbu (1) ko’rinishdagi differensial tenglamaning xususiy yechimini topishning aniqmas koeffitsiyentlar usuli bilan tanishamiz. bu yerda darajali ko’phad. teorema-1. 1) agar bo’lsa, u holda (1) differensial tenglamaning xususiy yechimi (2) ko’rinishda bo’ladi. 2) agar bo’lsa, u holda (1) differensial tenglamaning xususiy yechimi (3) ko’rinishda bo’ladi. bunda darajali ko’phad. isbot. berilgan (1) differensial tenglamaning yechimini (4) ko’rinishda izlaymiz. bu tenglikning ikki tomonini differensiallab (5) munosabatni topamiz. (4) va (5) tengliklarga asosan (1) differensial tenglamani quyidagicha yozish mumkin: oxirgi tenglikning ikki tomonini ga bo’lib (6) differensial tenglamani hosil qilamiz. bu tenglamada bo’lsa, (6) differensial tenglama (7) ko’rinishni oladi. tenglamaning o’ng tomonidagi ko’phad ko’rinishda bo’lgani uchun (7) differensial tenglamaning yechimi ko’rinishni oladi. bundan va yuqoridagi (4) almashtirishga asosan (1) differensial tenglamaning yechimi holda ko’rinishda bo’lishi kelib chiqadi. agar (6) differensial tenglamada bo’lsa, u holda uning yechimini (8) ko’rinishda izlaymiz. bu yerda hozircha noma’lum sonlar. (8) tenglikni differensiallab (9) munosabatni hosil …

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5-§. birinchi tartibli chiziqli differensial tenglama. ushbu (1) ko’rinishdagi tenglamaga birinchi tartibli chiziqli differensial tenglama deyiladi. bu yerda va funksiyalar biror oraliqda aniqlangan va uzluksiz deb qaraladi. agar bo’lsa, (1) tenglamaga chiziqli bir jinsli bo’lmagan differensial tenglama deyiladi. agar bo’lsa, (1) tenglamaga chiziqli bir jinsli differensial tenglama deyiladi va ushbu (2) ko’rinishni oladi. bu esa o’zgaruvchilari ajraladigan differensial tenglamadir. ko’rinib turibdiki funksiya (2) differensial tenglamaning yechimidan iborat. agar bo’lsa, (2) differensial tenglamani quyidagicha yozish mumkin: . bu tenglikning ikkala tomonini integrallab quyidagi (3) tenglikni olamiz. ushbu belgilashdan foydanib (3) tenglikdan (4) formulani hosil qilamiz. bu yerda -ixtiyoriy ...

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