Isbot. Krum almashtirishi yordamida (3.1) chegaraviy masalani quyidagi
(3.32)
Dirixle masalasiga keltiramiz. Bu yerda
(3.33)
funksiya (3.1) chegaraviy masalaning xos qiymatga mos keluvchi xos funksiyasi. funksiya esa quyidagi shartlarni qanoatlantiradi:
(3.34)
(3.35)
Agar (3.1) chegaraviy masalaning xos qiymatlari bo`lsa, u holda (3.32) masalaning xos qiymatlari bo`ladi. (3.33) va (3.34) tengliklardan ushbu
(3.36)
formula kelib chiqadi. (3.32) chegaraviy masala uchun regulyarlashtirilgan izlar formulasini yozamiz:
(3.37)
Bu yerda
(3.38)
(3.33) ifodani (3.38) tenglikka qo`yib, (3.35) formulalarni inobatga olsak,
kelib chiqadi. Endi (3.36) tenglikni (3.37) formulaga qo`yib, ushbu
izlar formulasi kelib chiqadi.
4-§. Masalalar yechish
Misol.№1
Ushbu Dirixle chegaraviy masalasi berilgan bo`lsin
(4.1)
(4.2)
holida (4.1)+(4.2) chegaraviy masalaning xos qiymatlarini va ortonormallangan xos funksiyalarni topib olamiz:
![](data:image/png;base64,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)
so`ngra Laks teoremasidagi ni hisoblaymiz
(4.1) tenglikni oraliqda qarab o`rniga ni qo`yamiz:
bundan foydalanib (4.6) tenglikni quyidagi ko`rinishda yozish mumkin:
() tenglikdan quyidagi
tenglik kelib chiqadi.
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