1-Mavzu: Mapleda Differentsiallash va integrallash buyruqlari Reja

Differensial tenglamalar sistemasi
dsolve buyrug’i differensial tenglamalar sistemasi (yoki Koshi masalasi) yechimini topishi mumkin, agar unda quyidagilar ko’rsatilsa: dsolve({sys},{x(t),y(t),…}), bu yerda sys - differensial tenglamalar sistemasi, x(t),y(t),… - noma’lum funksiyalar majmuasi.
Misol
1. Differensial tenglamalar sistemasi yechimini toping:

> sys:=diff(x(t),t)=-4*x(t)-2*y(t)+2/(exp(t)-1), diff(y(t),t)=6*x(t)+3*y(t)-3/(exp(t)-1):
> dsolve({sys},{x(t),y(t)});

Ikkita _S1 i _S2 doimiy o’zgarmaslarga bog’liq bo’lgan x(t) va y(t) funksiyalar topildi.
Darajali qatorlar yordamida differensial tenglamalarni yechish.
Ko’p turdagi differensial tenglamalarning aniq analitik yechimini topish qiyin. Bunday holda differensial tenglamani taqribiy metodlar orqali yechish mumkin, xususan, noma’lum funksiyani darajali qatorga yoyish orqali.
Differensial tenglamaning yechimini darajali qator ko’rinishida yechish uchun dsolve buyrug’ida o’zgaruvchidan keyin type=series (yoki oddiy series) parametrni ko’rsatish kerak. Qator yoyish darajasi n, ya’ni yoyish amalga oshiriladigan daraja ko’rsatkichini ko’rsatish uchun, dsolve buyrug’ini oldiga daraja tartibini aniqlash buyrug’i Order:=n yoziladi.
Xususiy yechimlarni ajratish uchun boshlang’ich shartlarni y(0)=u1, D(y)(0)=u2, (D@@2)(y)(0)=u3 va hokozalarni berish kerak bo’ladi.
Darajali qatorga yoyish turi series bo’ladi, shuning uchun keyinchalik bu qator bilan ishlash uchun uni convert(%,polynom) buyrug’i bilan polinom ajratish, so’ngra esa rhs(%) buyrug’i bilan olingan natijani o’ng tomonini ajratish kerak bo’ladi.


Misollar
1. y''(x)- y³(x)=ye ^{- x}cosx differensial tenglamaning umumiy yechimini 4-tartibli darajali qatorga yoyish ko’rinishida toping. Yoyishni y(0)=1, y'(0)=0 boshlang’ich shartlarda amalga oshiring..
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