Oddiy differensial tenglamalarning analitik yechimini maple dasturi yordamida topish

1-rasm. Koshi masalasi sonli yechimining grafigi.

Koshi masalasi yoki chegaraviy masalaning yechilishi. Dsolve komanda Koshi masalasi yoki chegaraviy masalaning yechimini topishi mumkin, agarda berilgan differensial tenglama uchun noaniq funksiyaning boshlang’ich hamda chegaraviy shartlari berilsa. Boshlang’ich yoki chegaraviy shartlarda hosilalarni belgilash uchun differensial operator ishlatiladi masalan, y''(0)=2 shartni kabi berishga to’g’ri keladi yoki y'(1)=0 shartni: . Eslatib o’tamiz, n-chi tartibli hosila kabi yoziladi.

1). Muammoni oydinlashtirishni mashqlarda bajarib ko’raylik va quyidagi tadbiqlarni bajaraylik, ya’ni Koshi masalasining yechimini topaylik :
y⁽⁴⁾+y''=2cosx, y(0)=2, y'(0)=1, y''(0)=0, y'''(0)=0.
Yechish:
> de:=diff(y(x),x$4)+diff(y(x),x$2)=2*cos(x);


> cond:=y(0)=-2, D(y)(0)=1, (D@@2)(y)(0)=0,
(D@@3)(y)(0)=0;
cond:=y(0)=2, D(y)(0)=1, (D⁽²⁾)(y)(0)=0, (D⁽³⁾)(y)(0)=0
> dsolve({de,cond},y(x));
y(x) = 2cos(x)xsin(x)+x.
2). Boshqa turdagi oddiy differensial tenglamaning yechimini turli analitik usullar yordamida Maple dasturidan foydalanib yeching:
.
Yechish:
> ode_L:=sin(x)*diff(y(x),x)-cos(x)*y(x)=0;

> dsolve(ode_L,[linear],useInt);

> value(%);

> dsolve(ode_L,[separable],useInt);

> value(%);

Ko’pchilik differensial tenglamalar turlarining aniq analitik yechimi topilmaydi. Bu holda differensial tenglamalarning yechimini yaqinlashuvchi metodlar yordamida topish mumkin, ya’ni noaniq funksiyani darajali qatorga yoyish orqali topish.

Differensial tenglamaning yechimini darajali qator ko’rinishida topish uchun dsolve komandada o’zgaruvchilardan keyin type=series (yoki shunchaki series) parametrini ko’rsatish kerak. n-chi yoyilma tartibini ko’rsatish uchun, ya’ni daraja tartibini yoyilma tugaguncha, dsolve komandadan oldin tartibni aniqlaydigan Order:=n komandani qo’yish kerak.