1-Mavzu: Mapleda Differentsiallash va integrallash buyruqlari Reja

Misollar
1 y'+ycosx=sinx cosx differensial tenglamaning umumiy yechimini toping.
> restart;
> de:=diff(y(x),x)+y(x)*cos(x)=sin(x)*cos(x);

> dsolve(de,y(x));

2. y''= 2y'+y=sinx+e ^{– x} ikkinchi tartibli differensial tenglamaning umumiy yechimini toping.
> restart;
> deq:=diff(y(x),x$2)-2*diff(y(x),x)+y(x)=sin(x)+exp(-x);

> dsolve(deq,y(x));

3. q> k va q=k (rezonans) ikki holda y''+k²y=sin(qx) differensial tenglamaning umumiy yechimini toping.
> restart; de:=diff(y(x),x$2)+k^2*y(x)=sin(q*x);
de:=
> dsolve(de,y(x));

Endi rezonans holida yechimni topamiz. Buning uchun dsolve buyrug’ining oldida q=k ni yozish kerak.
> q:=k: dsolve(de,y(x));

Yechimning fundamental (bazis) sistemasi.
dsolve buyrug’i yechimning fundamental sistemasini topish imkoniyatini yaratadi. Buning uchun dsolve buyrug’i parametrida output=basis deb ko’rsatish kerak.
Misol
1. Diffrensial tenglama yechimining fundamental sistemasini toping: y⁽⁴⁾+2y''+y=0.
> de:=diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=0;

> dsolve(de, y(x), output=basis);

Koshi masalasi yoki chegaraviy masalani yechish.
dsolve buyrug’i Koshi masalasi yoki chegaraviy masalani yechadi, agar differensial tenglama bilan birga noma’lum funksiya uchun boshlang’ich yoki chegaraviy shartlar qo’yilgan bo’lsa. Boshlang’ich yoki chegaraviy shartlarda hosilani belgilash uchun differensial operator ishlatiladi, masalan, y''(0)=2 shartni quyidagicha yozish kerak bo’ladi : , yoki y'(1)=0 shart quyidagicha yoziladi: . Eslatib qtamizki, n- tartibli hosila ko’rinishda yoziladi.
Misollar
1. Koshi masalasi yechimini toping: y⁽⁴⁾+y''=2cosx, y(0)=- 2, y'(0)=1, y''(0)=0, y'''(0)=0.
> 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. Chegaraviy masalani yeching: , , . Yechim grafigini yasang.
> restart; de:=diff(y(x),x$2)+y(x)=2*x-Pi;

> cond:=y(0)=0,y(Pi/2)=0;

> dsolve({de,cond},y(x));

Izoh: Yechimni grafigini yasash uchun olingan ifodaning o’ng tomonini ajratish kerak bo’ladi.
> y1:=rhs(%):plot(y1,x=-10..20,thickness=2);