8-mavzu. Maple tizimi. Matematik ifodalar va funksiyalar. Algebra va sonlar nazariyasi masalalarini yechish

Fundamental (bazis) yechimlar sistemasi

dsolve komandasi ODT ning bazis yechimlar sistemasini ham topishda ishlatiladi. Uning uchun parametrlar bo’limida output=basis deb ko’rsatish kerak . Masalan, ODT ning bazis yechimlar sistemasini topaylik.

> de:=diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=0; \\
> dsolve(de, y(x), output=basis); \\[cos(x), sin(x), xcos(x), xsin(x)]


Koshi yoki chegara masalani yechish

dsolve komandasi yordamida Koshi yoki chegara masalani ham yechish mumkin. Buning uchun blshlang’ich yoki chegara shartlarni qo’shimcha ravishda berish kerak. Qo’shimcha shartlarda hosila differensial operator D bilan beriladi. Masalan, shart ko’rinishda, shart ko’rinishda, shart ko’rinishda yozilishi kerak.

Misollar 1. Koshi masalasi yechilsin.
> 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; \\
> dsolve({de,cond},y(x)); \\
2. chegara masala yechilsin.
> 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)); \\

Yechim grafigini chizish uchun tenglama щng tomonini ajratib olish kerak:
> y1:=rhs(%):plot(y1,x=-10..20,thickness=2);


ODT sistemasi

dsolve komandasi yordamida LN sistemasini ham yechish mumkin. Buning uchun uni dsolve({sys},{x(t),y(t),…}), ko’rinishda yozib olish kerak, sys-ODT lar sistemasi, x(t), y(t) ,...-no’malum funksiyalar sistemasi.

Misollar 1.

> 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)}); \\



ODT ni qator yordamida taqribiy yechish
dsolve komandasi yordamida ODT yechimini taqribiy usulda qator yordamida topish mumkin. Buning uchun dsolve komandasida output=series va Order:=n parametrlarni kiritish kerak . Bishlang’ich qiymatlar y(0)=u1, D(y)(0)=u2, (D@@2)(y)(0)=u3 i hokazo ko’rinishda beriladi. Yechimni ko’phadga aylantirish uchun convert(%,polynom) komandasini berish kerak. Yechimning grafik ko’rinishda chiqarish uchun tenglama o’ng toioning rhs(%) komandasi bilan ajratib olish kerak.
Misollar 1. Koshi masalasining taqribiy yechimi 5-darajali ko’phad ko’rinishda olinsin.
> restart; Order:=5:
> dsolve({diff(y(x),x)=y(x)+x*exp(y(x)), y(0)=0}, y(x), type=series);
\\
2. Koshi masalasining taqribiy yechimi 4-tartibli qator uo’rinishda topilsin.
> restart; Order:=4: de:=diff(y(x),x$2)-y(x)^3=exp(-x)*cos(x):
> f:=dsolve(de,y(x),series);
\\
3. Koshi masalasining taqribiy yechimi 6 tartibli ko’phad ko’rinishda topilsin.
> restart; Order:=6:
> de:=diff(y(x),x$3)-diff(y(x),x)= 3*(2-x^2)*sin(x);
\\
> cond:=y(0)=1, D(y)(0)=1, (D@@2)(y)(0)=1;
\\cond:=y(0)=1, D(y)(0)=1, D(2)(y)(0)=1
> dsolve({de,cond},y(x)); \\
> y1:=rhs(%):
> dsolve({de,cond},y(x), series);\\
Aniq va taqribiy yechim grafigini chiqarish uchun quyidagi komandalarni berish kerak:
> convert(%,polynom): y2:=rhs(%):
> p1:=plot(y1,x=-3..3,thickness=2,color=black):
> p2:=plot(y2,x=-3..3, linestyle=3,thickness=2, color=blue):
> with(plots): display(p1,p2);

ODT ni sonli usulda yechish

dsolve komandasi ODT ni taqribiy yechish uchun ham ishlatiladi, faqatgina parametrlar safida type=numeric deb ko’rsatish kerak, undan tashqari options bo’limida sonli usullar turini ham ko’rsatish kerak: dsolve(eq, vars, type=numeric, options). Quyidagi sonli usullar ishlatilishi mumkin:

method=rkf45- 4-5-tartibli Runge-Kutta usuli,
method=dverk78-,7-8-tartibli Runge-Kutta usuli,
mtthod=classical-,3-4-tartibli klassik Runge-Kutta usuli,
method=gear- Girning bir qadamli usuli,
method=mgear- Girning ko’p qadamli usuli.
ODT ning yechimini grafik usulda yechish uchun odeplot(dd, [x,y(x)], x=x1..x2), komandasi ishlatiladi, bu yerda dd:=dsolve({eq,cond}, y(x), numeric).