ma’ruza 14. differensial tenglamalarni yechish

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ma’ruza 14. differensial tenglamalarni yechish. reja: 1. dsolve(t,f,options) buyrug’i. 2. yechimning fundamental (bazis) sistemasi. 3. differensial tenglamalarni sonli yechish. tayanch iboralar: differensial tenglama, koshi masalasi, chegaraviy masala. maple muhitida differensial tenglamalarni analitik yechish uchun dsolve(t,f,options) buyrug’i ishlatiladi, bu yerda t – differensial tenglama, f – noma’lum funksiya, options – parametrlar. parametrlar yechish metodlarini ko’rsatadi, masalan, jimlikda analitik yechim quyidagicha izlanadi: type=exact. differensial tenglamalarni tuzishda hosilalarni belgilash uchun diff buyrug’i ishlatiladi, masalan, y''+y=x differensial tenglama quyidagicha yoziladi: diff(y(x),x$2)+y(x)=x. differensial tenglamalarning umumiy yechimi ixtiyoriy o’zgarmasdan, ya’ni differensial tenglama tartibini bildiruvchidan sondan bog’liq bo’ladi. maple da bunday o’zgarmaslar, odatda , _s1, _s2, va hokazo ko’rinishlarda belgilanadi dsolve buyrug’i differensial tenglamalar yechimini hisoblanmaydigan formatda chiqarishni amalga oshiradi. yechim bilan keyinchalik ishlash kerak bo’lsa, (masalan , yechimni grafigini qurish kerak bo’lsa) olingan yechimning chap tomonini rhs(%) buyrug’i bilan ajratish kerak bo’ladi. 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); > …

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ratadi. 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); differensial tenglamalar sistemasi dsolve buyrug’i …

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rdagi 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 - xcosx differensial tenglamaning umumiy yechimini 4-tartibli darajali qatorga yoyish ko’rinishida toping. yoyishni y(0)=1, y'(0)=0 boshlang’ich shartlarda amalga oshiring.. > restart; order:=4: de:=diff(y(x),x$2)-y(x)^3=exp(-x)*cos(x): > f:=dsolve(de,y(x),series); izoh: olingan yoyilmada d(y)(0) noldagi hosilani bildiradi: y'(0). xususiy yechimni topish …

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$ordamida albatta ko’phad ko’rinishiga keltirish 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); rasmda ko’rinib turibdiki, darajali qatorning aniq yechimiga yaqinlashishi taxminan - 1 restart; ordev=6: > eq:=diff(y(x),x$2)-x*sin(y(x))=sin(2*x): > cond:=y(0)=0, d(y)(0)=1: > de:=dsolve({eq,cond},y(x),numeric); de:=proc(rkf45_x)...end izoh: agar x o’zgaruvchining biror fiksirlangan qiymatida yechimni topish kerak bo’lsa , shu qiymat oldindan berilishi kerak, masalan, x=0.5 da quyidagi teriladi: > de(0.5); > with(plots): > odeplot(de,[x,y(x)],-10..10,thickness=2); endi koshi masalasining darajali qator ko’rinishida taqribiy yechimini topamiz va grafigini yasaymiz. > dsolve({eq, cond}, y(x), series) > convert(%, polynom):p:=rhs(%): > p1:=odeplot(de,[x,y(x)],-2..3, thickness=2,color=black): > p2:=plot(p,x=-2..3,thickness=2,linestyle=3,color=blue): > display(p1,p2); yechimning darajali qatorga yaqinlashuvi taxminan -1 restart; cond:=x(0)=1,y(0)=2: sys:=diff(x(t),t)=2*y(t)*sin(t)-x(t)-t,diff(y(t),t)=x(t): f:=dsolve({sys,cond},[x(t),y(t)],numeric): > with(plots): p1:=odeplot(f,[t,x(t)],-3..7, color=black, thicness=2,linestyle=3): p2:=odeplot(f,[t,y(t)],-3..7,color=green,thickness=2): > p3:=textplot([3.5,8,"x(t)"], font=[times, italic, 12]): > p4:=textplot([5,13,"y(t)"], font=[times, italic, 12]): > display(p1,p2,p3,p4); nazorat savollari 1. mapleda differensial tenglamalar qanday yechiladi? 2. yuqori tartibli differensial tenglamalar qanday yechiladi? 3. yechimning fundamental sistemasi qanday topiladi? 4. koshi masalasi qanday yechiladi? 5. differensial tenglamaning …$

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ma’ruza 14. differensial tenglamalarni yechish. reja: 1. dsolve(t,f,options) buyrug’i. 2. yechimning fundamental (bazis) sistemasi. 3. differensial tenglamalarni sonli yechish. tayanch iboralar: differensial tenglama, koshi masalasi, chegaraviy masala. maple muhitida differensial tenglamalarni analitik yechish uchun dsolve(t,f,options) buyrug’i ishlatiladi, bu yerda t – differensial tenglama, f – noma’lum funksiya, options – parametrlar. parametrlar yechish metodlarini ko’rsatadi, masalan, jimlikda analitik yechim quyidagicha izlanadi: type=exact. differensial tenglamalarni tuzishda hosilalarni belgilash uchun diff buyrug’i ishlatiladi, masalan, y''+y=x differensial tenglama quyidagicha yoziladi: diff(y(x),x$2)+y(x)=x. differensial tenglamalarning umumiy yechimi ixtiyoriy o’zgarmasdan, ya’...

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