differensial tenglama yechish

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amaliy mashg’ulot 9. differensial tenglamalarni yechish funksiyalari. differensial tenglamalarning umumiy yechimi. 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); > dsolve(de,y(x)); 2. y''= 2y'+y=sinx+e – x ikkinchi tartibli differensial tenglamaning umumiy yechimini toping. > restart; > …

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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 differensial tenglamalar sistemasi (yoki koshi masalasi) yechimini topishi mumkin, agar unda quyidagilar ko’rsatilsa: dsolve({sys},{x(t),y(t),…}), bu yerda …

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hish 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 uchun boshlang’ich shartlarni berish qoldi: > y(0):=1: d(y)(0):=0:f; 2. koshi masalasining taqribiy yechimini 5-tartibli aniqlikgacha …

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$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$2)-x*sin(y(x))=sin(2*x): > cond:=y(0)=0, d(y)(0)=1: > de:=dsolve({eq,cond},y(x),numeric); de:=proc(rkf45_x)...end izoh: agar x o’zgaruvchining biror fiksirlangan qiymatida yechimni topish kerak bo’lsa , shu qiymat oldindan berilishi kerak, masalan, x=0.5 da quyidagi teriladi: > de(0.5); > with(plots): > odeplot(de,[x,y(x)],-10..10,thickness=2); endi koshi masalasining darajali qator ko’rinishida taqribiy yechimini topamiz va grafigini yasaymiz. > dsolve({eq, cond}, y(x), series) > convert(%, polynom):p:=rhs(%): > p1:=odeplot(de,[x,y(x)],-2..3, thickness=2,color=black): > p2:=plot(p,x=-2..3,thickness=2,linestyle=3,color=blue): > display(p1,p2); yechimning darajali qatorga yaqinlashuvi taxminan -1 restart; cond:=x(0)=1,y(0)=2: sys:=diff(x(t),t)=2*y(t)*sin(t)-x(t)-t,diff(y(t),t)=x(t): f:=dsolve({sys,cond},[x(t),y(t)],numeric): > with(plots): p1:=odeplot(f,[t,x(t)],-3..7, color=black, thicness=2,linestyle=3): p2:=odeplot(f,[t,y(t)],-3..7,color=green,thickness=2): > p3:=textplot([3.5,8,"x(t)"], font=[times, italic, 12]): > p4:=textplot([5,13,"y(t)"], font=[times, italic, 12]): > display(p1,p2,p3,p4); mustaqil topshiriqlar 1-topshiriq differensial tenglamalarni yeching 1. 6. 2. 7. 3. 8. 4. 9. 5. 10. 2-topshiriq differensial tenglamalar sistemasini yeching 1. 6. 2. 7. 3. 8. 4. 9. 5. 10. 3-topshiriq 1. differensial tenglamani 7-tartibli darajali qator ko’rinishida yechimini toping …$

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nday topiladi? 4. koshi masalasi qanday yechiladi? 5. differensial tenglamaning darajali qator ko’rinishida yechimini topish va grafigini chizish qanday bajariladi? 5. mapleda differensial tenglamalar tenglamalar sistemasining sonli yechimi qanday topiladi? 6. mapleda differensial tenglamalarni yechish uchun qaysi metodlardan foydalanish imkoni bor? image5.png oleobject13.bin image54.wmf oleobject14.bin image55.wmf oleobject15.bin image56.wmf oleobject16.bin image57.wmf oleobject17.bin image58.wmf oleobject18.bin image59.wmf oleobject19.bin image60.wmf oleobject20.bin image61.wmf oleobject21.bin image62.wmf oleobject22.bin image63.wmf image6.wmf oleobject23.bin image7.wmf image8.wmf image9.png image10.png image11.png image12.png image13.png image14.png image15.png image16.png image17.png image18.wmf image19.wmf image20.wmf image21.png image22.png image23.wmf image24.wmf image25.png image26.png image27.png image28.png image1.wmf image29.png image30.png image31.png image32.png image33.png image2.wmf image34.png image35.png image36.png image37.png image38.png image3.wmf image39.png image40.png image41.wmf oleobject1.bin image42.wmf oleobject2.bin image43.wmf image4.png oleobject3.bin image44.wmf oleobject4.bin image45.wmf oleobject5.bin image46.wmf oleobject6.bin image47.wmf oleobject7.bin image48.wmf oleobject8.bin image49.wmf oleobject9.bin image50.wmf oleobject10.bin image51.wmf oleobject11.bin image52.wmf oleobject12.bin image53.wmf ï ï î ï ï í ì - - = + = . 3 , 5 z y dx dz z y dx …

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amaliy mashg’ulot 9. differensial tenglamalarni yechish funksiyalari. differensial tenglamalarning umumiy yechimi. 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’ri...

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