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Консультация № 132350
16.04.2008, 10:08
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Дорогие эксперты!
Прошу Вас, уважаемые, о помощи в решении задания Коши:
y^3 y’’+ 49 = 0, y(3) = -7, y’(3) = -1
Заранее Вам благодарна!
Прошу Вас, уважаемые, о помощи в решении задания Коши:
y^3 y’’+ 49 = 0, y(3) = -7, y’(3) = -1
Заранее Вам благодарна!
Обсуждение
Неизвестный
16.04.2008, 16:33
общий
это ответ
Здравствуйте, Маргарита Левса!
y^3 y’’+ 49 = 0 дифференциальное уравнение 2го порядка, допускающее снижение порядка, для чего введем замену
y‘=p (р зависит от у)
y‘‘=pp‘
y^3*p*p‘+49=0
pp‘=-49/y^3
pdp=-49dy/y^3
Int[pdp]=Int[-49dy/y^3]
p^2 /2=49/(2*y^2) +C1
p^2=49/y^2 +C1
p=sqrt(49/y^2 +C1)
Возвращаемся к замене:
y‘=p
y‘=sqrt(49/y^2 +C1)
dy/dx=sqrt(49+C1*y^2) /y
ydy/(sqrt(49+C1*y^2))=dx
Int[ydy/(sqrt(49+C1*y^2))]=Int[dx]
1/(2C1)* Int[d(C1*y^2+49)/(sqrt(49+C1*y^2))]=Int[dx]
1/C1 *sqrt(C1*y^2+49)=x+C2 (1)
Выразив из (1) y, получим
y=sqrt(1/C1*((C1*x+C1*C2)^2 - 49)) (2)
y‘=sqrtC1*(C1*x+C1*C2)/sqrt((C1*x+C1*C2)^2 -49)
Теперь подставьте в первом уравнении вместо у - -7, вместо х - 3, а во втором вместо y‘ - -1, а вместо х - 3
Получите систему из двух уравнений с переменными С1 и С2, к-рые нужно найти и подставить в решение (2)
y^3 y’’+ 49 = 0 дифференциальное уравнение 2го порядка, допускающее снижение порядка, для чего введем замену
y‘=p (р зависит от у)
y‘‘=pp‘
y^3*p*p‘+49=0
pp‘=-49/y^3
pdp=-49dy/y^3
Int[pdp]=Int[-49dy/y^3]
p^2 /2=49/(2*y^2) +C1
p^2=49/y^2 +C1
p=sqrt(49/y^2 +C1)
Возвращаемся к замене:
y‘=p
y‘=sqrt(49/y^2 +C1)
dy/dx=sqrt(49+C1*y^2) /y
ydy/(sqrt(49+C1*y^2))=dx
Int[ydy/(sqrt(49+C1*y^2))]=Int[dx]
1/(2C1)* Int[d(C1*y^2+49)/(sqrt(49+C1*y^2))]=Int[dx]
1/C1 *sqrt(C1*y^2+49)=x+C2 (1)
Выразив из (1) y, получим
y=sqrt(1/C1*((C1*x+C1*C2)^2 - 49)) (2)
y‘=sqrtC1*(C1*x+C1*C2)/sqrt((C1*x+C1*C2)^2 -49)
Теперь подставьте в первом уравнении вместо у - -7, вместо х - 3, а во втором вместо y‘ - -1, а вместо х - 3
Получите систему из двух уравнений с переменными С1 и С2, к-рые нужно найти и подставить в решение (2)
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