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Консультация № 132352
16.04.2008, 10:13
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Уважаемые эксперты!
Помогите, пожалуйста с поиском общего решения диференциального уравнения
y’’ – 4y’ = 8e^(4x) + 8 e^(-4x)
Заранее благодарна вам за помощь и уделённое внимание!!! СПАСИБО!
Помогите, пожалуйста с поиском общего решения диференциального уравнения
y’’ – 4y’ = 8e^(4x) + 8 e^(-4x)
Заранее благодарна вам за помощь и уделённое внимание!!! СПАСИБО!
Обсуждение
Неизвестный
16.04.2008, 17:33
общий
это ответ
Здравствуйте, Маргарита Левса!
y’’ – 4y’ = 8e^(4x) + 8 e^(-4x) <i>линейное неоднородное дифференциальное уравнение 2го порядка с постоянными коэффициентами</i>
Решим сперва соответствующее однородное уравнение:
y‘‘-4y‘=0
Используем подстановку
y=e^(tx)
y‘=te^(tx)
y‘‘=t^2*e^(tx)
t^2*e^(tx) -4te^(tx)=0
Разделим обе части уравнения на e^(tx) не равное 0:
t^2-4t=0
t(t-4)=0
t1=0, t2=4
y1=e^(0*x)=1
y2=e^(4x)
y=C1*y1+C2*y2
y=C1+C2*e^(4x)
Метод вариации неопределенных коэффициентов:
C1‘(x)+C2‘(x)*e^(4x)=0
C1‘(x)*0+4*C2‘(x)*e^(4x)=8e^(4x) + 8 e^(-4x)
4*C2‘(x)*e^(4x)=8e^(4x) + 8 e^(-4x)
C2‘(x)=2*(1+e^(-8x))
C2(x)=Int[2*(1+e^(-8x)) dx]=2x-1/4*e^(-8x)+C2
C1‘(x)=-C2‘(x)*e^(4x)=-2*(1+e^(-8x))*e^(4x)=-2(e^(4x)+e^(-4x))
C1(x)=-2Int[(e^(4x)+e^(-4x)) dx]=-1/2*e^(4x) + 1/2*e^(-4x)+C1
y=C1(x)+C2(x)*e^(4x)
<b>y=-1/2*e^(4x) + 1/2*e^(-4x)+C1+e^(4x)*(2x-1/4*e^(-8x)+C2)</b>
y’’ – 4y’ = 8e^(4x) + 8 e^(-4x) <i>линейное неоднородное дифференциальное уравнение 2го порядка с постоянными коэффициентами</i>
Решим сперва соответствующее однородное уравнение:
y‘‘-4y‘=0
Используем подстановку
y=e^(tx)
y‘=te^(tx)
y‘‘=t^2*e^(tx)
t^2*e^(tx) -4te^(tx)=0
Разделим обе части уравнения на e^(tx) не равное 0:
t^2-4t=0
t(t-4)=0
t1=0, t2=4
y1=e^(0*x)=1
y2=e^(4x)
y=C1*y1+C2*y2
y=C1+C2*e^(4x)
Метод вариации неопределенных коэффициентов:
C1‘(x)+C2‘(x)*e^(4x)=0
C1‘(x)*0+4*C2‘(x)*e^(4x)=8e^(4x) + 8 e^(-4x)
4*C2‘(x)*e^(4x)=8e^(4x) + 8 e^(-4x)
C2‘(x)=2*(1+e^(-8x))
C2(x)=Int[2*(1+e^(-8x)) dx]=2x-1/4*e^(-8x)+C2
C1‘(x)=-C2‘(x)*e^(4x)=-2*(1+e^(-8x))*e^(4x)=-2(e^(4x)+e^(-4x))
C1(x)=-2Int[(e^(4x)+e^(-4x)) dx]=-1/2*e^(4x) + 1/2*e^(-4x)+C1
y=C1(x)+C2(x)*e^(4x)
<b>y=-1/2*e^(4x) + 1/2*e^(-4x)+C1+e^(4x)*(2x-1/4*e^(-8x)+C2)</b>
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