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Задать вопрос экспертам

Консультация № 185453

16.02.2012, 18:38

72.36 руб.

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Образование

Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос: Найдите общее решение дифференциального уравнения второго порядка:

Сформулируйте задачу Коши для этого уравнения, задав корректные начальные условия для
искомой функции.

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

16.02.2012, 19:46

общий

это ответ

Здравствуйте, Посетитель - 383833!
1) Решаем сначала однородное уравнение y''-4y'+4y=0.
Характеристическое уравнение [$955$]²-4[$955$]+4=0 имеет двукратный корень [$955$]=2, общее решение
y=C₁e^2x+C₂xe^2x
2) Находим частное решение для правой части f(x)=2x². Ищем его в виде y=Ax²+Bx+C.
Подставляя в уравнение, имеем
2A-8Ax-4B+4Ax²+4Bx+4C=2x²
Получаем систему
4A=2
-8A+4B=0
2A-4B+4C=0
Решая систему, находим A=1/2,B=1,C=3/4, т.е. y=(1/2)x²+x+3/4
3) Находим частное решение для правой части f(x)=-e^2x. Ищем его в виде y=Ax²e^2x.
Подставляя в уравнение, имеем
A(4x²+8x+2)e^2x-4A(2x²+2x)e^2x+4Ae^2x=-e^2x
Получаем уравнение 2A=-1 ---> A=-1/2, т.е. y=-(1/2)x²e^2x

Ответ: Общее решение y=C₁e^2x+C₂xe^2x+(1/2)x²+x+3/4-(1/2)x²e^2x

P.S. Задача Коши ставится следующим образом

y''-4y'+4y=2x²-e^2x
y(x₀)=y₀
y'(x₀)=y₁
где x₀ - произвольная фиксированная точка на оси абсцисс
y₀,y₁ - произвольные заданные числа

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