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Консультация № 189506
25.05.2016, 22:54
0.00 руб.
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:Найти общее решение дифференциального уравнения:
Прикрепленные файлы:
Обсуждение
28.05.2016, 13:42
общий
это ответ
Здравствуйте, ushtvan98!
Пусть дано дифференциальное уравнение второго порядка
Оно не содержит искомой функции
Стандартный приём решения таких уравнений заключается в решении дифференциального уравнения первого порядка с использованием подстановки 
Тогда 
Заданное уравнение принимает вид
и представляет собой линейное дифференциальное уравнение первого порядка. Решим его методом Лагранжа (методом вариации произвольной постоянной). Для этого рассмотрим соответствующее уравнение с нулевой правой частью
- линейное однородное дифференциальное уравнение первого порядка. Разделим в нём переменные: 
и получим 
где 
Постоянную 
в полученном решении заменим функцией 
и найдём решение уравнения (2) в виде 
Имеем 
(из уравнения (2)), 
Итак,
Значит,
Получили, что общим решением дифференциального уравнения (1) является функция
Не поленитесь проверить выкладки во избежание ошибок!
С уважением.
Пусть дано дифференциальное уравнение второго порядка
Оно не содержит искомой функции
Стандартный приём решения таких уравнений заключается в решении дифференциального уравнения первого порядка с использованием подстановки Тогда Заданное уравнение принимает види представляет собой линейное дифференциальное уравнение первого порядка. Решим его методом Лагранжа (методом вариации произвольной постоянной). Для этого рассмотрим соответствующее уравнение с нулевой правой частью
- линейное однородное дифференциальное уравнение первого порядка. Разделим в нём переменные: и получим где Постоянную в полученном решении заменим функцией и найдём решение уравнения (2) в виде Имеем (из уравнения (2)), Итак,Значит,
Получили, что общим решением дифференциального уравнения (1) является функция
Не поленитесь проверить выкладки во избежание ошибок!
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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