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Консультация № 189514
27.05.2016, 15:21
0.00 руб.
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1
Здравствуйте! Прошу помощи в следующем вопросе:
Необходимо найти все экстремали функционала, удовлетворяющие заданным граничным условиям :
(первое задание)
Необходимо найти все экстремали функционала, удовлетворяющие заданным граничным условиям :
(первое задание)
Обсуждение
27.05.2016, 18:26
общий
это ответ
Здравствуйте, RoflGuy!
Рассмотрим задание 1a.
Найти все экстремали функционала
удовлетворяющие заданным граничным условиям:
Здесь
так что 
Уравнение Эйлера - Лагранжа принимает вид 
или 
Решим это линейное однородное дифференциальное уравнение второго порядка с постоянными коэффициентами. Составим характеристическое уравнение:
Его корнями являются числа 
Запишем общее решение уравнения Эйлера - Лагранжа: 
В соответствии с заданными граничными условиями получим систему уравнений
решая которую, получим
Следовательно, экстремум может достигаться лишь на кривой
Проверьте, пожалуйста, выкладки во избежание ошибок.
С уважением.
Рассмотрим задание 1a.
Найти все экстремали функционала
удовлетворяющие заданным граничным условиям:Здесь
так что Уравнение Эйлера - Лагранжа принимает вид или Решим это линейное однородное дифференциальное уравнение второго порядка с постоянными коэффициентами. Составим характеристическое уравнение:
Его корнями являются числа Запишем общее решение уравнения Эйлера - Лагранжа: В соответствии с заданными граничными условиями получим систему уравненийрешая которую, получим
Следовательно, экстремум может достигаться лишь на кривой
Проверьте, пожалуйста, выкладки во избежание ошибок.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
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