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Консультация № 65371
03.12.2006, 16:01
0.00 руб.
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8
1
Помогите пожалуйста решить диф. уравнение y‘‘+tg(y)=2(y‘)^2
Пробовал заменой p=y‘, pp‘=y‘‘, получилось pp‘+tg(y)=2p^2. Получаем уравнение Бернулли, делаю замену z=p^2, получаю 1/2*dz/dy+tg(y)=2z.
Решение последнего нахожу в виде z=uv и вроде все выходит, но потом получаю int(1)du=-2int[(e^(-4y))*tgy]dy. Найти правый интеграл у меня не получается. Кто поможет? Может я сначала неправильно начал, не пойму никак.
Пробовал заменой p=y‘, pp‘=y‘‘, получилось pp‘+tg(y)=2p^2. Получаем уравнение Бернулли, делаю замену z=p^2, получаю 1/2*dz/dy+tg(y)=2z.
Решение последнего нахожу в виде z=uv и вроде все выходит, но потом получаю int(1)du=-2int[(e^(-4y))*tgy]dy. Найти правый интеграл у меня не получается. Кто поможет? Может я сначала неправильно начал, не пойму никак.
Обсуждение
05.12.2006, 08:58
общий
А почему Вы решили, что pp‘+tg(y)=2p^2 - уравнение Бернулли?
Об авторе:
Facta loquuntur.
Facta loquuntur.
Неизвестный
05.12.2006, 22:11
общий
Я предположил.В общем виде уравнения Бернулли имеют вид:dy/dx+P(x)y=Q(x)y^n.Где n ≠ 1 , n ≠ 0 и функции от x непрерывны
06.12.2006, 10:27
общий
Да, но Ваше уравнение не приводится к такому виду. В указанных обозначениях оно имеет видy*dy/dx+tg x=2*y^2, или dy/dx+(tg x)/y=2*y. Я бы не сказал, что получилось уравнение Бернулли. Или я ошибаюсь?
Об авторе:
Facta loquuntur.
Facta loquuntur.
06.12.2006, 14:57
общий
это ответ
Здравствуйте, Гаврилов Андрей Владимирович !
Указанный Вами интеграл берется по частям. Положим u=tg y, dv=e^(-4*y)*dy. Тогда du=dy/(cos y)^2, v=-(1/4)*e^(-4*y), и искомый интеграл I=-(1/4)*e^(-4*y)*tg y-(1/a)+(1/4)*∫(e^(-4*y)*dy/(cos y)^2. А получившийся интеграл можно взять по таблице интегралов, содержащих показательную функцию.
С уважением,
Mr. Andy.
Указанный Вами интеграл берется по частям. Положим u=tg y, dv=e^(-4*y)*dy. Тогда du=dy/(cos y)^2, v=-(1/4)*e^(-4*y), и искомый интеграл I=-(1/4)*e^(-4*y)*tg y-(1/a)+(1/4)*∫(e^(-4*y)*dy/(cos y)^2. А получившийся интеграл можно взять по таблице интегралов, содержащих показательную функцию.
С уважением,
Mr. Andy.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Неизвестный
07.12.2006, 20:25
общий
Спасибо за ответ.Но дело в том что уменя тоже самое и получилось с этим интегралом. Знаете, MathCad при его нахождении выдает функцию с букавками i и вот таким, видимо обозначающим интеграл, значением - indef_int. В связи с этим я понял что нормальных значений я там не получу никак), значит вначале надо другие замены делать или как-то преобразовывать уравнение.
08.12.2006, 09:55
общий
А мне кажется, что все более-менее нормально. Ведь ∫e^(-4y)*dy/(cos y)^2= (1/20)*e^(-4y)*cos y*(-4*cos y+2*sin y)+(1/10)*∫e^(-4y)dy. А уж последний интеграл взять несложно.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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