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Понизить порядок дифференциального уравнения y"tg y = (y')[sup]2[/sup].
Заданное дифференциальное уравнение явно не содержит независимой переменной x. Поэтому оно допускает понижение порядка подстановкой y' = p(y). Формальное отсутствие аргумента x позволяет считать неизвестную функцию p функцией аргумента y. Тогда y" = (p(y))' = p'(y) [$183$] y'(x) = p'p.
Значит, заданное уравнение равносильно уравнению p'p [$183$] tg y = p2, или p'tg y = p, где p = y'.
Не задавайте несколько разных вопросов в одном. Не задавайте несколько разных, не связанных с друг другом вопросов, в одном. Это не запрещено (если все вопросы относятся к теме рассылки), но вероятность того, что Вы получите на них ответы, будет гораздо выше, если Вы зададите их по отдельности. Например, мало кому из экспертов захочется отвечать на вопрос, в котором просто перечислено несколько задач из задачника. Отвечать на такие вопросы неудобно, ответы трудно читаются в выпусках рассылок, затрудняется обсуждение в форуме. Поэтому большинство экспертов просто игнорируют вопросы, в которых под видом одного дано несколько вопросов или задач. Гораздо лучше, если Вы в одном вопросе спросите про решение одной проблемы, особенно, если Вы покажете, что пытались решить ее самостоятельно, и укажете, что именно вызвало трудности. Тогда многие захотят Вам помочь.
И ещё: несмотря на то, что Вам так проще, не следует без большой нужды использовать графические файлы. Вы вполне могли набрать тексты заданий вручную.
Здравствуйте, Посетитель - 370501! 4) Уравнение Бернулли. Делим на y2 и полагаем z=1/y: -z'-z*tg x+cos x=0 (линейное уравнение) а) решаем однородное z'+z*tg x=0 dz/z=-tg xdx=-sin xdx/cos x=d(cos x)/cos x) ln|z|=ln|cos x|+const z=Ccos x б) применяем метод вариации: z=C(x)cos x C'(x)cos x=cos x C'(x)=1 C(x)=x+C Таким образом z=x*cos x+C*cos x y=1/z=1/(x*cos x+C*cos x) (и еще нужно добавить решение y=0, потерянное при делении на y2) Ответ: y=1/(x*cos x+C*cos x); y=0.
Вы имеете в виду задание без номера? Полностью - это как? Чего Вам не хватает в решении? Порядок уравнения понижен ведь. И в условии не говорится, что полученное уравнение нужно решить.
Об авторе:
Facta loquuntur.
Неизвестный
27.11.2011, 01:44
общий
27.11.2011, 01:44
Адресаты:
да,без номера. я не особо силен в этой матике,но преподу надо решить что то другое...
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