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Консультация № 182430
09.03.2011, 09:59
60.61 руб.
0
3
2
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти общее решение диф. уравнения:
xy' = y + sqrt(x^2 + y^2)
Найти общее решение диф. уравнения:
xy' = y + sqrt(x^2 + y^2)
Обсуждение
09.03.2011, 10:34
общий
это ответ
Здравствуйте, Ulitka71!
Будут вопросы обращайтесь в минифорум.
Удачи
Будут вопросы обращайтесь в минифорум.
Удачи
5
Неизвестный
09.03.2011, 10:42
общий
это ответ
Здравствуйте, Ulitka71!
Данное уравнение является однородным уравнением 1-го порядка. Это можно установить, заменив x и y на tx и ty. Легко видеть, что уравнение не изменится. Такие уравнения решаются подстановкой y(x)=xz(x). Тогда y'=z+xz'. Подставляя все это в уравнение, получим:
xz+x[sup]2[/sup]z'=xz+[$8730$](x[sup]2[/sup]+x[sup]2[/sup]z[sup]2[/sup]).
Теперь после сокращений, будем иметь уравнение xz'=[$8730$](1+z[sup]2[/sup]) с разделяющимися переменными. Разделяя переменные, получим:
.
После интегрирования, получим:
где C - произвольная постоянная. Отсюда
или, после возвращения к старой переменной y (z=y/x) и некоторых простых преобразований, получим общее решение
Данное уравнение является однородным уравнением 1-го порядка. Это можно установить, заменив x и y на tx и ty. Легко видеть, что уравнение не изменится. Такие уравнения решаются подстановкой y(x)=xz(x). Тогда y'=z+xz'. Подставляя все это в уравнение, получим:
xz+x[sup]2[/sup]z'=xz+[$8730$](x[sup]2[/sup]+x[sup]2[/sup]z[sup]2[/sup]).
Теперь после сокращений, будем иметь уравнение xz'=[$8730$](1+z[sup]2[/sup]) с разделяющимися переменными. Разделяя переменные, получим:
.После интегрирования, получим:
где C - произвольная постоянная. Отсюда
или, после возвращения к старой переменной y (z=y/x) и некоторых простых преобразований, получим общее решение
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