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Консультация № 139327
06.06.2008, 14:53
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Уважаемые Эксперты! Помогите пожалуйста найти общий интеграл нелинейной ссистемы:
dx/x(y+z)=dy/z(z-y)=dz/y(y-z)
Заранее благодарю.
dx/x(y+z)=dy/z(z-y)=dz/y(y-z)
Заранее благодарю.
Обсуждение
07.06.2008, 20:53
общий
это ответ
Здравствуйте, Иванов Алексей Александрович!
Предлагаю Вам следующее
Решение.
Имеем систему трех дифференциальных уравнений в симметрической форме. Построим две интегрируемые комбинации. Одной из них, очевидно, будет
dy/z = -dz/y,
ydy + zdz = 0,
откуда
(1/2)*(y^2 + z^2) = C,
y^2 + z^2 = 2C,
y^2 + z^2 = (C1)^2 (C1 = sqrt (2C)) - первый первый интеграл заданной системы. (1)
Для получения второй интегрируемой комбинации вычтем в заданной системе из числителя и знаменателя второй дроби соответственно числитель и знаменатель третьей дроби и выполним тождественные преобразования:
dx/(x(y + z)) = d(y - z)/(z^2 - y^2),
dx/(x(z + y)) = -d(z - y)/((z - y)(z + y)),
dx/x = -d(z - y)/(z - y),
откуда
ln |x| = - ln |z - y| + ln |C2|,
x = C2/(z - y) - второй первый интеграл заданной системы. (2)
Первые интегралы (1) и (2) образуют общий интеграл заданной системы.
Предлагаю Вам следующее
Решение.
Имеем систему трех дифференциальных уравнений в симметрической форме. Построим две интегрируемые комбинации. Одной из них, очевидно, будет
dy/z = -dz/y,
ydy + zdz = 0,
откуда
(1/2)*(y^2 + z^2) = C,
y^2 + z^2 = 2C,
y^2 + z^2 = (C1)^2 (C1 = sqrt (2C)) - первый первый интеграл заданной системы. (1)
Для получения второй интегрируемой комбинации вычтем в заданной системе из числителя и знаменателя второй дроби соответственно числитель и знаменатель третьей дроби и выполним тождественные преобразования:
dx/(x(y + z)) = d(y - z)/(z^2 - y^2),
dx/(x(z + y)) = -d(z - y)/((z - y)(z + y)),
dx/x = -d(z - y)/(z - y),
откуда
ln |x| = - ln |z - y| + ln |C2|,
x = C2/(z - y) - второй первый интеграл заданной системы. (2)
Первые интегралы (1) и (2) образуют общий интеграл заданной системы.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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