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Консультация № 181274
13.12.2010, 02:52
53.35 руб.
0
2
2
Здравствуйте, уважаемые эксперты! Прошу Вас ответить на следующий вопрос:
Решить дифференциальные уравнения:
1. y = xy' + e^y'
2. 2y - xy' + ln y' = 0
3. y = xy' + y^2
Решить дифференциальные уравнения:
1. y = xy' + e^y'
2. 2y - xy' + ln y' = 0
3. y = xy' + y^2
Обсуждение
13.12.2010, 09:59
общий
это ответ
Здравствуйте, zim-zum!
Предлагаю решение третьего уравнения:
Будут вопросы обращайтесь в минифорум.
Удачи
Предлагаю решение третьего уравнения:
Будут вопросы обращайтесь в минифорум.
Удачи
Неизвестный
13.12.2010, 14:32
общий
это ответ
Здравствуйте, zim-zum!
1. y=x*y'+ey'
Продифференцируем уравнение
y'=y'+x*y''+ey'*y''
y''*(x+ey')=0 => y''=0 и x+ey'=0
y''=0 => y=C1*x+C2; C1,C2 - const
y'=C1
Подставим в исходное уравнение:
C1*x+C2=x*C1+eC1 => C2=eC1
y=C[sub]1[/sub]*x+e[sup]C1[/sup], C1 - const
x+e[sup]y'[/sup]=0
ey'= -x => y'=ln(-x) => y=[$8747$]ln(-x)dx=x*ln(-x)-x + C3 , C3 - const
Подставим в исходное уравнение:
x*ln(-x)-x + C3=x*ln(-x)+eln(-x) => C3=0
y=x*ln(-x)-x
Окончательно получим:
y=x*ln(-x)-x, y=C*x+e[sup]C[/sup], C - const
1. y=x*y'+ey'
Продифференцируем уравнение
y'=y'+x*y''+ey'*y''
y''*(x+ey')=0 => y''=0 и x+ey'=0
y''=0 => y=C1*x+C2; C1,C2 - const
y'=C1
Подставим в исходное уравнение:
C1*x+C2=x*C1+eC1 => C2=eC1
y=C[sub]1[/sub]*x+e[sup]C1[/sup], C1 - const
x+e[sup]y'[/sup]=0
ey'= -x => y'=ln(-x) => y=[$8747$]ln(-x)dx=x*ln(-x)-x + C3 , C3 - const
Подставим в исходное уравнение:
x*ln(-x)-x + C3=x*ln(-x)+eln(-x) => C3=0
y=x*ln(-x)-x
Окончательно получим:
y=x*ln(-x)-x, y=C*x+e[sup]C[/sup], C - const
Форма ответа
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