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Консультация № 200883
20.05.2021, 11:50
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Уважаемые эксперты! Помогите, пожалуйста, решить уравнение
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Обсуждение
20.05.2021, 12:52
общий
Для начала сделаем замену z=y'. Получим: x*(z' + 1) + z =0 или x*z' + z = -x
Вместо него решим однородное уравнение (с нулевой правой частью): x*z' + z = 0
Получим x*dz + z*dx=0 или d(xz)=0. Откуда xz=C, где C - некая константа, т.е. z(x) = C/x.
Теперь рассматриваем это C как C (х) и подставляем в неоднородное уравнение (x*z' + z = -x).
После приведения подобных получаем: C' = -x. Откуда C(x) = -1/2*x^2 + D, где D - некая константа.
Значит, z(x) = (-1/2*x^2 + D)/X = -x/2 + D/x. Вспоминаем, что z(x) = y'.
Тогда y' = -x/2 + D/x. Здесь просто интегрируем и получаем:
y(x) = - 1/4*x^2 + D*ln|x| + E, где D и E - произвольные константы.
Вроде бы, так.
Вместо него решим однородное уравнение (с нулевой правой частью): x*z' + z = 0
Получим x*dz + z*dx=0 или d(xz)=0. Откуда xz=C, где C - некая константа, т.е. z(x) = C/x.
Теперь рассматриваем это C как C (х) и подставляем в неоднородное уравнение (x*z' + z = -x).
После приведения подобных получаем: C' = -x. Откуда C(x) = -1/2*x^2 + D, где D - некая константа.
Значит, z(x) = (-1/2*x^2 + D)/X = -x/2 + D/x. Вспоминаем, что z(x) = y'.
Тогда y' = -x/2 + D/x. Здесь просто интегрируем и получаем:
y(x) = - 1/4*x^2 + D*ln|x| + E, где D и E - произвольные константы.
Вроде бы, так.
20.05.2021, 14:24
общий
это ответ
Здравствуйте, finder!
Решим теперь методом Лагранжа уравнение
Получим
Определим решение уравнения (1) как функцию вида
Имеем
где 
Значит,
где 
Решим теперь методом Лагранжа уравнение
ПолучимОпределим решение уравнения (1) как функцию вида
Имеем где Значит,
где
Об авторе:
Facta loquuntur.
Facta loquuntur.
, тогда 
не является решением уравнения, поэтому разделим обе части уравнения на 
, тогда 
Форма ответа
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