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Консультация № 180707
13.11.2010, 03:04
0.00 руб.
13.11.2010, 12:58
0
5
3
Здравствуйте эксперты! Помогите решить пример.
Линейное уравнение:
1)(2x+1)*y'=4x+2y;
2)(xy+e^x)*dx-xdy=0;
Уравнение Бернулли:
2y'-x/y=xy/(x^2-1)
Заранее Спасибо!!!!!
Линейное уравнение:
1)(2x+1)*y'=4x+2y;
2)(xy+e^x)*dx-xdy=0;
Уравнение Бернулли:
2y'-x/y=xy/(x^2-1)
Заранее Спасибо!!!!!
Обсуждение
13.11.2010, 10:44
общий
это ответ
Здравствуйте, Magma!
Решение 2:
Перепишем уравнение в производных (разделив уравнение на x и dx):
y'=y+ex/x
1) решаем однородное уравнение
y'=y; dy/y=dx; ln|y|=x+const; y=Cex
2) применяем метод вариации:
y=C(x)ex ----> C'(x)ex=ex/x;
C'(x)=1/x ----> C(x)=ln|x|+C
Ответ: y=ex(ln|x|+C)
P.S. Так как уравнение записано в симметричной форме, то его часто рассматривают как уравнение кривой, а не функции. В этом случае нужно добавить решение x=0, потерянное при делении уравнения на x.
Решение 2:
Перепишем уравнение в производных (разделив уравнение на x и dx):
y'=y+ex/x
1) решаем однородное уравнение
y'=y; dy/y=dx; ln|y|=x+const; y=Cex
2) применяем метод вариации:
y=C(x)ex ----> C'(x)ex=ex/x;
C'(x)=1/x ----> C(x)=ln|x|+C
Ответ: y=ex(ln|x|+C)
P.S. Так как уравнение записано в симметричной форме, то его часто рассматривают как уравнение кривой, а не функции. В этом случае нужно добавить решение x=0, потерянное при делении уравнения на x.
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