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Консультация № 184825
17.12.2011, 15:56
80.00 руб.
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1
Уважаемые эксперты! Пожалуйста, ответьте на вопросы:
Решить дифференциальные уравнения:
1.6xdx-6ydy=2x2ydy-3xy2dx
2.y'=y2/x2+6y/x+6
Решить дифференциальные уравнения:
1.6xdx-6ydy=2x2ydy-3xy2dx
2.y'=y2/x2+6y/x+6
Обсуждение
17.12.2011, 16:07
общий
это ответ
Здравствуйте, Дмитрий!
1
Уравнение с разделяющимися переменными.
(6x+3xy^2)dx=(2x^2y+6y)dy -> 3x(2+y^2)dx=2y(x^2+3)dy -> 3xdx/x^2+3 = 2ydy/(y^2+2)
После интегрирования:
3/2 ln(x^2+3)=ln(y^2+2)+lnC -> ln(x^2+3)^(3/2)=lnC(y^2+2) -> (x^2+3)^(3/2)=C(y^2+2)
2
Уравнение с однородными функциями.
Замена y=ux, y'=u'x+u:
u'x+u=u^2+6u+6 -> xdu/dx=u^2+5u+6 -> du/(u^2+5u+6)=dx/x -> du/((u+2)(u+3))=dx/x -> du/(u+2)-du/(u+3)=dx/x
После интегрирования:
ln(u+2)-ln(u+3)=lnx + lnC
(u+2)/(u+3)=Cx
(y/x+2)/(y/x+3)=Cx
(y+2x)/(y+3x)=Cx -> y=x(3Cx-2)/(1-Cx)
1
Уравнение с разделяющимися переменными.
(6x+3xy^2)dx=(2x^2y+6y)dy -> 3x(2+y^2)dx=2y(x^2+3)dy -> 3xdx/x^2+3 = 2ydy/(y^2+2)
После интегрирования:
3/2 ln(x^2+3)=ln(y^2+2)+lnC -> ln(x^2+3)^(3/2)=lnC(y^2+2) -> (x^2+3)^(3/2)=C(y^2+2)
2
Уравнение с однородными функциями.
Замена y=ux, y'=u'x+u:
u'x+u=u^2+6u+6 -> xdu/dx=u^2+5u+6 -> du/(u^2+5u+6)=dx/x -> du/((u+2)(u+3))=dx/x -> du/(u+2)-du/(u+3)=dx/x
После интегрирования:
ln(u+2)-ln(u+3)=lnx + lnC
(u+2)/(u+3)=Cx
(y/x+2)/(y/x+3)=Cx
(y+2x)/(y+3x)=Cx -> y=x(3Cx-2)/(1-Cx)
Форма ответа
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