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Консультация № 182432
09.03.2011, 12:15
106.72 руб.
0
10
2
Здравствуйте! Прошу помощи в следующем вопросе:
Найти общее решение диф. уравнения:
y'cos(x) + y sin(x) = 1
Найти общее решение диф. уравнения:
y'cos(x) + y sin(x) = 1
Обсуждение
Неизвестный
09.03.2011, 14:08
общий
09.03.2011, 15:57
это ответ
Здравствуйте, Ulitka71!
y'cos(x) + y sin(x) = 1 линейное уравнение 1-го порядка
Выполним подстановку y=uv
y'=u'v+uv'
Подставим в уравнение
u'vcosx+uv'cosx+uvsinx=1
u'vcosx+u(v'cosx+vsinx)=1
положим
v'cosx+vsinx=0
dv/ v= - sin(x)dx/ cos(x)
∫ dv/ v=∫ -sin(x)dx/ cos(x)
Ln v=∫d(sinx)/cos x
Ln v=ln cos x
v=cos x
подставим в уравнение значение v
u'cos2x=1
du=dx/ (cosx)^2
∫du = ∫dx/ (cosx)^2
u=tgx+C
Общее решение уравнения
y=cosx(tgx+C)
y'cos(x) + y sin(x) = 1 линейное уравнение 1-го порядка
Выполним подстановку y=uv
y'=u'v+uv'
Подставим в уравнение
u'vcosx+uv'cosx+uvsinx=1
u'vcosx+u(v'cosx+vsinx)=1
положим
v'cosx+vsinx=0
dv/ v= - sin(x)dx/ cos(x)
∫ dv/ v=∫ -sin(x)dx/ cos(x)
Ln v=∫d(sinx)/cos x
Ln v=ln cos x
v=cos x
подставим в уравнение значение v
u'cos2x=1
du=dx/ (cosx)^2
∫du = ∫dx/ (cosx)^2
u=tgx+C
Общее решение уравнения
y=cosx(tgx+C)
5
Неизвестный
09.03.2011, 14:21
общий
Здравствуйте, Наталья!
Это линейное уравнение. Его решают подстановкой y=u(x)v(x).
В правой части не 0, а 1. Времени решать нет, бегу на работу.
С уважением.
Это линейное уравнение. Его решают подстановкой y=u(x)v(x).
В правой части не 0, а 1. Времени решать нет, бегу на работу.
С уважением.
Неизвестный
09.03.2011, 15:26
общий
09.03.2011, 15:54
это ответ
Здравствуйте, Ulitka71!
Успехов,
Кривов Максим
Успехов,
Кривов Максим
Неизвестный
09.03.2011, 15:32
общий
Извините, нельзя параллельно двумя делами заниматься(((
Действительно
y'cos(x) + y sin(x) = 1 линейное уравнение 1-го порядка
Выполним подстановку y=uv
y'=u'v+uv'
Подставим в уравнение
u'vcosx+uv'cosx+uvsinx=1
u'vcosx+u(v'cosx+vsinx)=1
положим
v'cosx+vsinx=0
dv/ v= - sin(x)dx/ cos(x)
∫ dv/ v=∫ -sin(x)dx/ cos(x)
Ln v=∫d(sinx)/cos x
Ln v=ln cos x
v=cos x
подставим в уравнение значение v
u'cos2x=1
du=dx/ (cosx)^2
∫du = ∫dx/ (cosx)^2
u=tgx+C
Общее решение уравнения
y=cosx(tgx+C)
Действительно
y'cos(x) + y sin(x) = 1 линейное уравнение 1-го порядка
Выполним подстановку y=uv
y'=u'v+uv'
Подставим в уравнение
u'vcosx+uv'cosx+uvsinx=1
u'vcosx+u(v'cosx+vsinx)=1
положим
v'cosx+vsinx=0
dv/ v= - sin(x)dx/ cos(x)
∫ dv/ v=∫ -sin(x)dx/ cos(x)
Ln v=∫d(sinx)/cos x
Ln v=ln cos x
v=cos x
подставим в уравнение значение v
u'cos2x=1
du=dx/ (cosx)^2
∫du = ∫dx/ (cosx)^2
u=tgx+C
Общее решение уравнения
y=cosx(tgx+C)
Неизвестный
09.03.2011, 15:37
общий
Прошу прощения. Дал неверную ссылку. . Наталья дала правильное начало решения, но ее решение в том виде как есть - решение однородного уравнения. А у Вас неоднородное
Прикрепленные файлы:
09.03.2011, 15:59
общий
Ссылку на изображение надо давать в виде
[img ]https://rfpro.ru/d/4848.gif[/img] (без пробела)
[img ]https://rfpro.ru/d/4848.gif[/img] (без пробела)
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
09.03.2011, 16:01
общий
Заменил решение. Будьте внимательны
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
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