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Консультация № 185688
27.03.2012, 19:11
110.73 руб.
0
4
4
Здравствуйте! Прошу помощи в следующих вопросах:
Найти общее решение дифференциального уравнения:
а) y' - y*tg x = exp^(2*x)/cos x; (exp(2*x) - e в степени два икс)
б) y' = y/x - 2*sin (y/x);
в) y'''' + 4*y'' = 0; (y'''' - четвертая производная от игрека)
г) y'' + 16*y = x*sin 4x
Найти общее решение дифференциального уравнения:
а) y' - y*tg x = exp^(2*x)/cos x; (exp(2*x) - e в степени два икс)
б) y' = y/x - 2*sin (y/x);
в) y'''' + 4*y'' = 0; (y'''' - четвертая производная от игрека)
г) y'' + 16*y = x*sin 4x
Обсуждение
27.03.2012, 19:16
общий
это ответ
Здравствуйте, Aleksandrkib!
в
Составляем характеристическое уравнение: k^4+4k^2=0 -> k^2(k^2+4)=0 -> k1=k2=0, k3=-2i, k4=2i.
Общее решение однородного линейного уравнения с постоянными коэффициентами: y=C1+C2*x+C3*cos2x+C4*sin2x
в
Составляем характеристическое уравнение: k^4+4k^2=0 -> k^2(k^2+4)=0 -> k1=k2=0, k3=-2i, k4=2i.
Общее решение однородного линейного уравнения с постоянными коэффициентами: y=C1+C2*x+C3*cos2x+C4*sin2x
27.03.2012, 20:49
общий
это ответ
Здравствуйте, Aleksandrkib!
б) Положим y = ux. Тогда y' = u'x + u, а исходное уравнение принимает вид
откуда находим
Выражение (1) представляет собой общий интеграл заданного уравнения.
С уважением.
б) Положим y = ux. Тогда y' = u'x + u, а исходное уравнение принимает вид
u'x + u = u - 2 sin u,
откуда находим
u'x=-2 sin u,
xdu/dx = -2 sin u,
du/sin u = -2dx/x,
[$8747$]du/sin u = -2[$8747$]dx/x,
ln |tg (u/2)| = -2 ln |x| + ln |C|,
tg (u/2) = C/x2,
tg (y/(2x)) = C/x2. (1)
Выражение (1) представляет собой общий интеграл заданного уравнения.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
27.03.2012, 20:57
общий
это ответ
Здравствуйте, Aleksandrkib!
г
Общее решение однородного уравнения y''+16y=0 будет c1sin 4x+c2cos 4x
Частное решение неоднородного уравнения ищем в виде ax^2cos4x+bxsin4x
y''=-16ax^2cos 4x+16axsin4x+2acos4x-16bxsin4x-8bcos4x
y''+16y=16axsin4x+2acos4x-8bcos4x=xsin 4x
2a-8b=0
16a=1
a=1/16
b=1/64
y=c1sin 4x+c2cos 4x+1/16x^2cos4x+1/64xsin 4x
г
Общее решение однородного уравнения y''+16y=0 будет c1sin 4x+c2cos 4x
Частное решение неоднородного уравнения ищем в виде ax^2cos4x+bxsin4x
y''=-16ax^2cos 4x+16axsin4x+2acos4x-16bxsin4x-8bcos4x
y''+16y=16axsin4x+2acos4x-8bcos4x=xsin 4x
2a-8b=0
16a=1
a=1/16
b=1/64
y=c1sin 4x+c2cos 4x+1/16x^2cos4x+1/64xsin 4x
Форма ответа
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