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Консультация № 187872
07.06.2014, 08:19
78.80 руб.
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1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
http://dfiles.ru/files/bxev77wwo
Задача 3.
http://dfiles.ru/files/bxev77wwo
Задача 3.
Обсуждение
07.06.2014, 23:03
общий
Адресаты:
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07.06.2014, 23:46
общий
это ответ
Здравствуйте, Aleksandrkib!
(y')2=y''-1; y(0)=1,y'(0)=0.
Сделаем замену y'=z(x), тогда y''=z'(x).
Уравнение примет вид:
z2=z'-1,
z2+1=dz/dx,
dz/(z2+1)=dx.
Интегрируя, находим: arctgz=x+C1 [$8658$] z=tg(x+C1).
Далее y'=tg(x+C1) [$8658$] y=[$8747$]tg(x+C1)dx=[$8747$]sin(x+C1)dx/cos(x+C1)=
=-[$8747$]d(cos(x+C1))/cos(x+C1)=-ln|cos(x+C1)|+C2.
Определим С1, С2 из начальных условий:
y'(0)=0 [$8658$] tgС1=0 [$8658$] С1=0.
y(0)=1 [$8658$] -ln|cos0|+C2=1 [$8658$] С2=1.
Искомое частное решение: y=-ln|cosx|+1.
(y')2=y''-1; y(0)=1,y'(0)=0.
Сделаем замену y'=z(x), тогда y''=z'(x).
Уравнение примет вид:
z2=z'-1,
z2+1=dz/dx,
dz/(z2+1)=dx.
Интегрируя, находим: arctgz=x+C1 [$8658$] z=tg(x+C1).
Далее y'=tg(x+C1) [$8658$] y=[$8747$]tg(x+C1)dx=[$8747$]sin(x+C1)dx/cos(x+C1)=
=-[$8747$]d(cos(x+C1))/cos(x+C1)=-ln|cos(x+C1)|+C2.
Определим С1, С2 из начальных условий:
y'(0)=0 [$8658$] tgС1=0 [$8658$] С1=0.
y(0)=1 [$8658$] -ln|cos0|+C2=1 [$8658$] С2=1.
Искомое частное решение: y=-ln|cosx|+1.
5
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