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Консультация № 175054
09.12.2009, 04:31
0.00 руб.
10.12.2009, 00:52
0
5
1
Здравствуйте! Помогите пожалуйста решить задачку! Спасибо заранее
Решить дифференциальное уравнение и найти частное решение, удовлетворяющее условию y=4 при x=0. (1-x^2)dy+xydx=0
Решить дифференциальное уравнение и найти частное решение, удовлетворяющее условию y=4 при x=0. (1-x^2)dy+xydx=0
Обсуждение
Неизвестный
09.12.2009, 14:23
общий
Тихомирова Настя:
Добрый день! По-моему, у Вас ошибка в задании. Должно быть или (1-x^2)dx+xydy=0 или (1-x^2)dy+xydx=0 или ...
Добрый день! По-моему, у Вас ошибка в задании. Должно быть или (1-x^2)dx+xydy=0 или (1-x^2)dy+xydx=0 или ...
Неизвестный
09.12.2009, 15:32
общий
Добрый день! Извиняюсь... Неправильно переписала (1-x^2)dy+xydx=0 Вот так правильно...
Неизвестный
09.12.2009, 17:47
общий
это ответ
Здравствуйте, Тихомирова Настя.
Это дифференциальное уравнение первого порядка с разделяющимися переменными.
Преобразуем:
dy/y=xdx/(x^2-1)=d(x^2-1)/2(x^2-1).
Интегрируем обе части:
lny=ln(x^2-1)/2+c.
Потенцируем:
y=c'*sqrt(x^2-1).
Находим с' из начальных условий:
y(0)=-c'=4, отсюда c'=-4.
Ответ:
y=4*(1-x^2).
Это дифференциальное уравнение первого порядка с разделяющимися переменными.
Преобразуем:
dy/y=xdx/(x^2-1)=d(x^2-1)/2(x^2-1).
Интегрируем обе части:
lny=ln(x^2-1)/2+c.
Потенцируем:
y=c'*sqrt(x^2-1).
Находим с' из начальных условий:
y(0)=-c'=4, отсюда c'=-4.
Ответ:
y=4*(1-x^2).
Неизвестный
09.12.2009, 19:32
общий
schurik:
Лихо Вы корень из -1 нашли! Должны быть не lny и ln(x^2-1), а ln|y| и ln|x^2-1|, да и дальше как-то sqrt "пропал"
Лихо Вы корень из -1 нашли! Должны быть не lny и ln(x^2-1), а ln|y| и ln|x^2-1|, да и дальше как-то sqrt "пропал"
Неизвестный
10.12.2009, 13:18
общий
Тихомирова Настя:
Извиняюсь, еще не научился печатать формулы в одну строку. Исправленное решение:
Преобразуем:
dy/y=-xdx/(1-x^2)=d(1-x^2)/(2*(1-x^2)).
Интегрируем обе части:
lny=ln(x^2-1)/2+c.
Потенцируем:
y=c'*sqrt(1-x^2).
Находим с' из начальных условий:
y(0)=c'=4, отсюда c'=4.
Ответ:
y=4*sqrt(1-x^2).
Извиняюсь, еще не научился печатать формулы в одну строку. Исправленное решение:
Преобразуем:
dy/y=-xdx/(1-x^2)=d(1-x^2)/(2*(1-x^2)).
Интегрируем обе части:
lny=ln(x^2-1)/2+c.
Потенцируем:
y=c'*sqrt(1-x^2).
Находим с' из начальных условий:
y(0)=c'=4, отсюда c'=4.
Ответ:
y=4*sqrt(1-x^2).
Форма ответа
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