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Консультация № 197738
10.02.2020, 15:10
0.00 руб.
11.02.2020, 18:50
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Здравствуйте! Прошу помощи в следующем вопросе:
Решить дифференциальное уравнение первого порядка.
х3у'-у2+2у-10=0
Решить дифференциальное уравнение первого порядка.
х3у'-у2+2у-10=0
Прикрепленные файлы:
Обсуждение
15.02.2020, 10:47
общий
Адресаты:
Краткий Ответ : y = 1 + 3·tg(C - 1,5 / x2)
Через пару часов я отправлю полный Ответ с мат-выкладками.
Через пару часов я отправлю полный Ответ с мат-выкладками.
15.02.2020, 12:35
общий
это ответ
Здравствуйте, Гаяна!
Дано : Решить дифференциальное уравнение х3·у' - у2 + 2·у - 10 = 0
Решение : Разделяем переменные : все "игреки" группируем в левую часть уравнения, все "иксы" - в правую :
х3·у' = у2 - 2·у + 10
у'/(у2 - 2·у + 10) = 1/х3
Переписываем производную в "диффурный" вид (y' ==> dy/dx согласно рекомендациям статьи "Как быстро освоить высшую математику?" mathprofi.com/knigi_i_kursy ) :
dу/(у2 - 2·у + 10) = dx/х3
Из левого знаменателя выделяем полный квадрат:
у2 - 2·у + 10 = (у2 - 2·у + 1) + 9 = (у-1)2 + 32
Заменяем dy на dy-1 и интегрируем левую и правую части уравнения отдельно.
К первообразной приписываем константу : arctg[(y-1)/3] / 3 = -1 / (2·x2) + C1
Умножаем обе части уравнения на 3 : arctg[(y-1)/3] = -3 / (2·x2) + C
Тангенцируем : (y-1)/3 = tg(C - 1,5 / x2)
Умножаем на 3 : y-1 = 3·tg(C - 1,5 / x2)
Ответ : y = 1 + 3·tg(C - 1,5 / x2)
Проверяем в Маткаде (скриншот прилагаю)
Проверка успешна!
Дано : Решить дифференциальное уравнение х3·у' - у2 + 2·у - 10 = 0
Решение : Разделяем переменные : все "игреки" группируем в левую часть уравнения, все "иксы" - в правую :
х3·у' = у2 - 2·у + 10
у'/(у2 - 2·у + 10) = 1/х3
Переписываем производную в "диффурный" вид (y' ==> dy/dx согласно рекомендациям статьи "Как быстро освоить высшую математику?" mathprofi.com/knigi_i_kursy ) :
dу/(у2 - 2·у + 10) = dx/х3
Из левого знаменателя выделяем полный квадрат:
у2 - 2·у + 10 = (у2 - 2·у + 1) + 9 = (у-1)2 + 32
Заменяем dy на dy-1 и интегрируем левую и правую части уравнения отдельно.
К первообразной приписываем константу : arctg[(y-1)/3] / 3 = -1 / (2·x2) + C1
Умножаем обе части уравнения на 3 : arctg[(y-1)/3] = -3 / (2·x2) + C
Тангенцируем : (y-1)/3 = tg(C - 1,5 / x2)
Умножаем на 3 : y-1 = 3·tg(C - 1,5 / x2)
Ответ : y = 1 + 3·tg(C - 1,5 / x2)
Проверяем в Маткаде (скриншот прилагаю)
Проверка успешна!
5
Спасибо за такой развернутый ответ)
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