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Задать вопрос экспертам

Консультация № 178958

06.06.2010, 20:46

0.00 руб.

0 2 1

Образование

Здравствуйте. Нужно решить уравнение.
Y’’=2(1+y’)1/2; y(0)=5, y’(0)=-1

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

07.06.2010, 20:58

общий

это ответ

Здравствуйте, Ананьев Рудольф Олегович.
1) Делаем замену z=1+y', получаем уравнение z'=2[$8730$]z. Это уравнение с разделяющимися переменными:
dz/(2[$8730$]z)=dx, [$8730$]z=x+C₁, z=(x+C₁)². Возвращаясь к y, получаем
y'=(x+C₁)²-1.
Интегируя, находим общее решение
y=(1/3)(x+C₁)³-x+C₂

2) Подставляя начальные условия, имеем
y(0)=(1/3)C₁³+C₂=5
y'(0)=C₁²-1=-1
Отсюда находим C₁=0, C₂=5

Ответ: y=(1/3)x³-x+5

5

Неизвестный

07.06.2010, 23:53

общий

Ананьев Рудольф Олегович:
Здравствуйте!

Это уравнение имеет также второе решение: y=5-x, при котором обе части уравнения равны 0.

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