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Консультация № 198514
11.05.2020, 09:01
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Здравствуйте! У меня возникли сложности с таким вопросом:
Как определить тип и решить дифференциальное уравнение:
(x-y) ydx - x²dy=0.
Как определить тип и решить дифференциальное уравнение:
(x-y) ydx - x²dy=0.
Обсуждение
16.05.2020, 05:32
общий
это ответ
Здравствуйте, master87!
Дифференциальное уравнение y' = f(x, y), которое можно преобразовать к виду y' = f(y/x) или y' = f(x/y) (где функция зависит не от переменных x и y, а только от их отношения), называется однородным и решается путём введения новой переменной u = y/x, при котором y = ux и y' = u'x + u.
В данном случае, разделив исходное уравнение на x[sup]2[/sup]dx, получаем
или
то есть однородное уравнение. После замены u = y/x оно примет вид
откуда
или
Интегрируя, получаем
откуда, с учётом обратной замены y = ux,
Дифференциальное уравнение y' = f(x, y), которое можно преобразовать к виду y' = f(y/x) или y' = f(x/y) (где функция зависит не от переменных x и y, а только от их отношения), называется однородным и решается путём введения новой переменной u = y/x, при котором y = ux и y' = u'x + u.
В данном случае, разделив исходное уравнение на x[sup]2[/sup]dx, получаем
или
то есть однородное уравнение. После замены u = y/x оно примет вид
откуда
или
Интегрируя, получаем
откуда, с учётом обратной замены y = ux,
5
Спасибо
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