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Консультация № 198516
11.05.2020, 09:02
0.00 руб.
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1
Здравствуйте! Прошу помощи в следующем вопросе:
Определить тип и решить дифференциальное уравнение:
eʸ dx+(cos y+xeʸ)dy=0.
Определить тип и решить дифференциальное уравнение:
eʸ dx+(cos y+xeʸ)dy=0.
Обсуждение
16.05.2020, 06:03
общий
это ответ
Здравствуйте, master87!
Дифференциальное уравнение вида
для которого выполняется условие
называется уравнением в полных дифференциалах. Его левая часть представляет собой полный дифференциал некоторой функции u(x, y), для которой
откуда
где [$966$] и [$968$] - некоторые функции.
В данном случае P(x, y) = e[sup]y[/sup], Q(x, y) = cos y + xe[sup]y[/sup],
то есть дифференциальное уравнение является уравнением в полных дифференциалах. Тогда
и
Сравнивая эти два выражения, легко определить, что [$966$](y) = sin y и [$968$](x) = 0, то есть u(x, y) = sin y + xe[sup]y[/sup], и решение дифференциального уравнения можно записать в виде общего интеграла:
откуда
Дифференциальное уравнение вида
для которого выполняется условие
называется уравнением в полных дифференциалах. Его левая часть представляет собой полный дифференциал некоторой функции u(x, y), для которой
откуда
где [$966$] и [$968$] - некоторые функции.
В данном случае P(x, y) = e[sup]y[/sup], Q(x, y) = cos y + xe[sup]y[/sup],
то есть дифференциальное уравнение является уравнением в полных дифференциалах. Тогда
и
Сравнивая эти два выражения, легко определить, что [$966$](y) = sin y и [$968$](x) = 0, то есть u(x, y) = sin y + xe[sup]y[/sup], и решение дифференциального уравнения можно записать в виде общего интеграла:
откуда
5
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