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Консультация № 201171
16.06.2021, 11:58
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
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Обсуждение
16.06.2021, 23:59
общий
это ответ
Здравствуйте, ushatalal!
Пусть требуется решить дифференциальное уравнение
Рассмотрим сначала однородное уравнение
Корнями его характеристического уравнения
являются числа 
и 
Поэтому общее решение заданного уравнения -- функция
Чтобы определить частное решение заданного уравнения. воспользуемся методом вариации произвольных постоянных:
где
-- фундаментальная система решений однородного уравнения, а 
-- решения системы дифференциальных уравнений
или 
Из второго уравнения системы получим, что
а после подстановки этого выражения в первое уравнение получим, что
Тогда
(постоянную интегрирования опускаем),
(постоянную интегрирования опускаем)
-- частное решение заданного уравнения;
-- общее решение заданного уравнения.
Пусть требуется решить дифференциальное уравнение
Рассмотрим сначала однородное уравнение
Корнями его характеристического уравнения
являются числа и Поэтому общее решение заданного уравнения -- функцияЧтобы определить частное решение заданного уравнения. воспользуемся методом вариации произвольных постоянных:
где
-- фундаментальная система решений однородного уравнения, а -- решения системы дифференциальных уравнений или Из второго уравнения системы получим, что
а после подстановки этого выражения в первое уравнение получим, что
Тогда
(постоянную интегрирования опускаем),
(постоянную интегрирования опускаем; вычисление интеграла смотрите здесь: 
;
(постоянную интегрирования опускаем; вычисление интеграла смотрите здесь: (постоянную интегрирования опускаем)-- частное решение заданного уравнения;
-- общее решение заданного уравнения.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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