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Консультация № 185781
08.04.2012, 12:30
62.88 руб.
0
1
1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найти частное решение дифференциального уравнения:
y'' - 2y' + 2y = 4*e^x*cos x
y(пи) = пи*e^пи, y' (пи) = e^пи.
Найти частное решение дифференциального уравнения:
y'' - 2y' + 2y = 4*e^x*cos x
y(пи) = пи*e^пи, y' (пи) = e^пи.
Обсуждение
08.04.2012, 16:07
общий
это ответ
Здравствуйте, Aleksandrkib!
Решением линейного неоднородного дифференциального уравнения является сумма общего решения соответствующего однородного дифференциального уравнения и частного решения данного неоднородного.
Для однородного уравнения
запишем соответствующее характеристическое уравнение
Оно имеет пару комплексно-сопряжённых корней k[sub]1,2[/sub] = 1[$177$]i. Следовательно, общее решение однородного уравнения будет
Правой части неоднородного уравнения 4e[sup]x[/sup]cos x соответствуют значения 1[$177$]i, как раз являющиеся корнями характеристического уравнения кратности 1. Поэтому частное решение неоднородного уравнения ищем в форме с резонансным сомножителем степени 1:
Тогда
откуда -2B = 0 и 2A = 4, то есть частное решение имеет вид
а общим решением исходного уравнения будет
Частное решение находим, использовав начальные условия:
откуда C[sub]2[/sub] = -[$960$], C[sub]1[/sub] = -1-[$960$] и
Решением линейного неоднородного дифференциального уравнения является сумма общего решения соответствующего однородного дифференциального уравнения и частного решения данного неоднородного.
Для однородного уравнения
запишем соответствующее характеристическое уравнение
Оно имеет пару комплексно-сопряжённых корней k[sub]1,2[/sub] = 1[$177$]i. Следовательно, общее решение однородного уравнения будет
Правой части неоднородного уравнения 4e[sup]x[/sup]cos x соответствуют значения 1[$177$]i, как раз являющиеся корнями характеристического уравнения кратности 1. Поэтому частное решение неоднородного уравнения ищем в форме с резонансным сомножителем степени 1:
Тогда
откуда -2B = 0 и 2A = 4, то есть частное решение имеет вид
а общим решением исходного уравнения будет
Частное решение находим, использовав начальные условия:
откуда C[sub]2[/sub] = -[$960$], C[sub]1[/sub] = -1-[$960$] и
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