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Задать вопрос экспертам

Консультация № 185121

09.01.2012, 09:06

51.74 руб.

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Образование

Здравствуйте, уважаемые эксперты! Прошу помощи в следующем вопросе: Найти частное решение дифференциального уравнения: y'' - 4y' + 13y = 26x +5, удовлетворяющее начальным условиям y(0) = 1, y'(0) = 0.
Спасибо.

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

09.01.2012, 11:07

общий

это ответ

Здравствуйте, lamed!
Решаем однородное уравнение. Характеристическое уравнение
[$955$]²-4[$955$]+13=0
[$955$]=2[$177$]3i
Общее решение однородного уравнения: y=C₁e^2xcos3x+C₂e^2xsin3x
Частное решение ищем в виде y=Ax+B. Подставляя в уравнение, получаем
-4A+13Ax+13B=26x+5

13A=26
-4A+13B=5

A=2; B=1

Общее решение исходного уравнения:
y=C₁e^2xcos3x+C₂e^2xsin3x+2x+1

Подставляя в начальные данные, получаем
С₁+1=1
2С₁+3С₂+2=0

С₁=0; С₂=-2/3

Ответ:
y=-(4/3)e^2xsin3x+2x+1

5

Большое спасибо! С уважением.

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