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Консультация № 187608
04.11.2013, 14:29
88.90 руб.
0
3
1
Здравствуйте! Прошу помощи в следующем вопросе:
Найти частное решение разностного уравнения
y_(n+2)-y_(n+1)-12y_n=(2n+3)*2^n
Удовлетворяющее начальным условиям y_1=0,y_2=0
Найти частное решение разностного уравнения
y_(n+2)-y_(n+1)-12y_n=(2n+3)*2^n
Удовлетворяющее начальным условиям y_1=0,y_2=0
Обсуждение
Неизвестный
04.11.2013, 18:28
общий
Адресаты:
Что значит "найти частное решение"? Из самого уравнения y_(n+2)-y_(n+1)-12y_n=(2n+3)*2^n и начальных условий получается рекуррентное соотношение позволяющее последовательно найти все члены последовательности:
y_(n+2) = y_(n+1) +12y_n + (2n+3)*2^n
y_1=0,y_2=0.
Это не является решением?
y_(n+2) = y_(n+1) +12y_n + (2n+3)*2^n
y_1=0,y_2=0.
Это не является решением?
04.11.2013, 18:43
общий
это ответ
Здравствуйте, Кириллова Анна Витальевна!
Разностное уравнение с постоянными коэффициентами решается по той же схеме, что и дифференциальное. Сначала решаем однородное уравнение
yn+2-yn+1-12yn=0
Для этого составляем характеристическое уравнение
[$955$]2-[$955$]-12=0
его корни [$955$]=4 и [$955$]=-3
Общее решение однородного уравнения
yn=С14n+C2(-3)n
Частное решение неоднородного ищем в виде yn=(An+B)2n
Подставляя в уравнение, получаем
(-10An+6A-10B)2n=(2n+3)2n
Сравнивая коэффициенты справа и слева, получаем систему
-10A=2
6A-10B=3
Решая систему, находим A=-1/5;B=-21/50
Общее решение разностного уравнения
yn=С14n+C2(-3)n-(10n+21)2n/50
Далее используем начальные данные
y1=4C1-3C2-(31/25)=0
y2=16C1+9C2-(82/25)=0
Решая систему, получаем
C1=1/4
C2=-2/25
Ответ:
yn=(1/4)4n-(2/25)(-3)n-(10n+21)2n/50
Разностное уравнение с постоянными коэффициентами решается по той же схеме, что и дифференциальное. Сначала решаем однородное уравнение
yn+2-yn+1-12yn=0
Для этого составляем характеристическое уравнение
[$955$]2-[$955$]-12=0
его корни [$955$]=4 и [$955$]=-3
Общее решение однородного уравнения
yn=С14n+C2(-3)n
Частное решение неоднородного ищем в виде yn=(An+B)2n
Подставляя в уравнение, получаем
(-10An+6A-10B)2n=(2n+3)2n
Сравнивая коэффициенты справа и слева, получаем систему
-10A=2
6A-10B=3
Решая систему, находим A=-1/5;B=-21/50
Общее решение разностного уравнения
yn=С14n+C2(-3)n-(10n+21)2n/50
Далее используем начальные данные
y1=4C1-3C2-(31/25)=0
y2=16C1+9C2-(82/25)=0
Решая систему, получаем
C1=1/4
C2=-2/25
Ответ:
yn=(1/4)4n-(2/25)(-3)n-(10n+21)2n/50
5
Спасибо вам большое
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