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Консультация № 182433
09.03.2011, 12:20
53.02 руб.
0
3
1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти частное решение диф. уравнения:
y" + 5 y' + 6y = 12 cos(2x)
удовлетворяющее начальным условиям
y(0) = 1, y'(0) = 3
Найти частное решение диф. уравнения:
y" + 5 y' + 6y = 12 cos(2x)
удовлетворяющее начальным условиям
y(0) = 1, y'(0) = 3
Обсуждение
Неизвестный
09.03.2011, 15:15
общий
это ответ
Здравствуйте, Ulitka71!
Найдем корни однородного характеристического уравнения
K2+5k+6=0
k1=-3
k2=-2
Решение соответствующего однородного уравнения имеет вид:
u= C1e-2x+C2e-3x
Общее решение неоднородного уравнения будем искать в виде
y1=A cos 2x+B sin 2x
Найдем y" и y'
y1'=-2A sin 2x+2B cos 2x
y1"=-4Acos2x-4B sin 2x
Подставляем полученные y1, y1" и y1' в исходное уравнение
-4Acos2x-4B sin 2x+5(-2A sin 2x+2B cos 2x)+6(A cos 2x+B sin 2x)=12 cos 2x
Приравниваем коэффициенты одинаковых тригонометрических функций
cos 2x: -4A+10B+6A=12
sin 2x: -4B-10A+6B=0
Решим систему уравнений
2A+10B=12
2B-10A=0
B=5A
2A+50A=12
A=3/13
B=15/13
Общее решения уравнения имеет вид
y= C1e-2x+C2e-3x+3/13cos 2x+15/13 sin 2x
Найдем частное решение уравнения
Определим постоянные интегрирования
y'=-2 C1e-2x-3 C2e-3x-6/13 sin 2x+30/13 cos 2x
y= C1e-2x+C2e-3x+3/13cos 2x+15/13 sin 2x
3=-2С1-3С2+30/13
1= C1+ C2+3/13
C1=3, C2=-29/13
Частное решение имеет вид:
y= 3e-2x+C2e-3x-29/13cos 2x+15/13 sin 2x
Найдем корни однородного характеристического уравнения
K2+5k+6=0
k1=-3
k2=-2
Решение соответствующего однородного уравнения имеет вид:
u= C1e-2x+C2e-3x
Общее решение неоднородного уравнения будем искать в виде
y1=A cos 2x+B sin 2x
Найдем y" и y'
y1'=-2A sin 2x+2B cos 2x
y1"=-4Acos2x-4B sin 2x
Подставляем полученные y1, y1" и y1' в исходное уравнение
-4Acos2x-4B sin 2x+5(-2A sin 2x+2B cos 2x)+6(A cos 2x+B sin 2x)=12 cos 2x
Приравниваем коэффициенты одинаковых тригонометрических функций
cos 2x: -4A+10B+6A=12
sin 2x: -4B-10A+6B=0
Решим систему уравнений
2A+10B=12
2B-10A=0
B=5A
2A+50A=12
A=3/13
B=15/13
Общее решения уравнения имеет вид
y= C1e-2x+C2e-3x+3/13cos 2x+15/13 sin 2x
Найдем частное решение уравнения
Определим постоянные интегрирования
y'=-2 C1e-2x-3 C2e-3x-6/13 sin 2x+30/13 cos 2x
y= C1e-2x+C2e-3x+3/13cos 2x+15/13 sin 2x
3=-2С1-3С2+30/13
1= C1+ C2+3/13
C1=3, C2=-29/13
Частное решение имеет вид:
y= 3e-2x+C2e-3x-29/13cos 2x+15/13 sin 2x
5
Форма ответа
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