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Консультация № 169758
22.06.2009, 22:21
0.00 руб.
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помогите решить дифференциальное уравнение:
Найти общее решение дифференциального уравнения y'' + y = 2cos3x - 3sin3x
Найти общее решение дифференциального уравнения y'' + y = 2cos3x - 3sin3x
Обсуждение
Неизвестный
23.06.2009, 10:24
общий
это ответ
Здравствуйте, Павел!Ы!.
Решение неоднородного уравнения состоит
y=Y+y_, где y-общее решение неоднородного уравнения
Y-общее решение однородного уравнения, соотв. данному неоднородному
y_-частное решение неоднородного уравнения
y''+y=0 - однородное уравнение, соотв. данному неоднородному
этому уравнению соотв. характеристическое уравнение
k^2+1=0
корни характеристического уравнения k=+-i, где +- это плюс минус, i- комплексная единица
отсюда общее решение однородного уравнения имеет вид
Y=C1*cos(x)+C2*sin(x), C1,C2- произвольные постоянные
Частное решение неоднородного уравнения будем искать в виде
y_=A*sin(3x)+B*cos(3x)
тогда
y_'=3A*cos(3x)-3B*sin(3x)
y_''=-9A*sin(3x)-9B*cos(3x)
отсюда
y_''+y_=-9A*sin(3x)-9B*cos(3x)+A*sin(3x)+B*cos(3x)=-8A*sin(3x)-8B*cos(3x)=2cos3x - 3sin3x
значит -3=-8A; 2=-8B
A=3/8; B=-1/4
следовательно, частное решение неоднородного уравнения
y_=3/8*sin(3x)-1/4*cos(3x)
Общее решение неоднородного уравнения
y=C1*cos(x)+C2*sin(x)+3/8*sin(3x)-1/4*cos(3x)
Решение неоднородного уравнения состоит
y=Y+y_, где y-общее решение неоднородного уравнения
Y-общее решение однородного уравнения, соотв. данному неоднородному
y_-частное решение неоднородного уравнения
y''+y=0 - однородное уравнение, соотв. данному неоднородному
этому уравнению соотв. характеристическое уравнение
k^2+1=0
корни характеристического уравнения k=+-i, где +- это плюс минус, i- комплексная единица
отсюда общее решение однородного уравнения имеет вид
Y=C1*cos(x)+C2*sin(x), C1,C2- произвольные постоянные
Частное решение неоднородного уравнения будем искать в виде
y_=A*sin(3x)+B*cos(3x)
тогда
y_'=3A*cos(3x)-3B*sin(3x)
y_''=-9A*sin(3x)-9B*cos(3x)
отсюда
y_''+y_=-9A*sin(3x)-9B*cos(3x)+A*sin(3x)+B*cos(3x)=-8A*sin(3x)-8B*cos(3x)=2cos3x - 3sin3x
значит -3=-8A; 2=-8B
A=3/8; B=-1/4
следовательно, частное решение неоднородного уравнения
y_=3/8*sin(3x)-1/4*cos(3x)
Общее решение неоднородного уравнения
y=C1*cos(x)+C2*sin(x)+3/8*sin(3x)-1/4*cos(3x)
Форма ответа
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