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Консультация № 178955
06.06.2010, 20:01
0.00 руб.
0
1
1
Здравствуйте. Нужно решить уравнение.
Y”+9y=36e3x; y(0)=0, y’(0)=0
Y”+9y=36e3x; y(0)=0, y’(0)=0
Обсуждение
Неизвестный
06.06.2010, 20:42
общий
это ответ
Здравствуйте, Ананьев Рудольф Олегович.
y''+9*y=36*e3x
Решим однородное уравнение:
y0''+9*y0=0
характеристическое уравнение:
k2+9=0
k=[$177$]3*i
общее решение однородного уравнения
y0=C1*cos(3*x)+C2*sin(3*x)
Частное решение будем искать в виде:
y1=A*e3x
y1'=3*A*e3x
y1''=9*A*e3x
Подставим в уравнение
9*A*e3x+9*A*e3x=36*e3x
Следовательно A=2
Общее решение исходного уравнения:
y=y0+y1=C1*cos(3*x)+C2*sin(3*x)+2*e3x
y(0)=C1*1+C2*0+2*1=0 => C1= -2
y'=(-2*cos(3*x)+C2*sin(3*x)+2*e3x)'=6*sin(3*x)+3*C2*cos(3*x)+6*e3x
y'(0)=3*C2*1+6*1=0 => C2= -2
Получим решение:
y=2*(e3x-cos(3*x)-sin(3*x))
y''+9*y=36*e3x
Решим однородное уравнение:
y0''+9*y0=0
характеристическое уравнение:
k2+9=0
k=[$177$]3*i
общее решение однородного уравнения
y0=C1*cos(3*x)+C2*sin(3*x)
Частное решение будем искать в виде:
y1=A*e3x
y1'=3*A*e3x
y1''=9*A*e3x
Подставим в уравнение
9*A*e3x+9*A*e3x=36*e3x
Следовательно A=2
Общее решение исходного уравнения:
y=y0+y1=C1*cos(3*x)+C2*sin(3*x)+2*e3x
y(0)=C1*1+C2*0+2*1=0 => C1= -2
y'=(-2*cos(3*x)+C2*sin(3*x)+2*e3x)'=6*sin(3*x)+3*C2*cos(3*x)+6*e3x
y'(0)=3*C2*1+6*1=0 => C2= -2
Получим решение:
y=2*(e3x-cos(3*x)-sin(3*x))
5
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