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Консультация № 164089
04.04.2009, 09:55
0.00 руб.
0
1
1
Найти частное решение дифференциального уравнения y''+py'+qy=f(x) , удовлетворяющее начальным условиям y(0)=y_0, y'(0)=y'_0
y''-4y'+13y=26x+8,y(0)=1;y' (0)=0
y''-4y'+13y=26x+8,y(0)=1;y' (0)=0
Обсуждение
Неизвестный
05.04.2009, 01:44
общий
это ответ
Здравствуйте, Hellphoenix!
Решение состоит из 2 частей : у1 и у2 , соответственно , по виду левой и правой частей равенства .
Из левой части делаем характерестическое уравнение и находим его корни : y->1 , y'->k , y"->k^2 .
(k^2)-4*k+13=0
D=(b^2)-4*a*c=16-4*13=16-52=-36 ; k1,2=(-b+-sqrtD)/2a .
k1=2+3i ; k2=2-3i .
y1=(e^2x)*(C1*cos3x+C2*sin3x) .
y2=(x^r)*(e^(alfa*x))*(P(m)*cos(betta*x)+Q(m)*sin(betta*x)) .
alfa=0=betta => r=0 .
{ y2=A*x+B , (y2)'=A , (y2)"=0 } => (y2)"-4*(y2)'+13*(y2)=26x+8
-4*A+13*A*x+13*B=26*x+8
{ A=2 , B=16/13 } => y2=2*x+(16/13) .
OTBET : Y(x)=2*x+(16/13)+(e^2x)*(C1*cos3x+C2*sin3x) .
Решение состоит из 2 частей : у1 и у2 , соответственно , по виду левой и правой частей равенства .
Из левой части делаем характерестическое уравнение и находим его корни : y->1 , y'->k , y"->k^2 .
(k^2)-4*k+13=0
D=(b^2)-4*a*c=16-4*13=16-52=-36 ; k1,2=(-b+-sqrtD)/2a .
k1=2+3i ; k2=2-3i .
y1=(e^2x)*(C1*cos3x+C2*sin3x) .
y2=(x^r)*(e^(alfa*x))*(P(m)*cos(betta*x)+Q(m)*sin(betta*x)) .
alfa=0=betta => r=0 .
{ y2=A*x+B , (y2)'=A , (y2)"=0 } => (y2)"-4*(y2)'+13*(y2)=26x+8
-4*A+13*A*x+13*B=26*x+8
{ A=2 , B=16/13 } => y2=2*x+(16/13) .
OTBET : Y(x)=2*x+(16/13)+(e^2x)*(C1*cos3x+C2*sin3x) .
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