推论5.51
需积分: 0 189 浏览量
2022-08-08
21:36:08
上传
评论
收藏 31KB DOCX 举报
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
0
0
0
0
0 0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0 0 0
; , , , ; , , ,
, ; , , ,
, ,
, ; , , ,
1
, ; , , , , ; , , ,
t
t
t
t
t
t
t
t
t t x x f t t x d
f t t x
f t x d
t x x
f t t x
d
x x x
f t t x f t t x
d
x
j l t j l l t
t j l l
j j
l t
t j l l
j j
t
t j l l t j l l
j j
t
l l l
= +
¶
¶ ¶
= - +
¶ ¶ ¶
¶
¶ ¶
= +
¶ ¶ ¶
¶ ¶
¶ ¶
= +
¶ ¶ ¶ ¶
ò
ò
ò
ò
求解
( ) ( ) ( )
0
sin sin , 0
dz
tz t t t z t
dt
m m j j m j
= - - =
首先求解齐次方程
( ) ( ) ( ) ( )
, , , , in
dz
A t z t A t t t
dt
m j m j m m j
= = -，
解得齐次解为：
( ) ( )
(
)
0
= exp , ,
t
t
z t c A t dt
m j
ò
使用常数变异法，令
( ) ( ) ( )
(
)
0
= exp , ,
t
t
z t c t A t dt
m j
ò
带入非齐次方程得到
( ) ( ) ( )
(
)
( )
( )
(
)
0 0 0
sin exp in exp in
t t t
t t t
z t t t dt d t t dt
tj mtj m m j t m m j
= - -
ò ò ò
( )
( )
(
)
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
(
)
( ) ( ) ( ) ( )
(
)
( )
( )
(
)
( )
( ) ( ) ( )
(
)
( )
( )
(
)
0
0 0
0 0 0
0 0 0
0
0
0
exp , , , , c , ,
sin
exp , ,
exp , , exp , ,
0
exp , , exp , ,
t
t
t t
t t
t t t
t t t
t t t
t t t
dc t
A t dt c t A t z t t A t z t f t
dt
f t t t
c t c t f t A t dt d
z t c t f t A t dt d A t dt
z t
z t f t A t dt d A t dt
m j m j m j
j m j
m j t
m j t m j
m j t m j
ü
+ = +
ï
Þ
ý
ï
= -
þ
= + - Þ
ü
= + -
ï
Þ
ý
ï
=
þ
= -
ò
ò ò
ò ò ò
ò ò ò
即
( ) ( )
(
)
( ) ( )
(
)
( )
0 0 0
exp in sin exp in
t t
t t t
z t d t t dt d
t
mt mtj t tj mtj m m j t
= - -
ò ò ò
评论0