常微分方程教程(丁同仁、李承治第二版)
2t 112t
dt 2t2 t 2 2t2 t 2dt2t2 t 2
1dt12tdt 222(t 2(t 4) 164) 16
2
) 171d[(t 11dt1dt] 222171717
2(t 14 (t 12 (t 1
) ) )
11171dt
ln(t ) 217
24164(t 1
)
1dt1
2 17
4 (t 14) (t 4164
dt
4)(t 4
44
4
)
4
121
(
4 t 4 12ln|t 1 t
1
4
|
1t 4
)dt
t 117
故 2t2dt 22ln(t ) 16 t 2
2
4
2ln|
t 1 t
1
44
1
|.
v e
C
1
t 4
4
t 4 4
(
t 4 t 4
)2
121 121 21 12
C(t )(t )
4444
令 4
1
2
4, 4
144
11
4
,v C(t )
1
44
114
1
1
22
1
,
v C(t )(t )
u
C( )4
v
14
u
( )4v
C
(u v)v
1
44
1
(u v)v
1
44
1
11
4411
44 1
44
11
1
44
1
故
1
1 C(u v)
(u v)
1 44 1
(u v)
C(u v) C(u v) C(u v)
1 44 141 44
(u v)4(u v)
1 44
(u v) C(u v)