1
53.1
2.
!
A =
2
4
2t3 +1 t2
t 4t2
3
5;
k9
i1? ?
￥?
T ?
T￥?
.
3:^?
dA
dt =
2
4
6t2 2t
1 8t
3
5;
yN ?
￥?
T148t3?2t.
6BZ
, V p
detA = 8t5 +4t2?t3;
# ?
T￥?
140t4?3t2 +8t.
3.
!x(t)
^ uWfi? t? fl
￥ ??f
, O?fi? t? fl
H,
jx(t)j? L+M
Z t
fi
jx(?)jd?;
?L,M
^dμè
.
k¨??/íE￡
ü
jx(t)j? LeM(t?fi); 8fi? t? fl:
￡
ü:?
>B??, V
!x(t)? 0.?L
! V?i uWfi? t? fl
￥ ??f
r(t)? 0
P¤
x(t) = r(t)+L+M
Z t
fi
x(?)d?:
/f
?
xk(t)g ?/
x
0(t) = r(t);
xk(t) = r(t)+L+MRtfi xk?1(?)d?;
?k = 1;2;:::. ?^B
,1￡
ü, ?i1 ?
k,f
xk(t) uW[fi;fl]
μ?li O ??.
?
DB
,E V￡
ü
jxk(t)?xk?1(t)j? LM
k?1
(k?1)!(t?fi)
k?1; fi? t? fl:
¨1
q
YE?)
1X
k=1
Mk
k! (t?fi)
k
uWfi? t? fl
^Bá
l ?￥.yN)
x0(t)+
1X
j=1
(xj(t)?xj?1(t)); fi? t? fl
uWfi? t? fl
^Bá
l ?￥,V7f
?
fxk(t)g uWfi? t? fl
^Bá
l ?￥,i O
?^ A?
fxk(t)gBá
l ??x(t)g.
2
6BZ
, ?i1 ?
n,?
￥9,á
ìμ
0? xn+1(t)?r(t) =
n+1X
k=1
(xk(t)?xk?1(t))
n+1X
k=1
LMk?1
(k?1)!(t?fi)
k?1? LeM(t?fi):
V7
xn+1(t)? LeM(t?fi) +r(t)? LeM(t?fi):
?N?8fi? t? fl,x(t)? LeM(t?fi).
53.2
3. ?T/
?_
f
2
4
sint
cost
3
5;
2
4
cost
sint
3
5
1
Q±sZ?F
dx
dt =
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5x
￥'3F,
k paij(t); i;j = 1;2.
3:^?
d
dt
2
4
sint cost
cost?sint
3
5=
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5
2
4
sint cost
cost?sint
3
5;
yN
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5=
2
4
cost?sint
sint?cost
3
5
2
4
sint cost
cost?sint
3
5
1;
?N V pa11(t) = a22(t) = 0,a12(t) = 1,a21(t) =?1.
4. ?¨3￥i ?B?(? ?1.1 )￡
ü? ?2.2￥?B?s.
￡
ü: s?
!Z?Fdxdt = A(t)x￥3Ffxk(t), k = 1;2;:::;ng￥Wronsky?
TdetX(t)
?t = t0 2 [fi;fl]) |′?1
,,5fxk(t0), k = 1;2;:::;ngL?í1,?? ?2.1￥￡
ü V?3
Ffxk(t), k = 1;2;:::;ngL?í1.
A1?¨Q￡E.
!Z?Fdxdt = A(t)x￥3Ffxk(t), k = 1;2;:::;ngL?í1?
uWt 2 [fi;fl]
￥Wronsky?
TdetX(t) · 0. uWt 2 [fi;fl]
|?t0 2 [fi;fl],5
fxk(t0), k = 1;2;:::;ngL?M1,'i ? ?1
,￥è
C1,...,Cn
P¤
C1x1(t0)+C2x2(t0)+￠￠￠+Cnxn(t0) = 0:
A ?C1x1(t) + C2x2(t) + ￠￠￠ + Cnxn(t)x(t) · 0?
^Z?Fdxdt = A(t)x￥
@′Hq
x(t0) = 0￥3,yN?3￥i ?B?? ??Aμ
C1x1(t)+C2x2(t)+￠￠￠+Cnxn(t) · 0:
#3Ffxk(t), k = 1;2;:::;ngL?M1,??DL
!
±.
53.1
2.
!
A =
2
4
2t3 +1 t2
t 4t2
3
5;
k9
i1? ?
￥?
T ?
T￥?
.
3:^?
dA
dt =
2
4
6t2 2t
1 8t
3
5;
yN ?
￥?
T148t3?2t.
6BZ
, V p
detA = 8t5 +4t2?t3;
# ?
T￥?
140t4?3t2 +8t.
3.
!x(t)
^ uWfi? t? fl
￥ ??f
, O?fi? t? fl
H,
jx(t)j? L+M
Z t
fi
jx(?)jd?;
?L,M
^dμè
.
k¨??/íE￡
ü
jx(t)j? LeM(t?fi); 8fi? t? fl:
￡
ü:?
>B??, V
!x(t)? 0.?L
! V?i uWfi? t? fl
￥ ??f
r(t)? 0
P¤
x(t) = r(t)+L+M
Z t
fi
x(?)d?:
/f
?
xk(t)g ?/
x
0(t) = r(t);
xk(t) = r(t)+L+MRtfi xk?1(?)d?;
?k = 1;2;:::. ?^B
,1￡
ü, ?i1 ?
k,f
xk(t) uW[fi;fl]
μ?li O ??.
?
DB
,E V￡
ü
jxk(t)?xk?1(t)j? LM
k?1
(k?1)!(t?fi)
k?1; fi? t? fl:
¨1
q
YE?)
1X
k=1
Mk
k! (t?fi)
k
uWfi? t? fl
^Bá
l ?￥.yN)
x0(t)+
1X
j=1
(xj(t)?xj?1(t)); fi? t? fl
uWfi? t? fl
^Bá
l ?￥,V7f
?
fxk(t)g uWfi? t? fl
^Bá
l ?￥,i O
?^ A?
fxk(t)gBá
l ??x(t)g.
2
6BZ
, ?i1 ?
n,?
￥9,á
ìμ
0? xn+1(t)?r(t) =
n+1X
k=1
(xk(t)?xk?1(t))
n+1X
k=1
LMk?1
(k?1)!(t?fi)
k?1? LeM(t?fi):
V7
xn+1(t)? LeM(t?fi) +r(t)? LeM(t?fi):
?N?8fi? t? fl,x(t)? LeM(t?fi).
53.2
3. ?T/
?_
f
2
4
sint
cost
3
5;
2
4
cost
sint
3
5
1
Q±sZ?F
dx
dt =
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5x
￥'3F,
k paij(t); i;j = 1;2.
3:^?
d
dt
2
4
sint cost
cost?sint
3
5=
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5
2
4
sint cost
cost?sint
3
5;
yN
2
4
a11(t) a12(t)
a21(t) a22(t)
3
5=
2
4
cost?sint
sint?cost
3
5
2
4
sint cost
cost?sint
3
5
1;
?N V pa11(t) = a22(t) = 0,a12(t) = 1,a21(t) =?1.
4. ?¨3￥i ?B?(? ?1.1 )￡
ü? ?2.2￥?B?s.
￡
ü: s?
!Z?Fdxdt = A(t)x￥3Ffxk(t), k = 1;2;:::;ng￥Wronsky?
TdetX(t)
?t = t0 2 [fi;fl]) |′?1
,,5fxk(t0), k = 1;2;:::;ngL?í1,?? ?2.1￥￡
ü V?3
Ffxk(t), k = 1;2;:::;ngL?í1.
A1?¨Q￡E.
!Z?Fdxdt = A(t)x￥3Ffxk(t), k = 1;2;:::;ngL?í1?
uWt 2 [fi;fl]
￥Wronsky?
TdetX(t) · 0. uWt 2 [fi;fl]
|?t0 2 [fi;fl],5
fxk(t0), k = 1;2;:::;ngL?M1,'i ? ?1
,￥è
C1,...,Cn
P¤
C1x1(t0)+C2x2(t0)+￠￠￠+Cnxn(t0) = 0:
A ?C1x1(t) + C2x2(t) + ￠￠￠ + Cnxn(t)x(t) · 0?
^Z?Fdxdt = A(t)x￥
@′Hq
x(t0) = 0￥3,yN?3￥i ?B?? ??Aμ
C1x1(t)+C2x2(t)+￠￠￠+Cnxn(t) · 0:
#3Ffxk(t), k = 1;2;:::;ngL?M1,??DL
!
±.
