1
53.3
2. I
nZ?F
dx
dt = A(t)x+f(t); (1)
?
A(t) =
2
4
cos2 t 12 sin2t?1
1
2 sin2t+1 sin
2 t
3
5; f(t) =
2
4
cost
sint
3
5:
(1)
k￡
Φ(t) =
2
4
et cost?sint
et sint cost
3
5
^?￥
QZ?F
dx
dt = A(t)x
￥3 ?.
(2)
k p(1)￥
@′Hq
x(0) =
2
4
1
2
3
5
￥3.
3:
(1)^n
dΦ(t)
dt =
2
4
et(cost?sint)?cost
et(sint+cost)?sint
3
5= A(t)Φ(t):
#Φ(t)
^?￥
QZ?F￥3 ?.??detΦ(t) = et 6= 0?Φ(t)
^?￥
QZ?F￥
3 ?.
(2) ?^ p
Φ(t)Φ?1(?)f(?) =
2
4
et cost?sint
et sint cost
3
5
2
4
e cos? e sin?
sin? cos?
3
5
2
4
cos?
sin?
3
5= et
2
4
cost
sint
3
5;
Φ(t)Φ?1(0)x(0) =
2
4
et cost?sint
et sint cost
3
5
2
4
1 0
0 1
3
5
2
4
1
2
3
5=
2
4
et cost?2sint
et sint+2cost
3
5;
?è
M^
T¤ed
QZ?F￥31
x(t) = Φ(t)Φ?1(0)x(0)+
Z t
0
Φ(t)Φ?1(?)f(?)d? =
2
4
(?et?1)cost?2sint
(?et?1)sint+2cost
3
5:
3.
!n￡n ?f
A(t) [fi;fl]
??,n?_
f
f(t;x) u×fi? t? fl;kxk < 1
??.
￡
ü′ù5
dx
dt = A(t)x+f(t;x); x(t0) = x0
?N? p3sZ?
x(t) = X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)f(?;x(?))d?;
2
?t;t0 2 [fi;fl],X(t)
^M?
QL?Z?F￥3 ?.
￡
ü:
!x(t)1
ósZ?￥3,5
dx
dt =
dX(t)
dt X
1(t0)x0 +X(t)X?1(t)f(t;x(t))+
Z t
t0
dX(t)
dt X
1(?)f(?;x(?))d?
= A(t)X(t)X?1(t0)x0 +f(t;x(t))+A(t)
Z t
t0
X(t)X?1(?)f(?;x(?))d?
= A(t)x(t)+f(t;x(t));
#x(t)91
ó±sZ?￥3.]
H^nx(t0) = x0,#x(t)1
ó′ù5￥3.
Q-,
!x(t)1
ó′ù5￥3,5μ
dx
dt = A(t)x+g(t); x(t0) = x0;
?g(t) = f(t;x(t))1X?f
.?è
M^
T¤
x(t) = X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)g(?)d?
= X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)f(?;x(?))d?;
#x(t)91
ósZ?￥3.
53.4
2.
!x(t)
^L?±sZ?
d2x
dt2 +a1(t)
dx
dt +a2(t)x = 0
￥d
,3,
k￡?x(t0) = 0
H,x0(t0) 6= 0.
￡
ü:¨Q￡E. ?x0(t0) = 0,5x(t)
^′ù5
d2x
dt2 +a1(t)
dx
dt +a2(t)x = 0; x(t0) = 0; x
0(t0) = 0
￥3.A ???′ù5μ
,3ˉx(t) · 0,yN?3￥i ?B?? ??Aμx(t) · 0,?Dx(t)
^
d
,3
±.#?x(t0) = 0
H,x0(t0) 6= 0.
3.￡x = sintt
^Z?
d2x
dt2 +
2
t
dx
dt +x = 0
￥3,i p?Z?￥Y3.
3:?x = 1t sint?
d2x
dt2 +
2
t
dx
dt +x =
2
t3 sint?
2
t2 cost?
1
t sint
+2t(? 1t2 sint+ 1t cost)+ 1t sint = 0
3
yNx = sintt
^eZ?￥3.? è4.2￥2T? Y31
x(t) = sintt
C1?C2
Z t2
sin2 te
2R t?1dtdt
= 1t(C1 sint+C2 cost);
?C1,C21 ?iè
.
5.
!x1(t),x2(t)
^=¨L?±sZ?
d2x
dt2 +a1(t)
dx
dt +a2(t)x = f(t); (2)
?￥
QZ?￥
?L?í1￥+3, ?a1(t)a2(t)
^ uWfi? t? fl
￥ ??f
,5Z?
(2) uWfi? t? fl
￥Y31
x(t) = c1x1(t)+c2x2(t)+
Z t
t0
x1(?)x2(t)?x1(t)x2(?)
x1(?)x02(?)?x01(?)x2(?)f(?)d?;
?c1,c21 ?iè
.
￡
ü:3Ffx1(t),x2(t)g
^?￥
QZ?￥'3F, Wronsky?
T1
W(t) = det
2
4
x1(t) x2(t)
x01(t) x02(t)
3
5= x1(t)x02(t)?x01(t)x2(t):
?W(t)??2??1
?2
í
í￥}
?0
TW1(t),W2(t)sY1W1(t) =?x2(t),W2(t) =
x1(t).#?è
M^
T?
ó=¨L?±sZ?μ+3
xp(t) =
Z t
t0
x1(?)x2(t)?x1(t)x2(?)
x1(?)x02(?)?x01(?)x2(?)f(?)d?;
yN
óY3
T? ?.
6. pZ?
d2x
dt2 +4x = tsin2t
￥Y3.X? ?￥
QL?Z?μ'3Fcos2t,sin2t.
3: ?^ p ?￥
QL?Z?￥'3Fcos2t,sin2t￥Wronsky?
TW(t) = 2,W(t)?
?2??1
#?2
í
í￥}
?0
TW1(t),W2(t)sY1W1(t) =?sin2t,W2(t) = cos2t.y
N?è
M^
T?eZ?μ+3
xp(t) = 12
Z t
0
(sin2tcos2cos2tsin2?)? sin2?d?
=?t
2
8 cos2t+
t
16 sin2t:
#eZ?￥Y31
x = C1 cos2t+C2 sin2t? t
2
8 cos2t+
t
16 sin2t;
?C1,C21 ?iè
.
53.3
2. I
nZ?F
dx
dt = A(t)x+f(t); (1)
?
A(t) =
2
4
cos2 t 12 sin2t?1
1
2 sin2t+1 sin
2 t
3
5; f(t) =
2
4
cost
sint
3
5:
(1)
k￡
Φ(t) =
2
4
et cost?sint
et sint cost
3
5
^?￥
QZ?F
dx
dt = A(t)x
￥3 ?.
(2)
k p(1)￥
@′Hq
x(0) =
2
4
1
2
3
5
￥3.
3:
(1)^n
dΦ(t)
dt =
2
4
et(cost?sint)?cost
et(sint+cost)?sint
3
5= A(t)Φ(t):
#Φ(t)
^?￥
QZ?F￥3 ?.??detΦ(t) = et 6= 0?Φ(t)
^?￥
QZ?F￥
3 ?.
(2) ?^ p
Φ(t)Φ?1(?)f(?) =
2
4
et cost?sint
et sint cost
3
5
2
4
e cos? e sin?
sin? cos?
3
5
2
4
cos?
sin?
3
5= et
2
4
cost
sint
3
5;
Φ(t)Φ?1(0)x(0) =
2
4
et cost?sint
et sint cost
3
5
2
4
1 0
0 1
3
5
2
4
1
2
3
5=
2
4
et cost?2sint
et sint+2cost
3
5;
?è
M^
T¤ed
QZ?F￥31
x(t) = Φ(t)Φ?1(0)x(0)+
Z t
0
Φ(t)Φ?1(?)f(?)d? =
2
4
(?et?1)cost?2sint
(?et?1)sint+2cost
3
5:
3.
!n￡n ?f
A(t) [fi;fl]
??,n?_
f
f(t;x) u×fi? t? fl;kxk < 1
??.
￡
ü′ù5
dx
dt = A(t)x+f(t;x); x(t0) = x0
?N? p3sZ?
x(t) = X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)f(?;x(?))d?;
2
?t;t0 2 [fi;fl],X(t)
^M?
QL?Z?F￥3 ?.
￡
ü:
!x(t)1
ósZ?￥3,5
dx
dt =
dX(t)
dt X
1(t0)x0 +X(t)X?1(t)f(t;x(t))+
Z t
t0
dX(t)
dt X
1(?)f(?;x(?))d?
= A(t)X(t)X?1(t0)x0 +f(t;x(t))+A(t)
Z t
t0
X(t)X?1(?)f(?;x(?))d?
= A(t)x(t)+f(t;x(t));
#x(t)91
ó±sZ?￥3.]
H^nx(t0) = x0,#x(t)1
ó′ù5￥3.
Q-,
!x(t)1
ó′ù5￥3,5μ
dx
dt = A(t)x+g(t); x(t0) = x0;
?g(t) = f(t;x(t))1X?f
.?è
M^
T¤
x(t) = X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)g(?)d?
= X(t)X?1(t0)x0 +
Z t
t0
X(t)X?1(?)f(?;x(?))d?;
#x(t)91
ósZ?￥3.
53.4
2.
!x(t)
^L?±sZ?
d2x
dt2 +a1(t)
dx
dt +a2(t)x = 0
￥d
,3,
k￡?x(t0) = 0
H,x0(t0) 6= 0.
￡
ü:¨Q￡E. ?x0(t0) = 0,5x(t)
^′ù5
d2x
dt2 +a1(t)
dx
dt +a2(t)x = 0; x(t0) = 0; x
0(t0) = 0
￥3.A ???′ù5μ
,3ˉx(t) · 0,yN?3￥i ?B?? ??Aμx(t) · 0,?Dx(t)
^
d
,3
±.#?x(t0) = 0
H,x0(t0) 6= 0.
3.￡x = sintt
^Z?
d2x
dt2 +
2
t
dx
dt +x = 0
￥3,i p?Z?￥Y3.
3:?x = 1t sint?
d2x
dt2 +
2
t
dx
dt +x =
2
t3 sint?
2
t2 cost?
1
t sint
+2t(? 1t2 sint+ 1t cost)+ 1t sint = 0
3
yNx = sintt
^eZ?￥3.? è4.2￥2T? Y31
x(t) = sintt
C1?C2
Z t2
sin2 te
2R t?1dtdt
= 1t(C1 sint+C2 cost);
?C1,C21 ?iè
.
5.
!x1(t),x2(t)
^=¨L?±sZ?
d2x
dt2 +a1(t)
dx
dt +a2(t)x = f(t); (2)
?￥
QZ?￥
?L?í1￥+3, ?a1(t)a2(t)
^ uWfi? t? fl
￥ ??f
,5Z?
(2) uWfi? t? fl
￥Y31
x(t) = c1x1(t)+c2x2(t)+
Z t
t0
x1(?)x2(t)?x1(t)x2(?)
x1(?)x02(?)?x01(?)x2(?)f(?)d?;
?c1,c21 ?iè
.
￡
ü:3Ffx1(t),x2(t)g
^?￥
QZ?￥'3F, Wronsky?
T1
W(t) = det
2
4
x1(t) x2(t)
x01(t) x02(t)
3
5= x1(t)x02(t)?x01(t)x2(t):
?W(t)??2??1
?2
í
í￥}
?0
TW1(t),W2(t)sY1W1(t) =?x2(t),W2(t) =
x1(t).#?è
M^
T?
ó=¨L?±sZ?μ+3
xp(t) =
Z t
t0
x1(?)x2(t)?x1(t)x2(?)
x1(?)x02(?)?x01(?)x2(?)f(?)d?;
yN
óY3
T? ?.
6. pZ?
d2x
dt2 +4x = tsin2t
￥Y3.X? ?￥
QL?Z?μ'3Fcos2t,sin2t.
3: ?^ p ?￥
QL?Z?￥'3Fcos2t,sin2t￥Wronsky?
TW(t) = 2,W(t)?
?2??1
#?2
í
í￥}
?0
TW1(t),W2(t)sY1W1(t) =?sin2t,W2(t) = cos2t.y
N?è
M^
T?eZ?μ+3
xp(t) = 12
Z t
0
(sin2tcos2cos2tsin2?)? sin2?d?
=?t
2
8 cos2t+
t
16 sin2t:
#eZ?￥Y31
x = C1 cos2t+C2 sin2t? t
2
8 cos2t+
t
16 sin2t;
?C1,C21 ?iè
.
