1
?
1c5D
± I
6. p3Z?F
(1) dxdt =?3x+48y?28z; dydt =?4x+40y?22z,
dz
dt =?6x+57y?31z.
(2) dxdt = y; dydt =?x.
(3) dxdt =?5x?10y?20z; dydt = 5x+5y +10z,
dz
dt = 2x+4y +9z.
(4) dxdt = 3x?y; dydt =?4x?y,
dz
dt = 4x?8y?2z.
3:
(1)?Z?F￥"
?1
A =
2
4
3 48?28
4 40?22
6 57?31
3
5:
μ3?4Nos￥+?1 = 1,?2 = 2,?3 = 3, ?￥+?_
sY1c1 = (3;2;3)T,
c2 = (4;1;1)T,c3 = (2;2;3)T.?N¤eZ?F￥Y31
8
<
:
x = 3C1et +4C2e2t +2C3e3t
y = 2C1et +C2e2t +2C3e3t
z = 3C1et +C2e2t +3C3e3t;
?C1,C2,C31 ?iè
.
(2)?Z?F￥"
?1
A =
2
4
0 1
1 0
3
5:
^n:
A2 =?T; A3 =?A; A4 = I; ￠￠￠ ;
yNeZ?Fμ'3 ?
exp(At) =
2
64 1?
t2
2! +
t4
4! +￠￠￠ 1?
t3
3! +
t5
5! +￠￠￠
1+ t33!? t55! +￠￠￠ 1? t22! + t44! +￠￠￠
3
75
=
2
4
cost sint
sint cost
3
5:
?N¤eZ?F￥Y31
8
<
:
x = C1 cost+C2 sint
y =?C1 sint+C2 cost;
?C1,C21 ?iè
.
(3)?Z?F￥"
?1
A =
2
4
5?10?20
5 5 10
2 4 9
3
5:
2
μ3?4Nos￥+?1 = 5,?2 = 2 + i,?3 = 2? i, ?￥+?_
sY1
c1 = (?2;0;1)T,c2 = (3+i;2?i;?2)T,c3 = (3?i;2+i;?2)T.yNeZ?Fμˉ'3
?
X(t) =
2
4
2e5t (3+i)e(2+i)t (3?i)e(2?i)t
0 (2?i)e(2+i)t (2+i)e(2?i)t
e5t?2e(2+i)t?2e(2?i)t
3
5:
I
nX(t)￥
L?¤eZ?F￥Y31
8>
<
>:
x =?2C1e5t +C2(3cost?sint)e2t
+C3(cost+3sint)e2t
y = C2(2cost+sint)e2t +C3(?cost+2sint)e2t
z = C1e5t?2C2e2t cost?2C3e2t sint;
?C1,C2,C31 ?iè
.
(4)?Z?F￥"
?1
A =
2
4
3?1 0
4?1 0
4?8?2
3
5:
μ?+?1 =?2=×+?2 = 1.?1 =?2, ?￥+?_
1c1 = (0;0;1)T.
?2 = 1 p(A2I)2c = 0￥d
üO3,' p3L?Z?F
2
4
0 0 0
0 0 0
28 44 9
3
5c = 0:
¤?
?L?í1￥3c20 = (1;1;?8)Tc30 = (5;?4;4)T.?N?w¤
c21 = (A2I)c20 = (3;?6;20)T;
c31 = (A2I)c30 = (6;?12;40)T:
Ka¤?'3 ?
X(t) = (c1e?1t;e?2t(c20 + t1!c21);e?2t(c30 + t1!c31))
=
2
4
0 (1+3t)et (5+6t)et
0 (1?6t)et (?4?12t)et
e?2t (?8+20t)et (4+40t)et
3
5:
?N¤eZ?F￥Y31
8
<
:
x = C2(1+3t)et +C3(5+6t)et
y = C2(1?6t)et +C3(?4?12t)et
z = C1e?2t +C2(?8+20t)et +C3(4+40t)et;
?C1,C2,C31 ?iè
.
7?.ó?
QZ?F˙x = Ax, ?A1è
′ ?.￡
ü
(1) ?A￥
μ+??
L??< 0,5
μ3?t ! +1
H t?0.
(2) ?A￥
μ+??
L? 0 O
,
L?￥+???
^e??,5B M38t? 0?μ?.
(3) ?AμB?+??
L?> 0,5μ3?t ! +1
H t_í k.
￡
ü:
! ?Aμo?M]￥+?1,￠￠￠,?s,×
sY1n1,￠￠￠,ns On1 +n2 +￠￠￠+ns = n,
5
QZ?F˙x = Ax￥ ?B3x(t) (μ?
T:
x(t) =
sX
j=1
e?jtPj(t);
?Pj(t)1[
T OdegPj(t)? nj?1.
3
(1) ?A￥
μ+??
L??< 0,5 ?Bj (1? j? s),Qˉ¨
ArE5¤
lim
t!+1
e?jtPj(t) =? lim
t!+1
P0j(t)
jejt = ￠￠￠ = 0:
yN?t ! +1
Hx(t) t?0.'Z?F˙x = Ax￥
μ3?t ! +1
H t?0.
(2) ?A￥
μ+??
L? 0 O
,
L?￥+???
^e??,?^
!?1,￠￠￠,?k￥
L?1
,,
k+1,￠￠￠,?s￥
L?1μ.5?L
!,n1 = ￠￠￠ = nk = 1,V7P1(t),￠￠￠,Pk(t) (1è
,
!
1C1,￠￠￠,Ck,yN ?Bj (1? j? k)μ
jje?jtPj(t)jj?jje?jtjj￠jjPj(t)jj = jCjj;
V7e?1tP1(t),￠￠￠,e?ktPk(t)8t? 0 (μ?.
6BZ
,?(1) V¤
lim
t!+1
e?k+1tPk+1(t) = ￠￠￠ = lim
t!+1
e?stPs(t) = 0:
#e?k+1tPk+1(t),￠￠￠,e?stPs(t)8t? 09 (μ?.yNx(t)8t? 0μ?.'Z?F
˙x = Ax￥
μ38t? 0?μ?.
(3) ?AμB?+??
L?> 0,?^
!?1 = fi + ifl￥
L?fi > 0.
!·
^AM?1￥+?_
,5Z?F˙x = Axμ3x(t) = e?1t·.A ?μ
lim
t!+1
jjx(t)jj = lim
t!+1
jje?1t·jj = lim
t!+1
efitjj·jj = +1:
yNZ?F˙x = Axμ3?t ! +1
H t_í k.