1
?
1c5D
± I
1. p
QL?Z?￥
LY3
(2) d3xdt3? d2xdt2 +2dxdt?2x = 0.
(4) d4xdt4?2d3xdt3 +2dxdt?x = 0.
3:
(2)?Z?￥+?[
T1
32 +22 = (1)(?2 +2);
yN+??11,§p2i.#eZ?μ
L'3Fet,cosp2t,sinp2t.?N¤
LY3
x(t) = C1et +C2 cosp2t+C3 sinp2t;
?C1,C2,C31 ?iè
.
(4)?Z?￥+?[
T1
4?2?3 +21 = (1)3(?+1);
yN+??11 ( ?×?),?1.#eZ?μ
L'3Fet,tet,t2et,e?t.?N¤
LY3
x(t) = et(C1 +C2t+C3t2)+C4e?t;
?C1,C2,C3,C41 ?iè
.
2?.s??Z?
d2x
dt2 +2–
dx
dt +!
2x = 0
￥+??ióY3.? ú–? 0,! > 0.
3:V???Z?￥+?Z?
2 +2–?+!2 = 0
p¤+??1
1;2 =?– §
p
–2?!2:
? –2?!2￥?| Vs1 ?/ ?? f ?:
(i)?– > !
H,μ=?Ms
L+?– §p–2?!2,Z?￥
LY31
x(t) = e?–t(C1e
p
–2?!2t +C2e?
p
–2?!2t);
?C1,C21 ?iè
.
(ii)?– = !
H,μB?
L=×+?–,Z?￥
LY31
x(t) = C1e?–t(C1 +C2t);
?C1,C21 ?iè
.
(iii)?– < !
H,μB
aˉ+?– §p!2?–2i,Z?￥
LY31
x(t) = e?–t(C1 cos
p
!2?–2t+C2 sin
p
!2?–2t);
?C1,C21 ?iè
.
3. pd
QL?Z?￥
LY3
(1) d2xdt2 + dxdt = 1+t2.
(3) d2xdt2 +4x = tsin2t.
2
(4) d3xdt3?4d2xdt2 +3dxdt = t2.
3:
(1)?Z??￥
QL?Z?￥+?[
T1?2 +?,yN+??10,?1.#eZ??￥
Q
L?Z?μ
L'3F1,e?t.?eZ?μ+3
x(t) = 1D2 +D ￠(1+t2) = 1D +1 1D ￠(1+t2)
= 1D +1 ￠(t+ 13t3)
= (1?D +D2?D3)(t+ 13t3)
= t
3
3?t
2 +3t?3:
?N¤eZ?￥
LY3
x(t) = C1 +C2e?t + t
3
3?t
2 +3t;
?C1,C21 ?iè
.
(3)?Z??￥
QL?Z?￥+?[
T1?2 +4,yN+??1§2i.#eZ??￥
QL
?Z?μ
L'3Fcos2t,sin2t.?eZ?μ+3
x(t) = 1D2 +4 ￠tsin2t;
I
n￡ùZ?(D2 +4)z = te2it,
μ+3
z(t) = 1D2 +4 ￠te2it = e2it 1(D +2i)2 +4 ￠t
= e2it 1D 1D +4i ￠t = e
2it
4i
1
D(1?
1
4iD)t
= e
2it
4i
1
D(t?
1
4i) =
e2it
4i (
t2
2?
t
4i);
|′?¤?eZ?￥+3
x(t) =?18t2 cos2t+ 116tsin2t:
?N¤eZ?￥
LY3
x(t) = C1 cos2t+C2 sin2t? 18t2 cos2t+ 116tsin2t;
?C1,C21 ?iè
.
(4)?Z??￥
QL?Z?￥+?[
T1?3?4?2 + 3?,yN+??10,1,3.#eZ?
?￥
QL?Z?μ
L'3F1,et,e3t.?eZ?μ+3
x(t) = 1D3?4D2 +3D ￠t2 = 13?D 11?D 1D ￠t2
= 13?D(1+D +D2 +D3)(13t3)
= 19(1+ 13D + 19D2 + 127D3)(t3 +3t2 +6t+6)
= 19t3 + 49t2 + 2627t+ 7481:
?N¤eZ?￥
LY3
x(t) = C1 +C2et +C3e3t + 19t3 + 49t2 + 2627t;
?C1,C2,C31 ?iè
.