推 topee:太...神了..謝謝! 01/31 18:11
※ 引述《topee (eason)》之銘言：
: 題目:
: y" + 4y'-21y = 2e^-2t sin3t y(0)=1 y'(0)=0
: s+4 1 6
: Y = ───── + ───── ──────
: (s+7)(s-3) (s+7)(s-3) (s+2)^2 + 9
: 24 61 1
: 答案 y = ── e^-7t + ── e^3t - ── e^-2t sint3t
: 85 85 17
: 怎麼拆 都拆不出答案.....
令
s+4 A B
───── = ── + ──
(s+7)(s-3) s+7 s-3
去分母得
s+4 = A(s-3) + B(s+7)
s=3代入得 7=10B , B=7/10
s=-7代入得 -3=-10A , A=3/10
令
6 C D Es+F
──────────── = ── + ── + ───────
(s+7)(s-3)[(s+2)^2 + 9] s+7 s-3 [(s+2)^2 + 9]
去分母得
6 = C(s-3)[(s+2)^2 + 9] + D(s+7)[(s+2)^2 + 9]+(Es+F)(s+7)(s-3)
s=3代入得 6=D*10*34 , D=3/170
s=-7代入得 6=C*(-10)*34 , C=-3/170
將式子展開得
6 = C(s-3)(s^2+4s+13)+D(s+7)(s^2+4s+13)+(Es+F)(s^2+4s-21)
= C(s^3+s^2+s-39) + D(s^3+11s^2+41s+91) +Es^3+4Es^2-21Es
+Fs^2+4Fs-21F
= (C+D+E)s^3 + (C+11D+4E+F)s^2 + (C+41D-21E+4F)s + (-39C+91D-21F)
比較係數得
C+D+E=0 C=-3/170
C+11D+4E+F=0 D= 3/170
C+41D-21E+4F=0 配合已解之C、D的值解得 E= 0
-39C+91D-21F=6 F=-3/17
所以
3/10 7/10 -3/170 3/170 0s-3/17
Y = ── + ── + ─── + ─── + ──────
s+7 s-3 s+7 s-3 (s+2)^2 + 9
24/85 61/85 1 3
= ─── + ─── - ── ──────
s+7 s-3 17 (s+2)^2 + 9
-1 24 61 1
y(t) = ￡ [Y(s)] = ── e^(-7t) + ── e^(3t) - ── e^(-2t) sin(3t)
85 85 17
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