Re: [理工] [工數]-高階ODE Euler cauchy方程
※ 引述《CRAZYAWIND (怒火燒不盡)》之銘言：
: 3 2 6x^3
: x y''' - 4x y'' + 8xy' - 8y = ──────── <<94中興化工>>
: (x^2 + 1 )^3/2
: 這題yp項= = 我算了好久 用部份分式 或是重積分法 加上 三角代換
: 都展不回他給的答案 3 5
: 2 4 2x + 4x + 2x
: y(x) = c1x + c2x + c3x - ─────────
: √(1+x^2)
sol:
t
令 x = e ,t = lnx
2 3 d
xy'=dy , xy"=D(D-1)y , x y"'=D(D-1)(D-2) ,其中 D= ──
dt
帶回原式可得
6x^3
D(D-1)(D-2)y-4D(D-1)y+8Dy-8 = ──────
(x^2+1)^3/2
3t
3 2 6e
即[D -7D +14D-8]y = ──────
(e^2t+1)^3/2
(1) mt mt 2 2 mt 3 3 mt
令y=e , Dy=me , D y=m e ,D y=m e 代入ODE中可得特性方程式
3 2
m -7m +14m-8=0 m=1,2,4
t 2t 4t 2 4
yh=c1e + c2e +c3e =c1x + c2x + c3x
3t
(2) 1 6e
yp = ──────── ──────
(D-1)(D-4)(D-2) (e^2t+1)^3/2
t 1 1
= 6e ──────── ──────
(D-1)(D+1)(D+2) (e^2t+1)3/2
2t
3t 1 -2t e
= 6e ───── e ∫──────dt
(D-1)(D+1) (e^2t+1)3/2
2t 2t
3t 1 -2t e d(e + 1)
= 6e ───── e ∫────── ─────
(D-1)(D+1) (e^2t+1)3/2 2e^2t
3t 1 -2t 2t -1/2
= -6e ───── e (e + 1)
(D+1)(D-1)
3t 1 1 -2t 2t -1/2
= -3e [── - ──] e (e + 1)
D-1 D+1
3t 1 -2t 2t -1/2 3t 1 -2t 2t -1/2
= -3e ── e (e + 1) + 3e ── e (e + 1)
D-1 D+1
3t t -t -2t 2t -1/2 3t -t t -2t 2t -1/2
= -3e e ∫e e (e + 1) dt + 3e e ∫e e (e + 1) dt
-3t -t
4t e 2t e
= -3e ∫────── dt + 3e ∫────── dt
(1+e^2t)^1/2 (1+e^2t)^1/2 -t
^^^^^^^^^^^^^^ ^^^^^^^^^^^^^ 上下同乘 e
-4t -2t
4t e 2t e -2t
= -3e ∫─────── dt + 3e ∫─────── dt 令1 + e = v
(1+e^-2t)^1/2 (1+e^-2t)^1/2
-2t
3 4t 1/2 -1/2 3 2t -1/2 -2e dt = dv
= ─ e ∫( v - v ) dv - ─ e ∫v dv
2 2
4t 3/2 1/2 2t 1/2
= e (v - 3v ) - 3e v
4t 3/2 4t 1/2 2t 1/2
= e (1+e^-2t) -3e (1+e^-2t) -3e (1+e^-2t)
-2t 4t -2t 4t 2t
(1+ e )[e (1+e )-3e -3e ]
= ───────────────
-2t 1/2
(1 + e )
-2 4 -2 4 2
(1+x )[x (1+x )-3x -3x ]
= ────────────── 上下同乘x
(1+x^-2)^1/2
-1 4 2 4 2
(x+x )[x + x -3x -3x ]
= ────────────
(1+x^2)^1/2
5 3 3
-2x -2x -2x -2x
= ─────────
(1+x^2)^1/2
3 5
2x + 4x + 2x
= - ───────
(1+x^2)^1/2
(3)
故ODE通解: 3 5
2 4 2x + 4x + 2x
y = yh + yp = c1x + c2x + c3x - ───────
(1+x^2)^1/2
小弟的拙見= =
--
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◆ From: 134.208.40.34
※ 編輯: zendla 來自: 134.208.40.34 (12/10 19:05)
※ 編輯: zendla 來自: 134.208.40.34 (12/10 19:06)
※ 編輯: zendla 來自: 134.208.40.34 (12/10 19:09)
→
12/10 19:23, , 1F
12/10 19:23, 1F
推
12/10 19:32, , 2F
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→
12/10 19:37, , 3F
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推
12/10 19:38, , 4F
12/10 19:38, 4F
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12/10 20:43, , 6F
12/10 20:43, 6F
推
12/10 22:16, , 7F
12/10 22:16, 7F
→
12/10 22:27, , 8F
12/10 22:27, 8F
→
12/10 22:28, , 9F
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12/10 22:30, , 10F
12/10 22:30, 10F
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12/10 22:30, , 11F
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12/10 22:31, , 12F
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12/10 22:35, , 14F
12/10 22:35, 14F
→
12/11 03:20, , 15F
12/11 03:20, 15F
※ 編輯: zendla 來自: 134.208.40.34 (12/11 15:51)
→
12/11 15:52, , 16F
12/11 15:52, 16F
→
12/12 03:16, , 17F
12/12 03:16, 17F
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