Re: [理工] [工數]-級數解
我去翻翻課本
發覺課本都是用你講的解法 XD
這題要這樣解的話
2 2
x y" + 6xy' + (6 - x )y = 0
∞ r+n ∞ r+n-1 ∞ r+n-2
Let y = ΣCn x , y' = ΣCn(r+n)x , y" = ΣCn(r+n)(r+n-1)x
n=0 n=0 n=0
代入得
∞ r+n ∞ r+n ∞ r+n ∞ r+n+2
ΣCn(r+n)(r+n-1)x + Σ6Cn(r+n)x + Σ6Cn x - ΣCn x = 0
n=0 n=0 n=0 n=0
整理後得
∞ r+n ∞ r+n
ΣCn[(r+n)(r+n-1)+6(r+n)-6]x + ΣCn-2 x = 0
n=0 n=2
當n=0:
2
C0(r +5r+6) = C0(r+2)(r+3) = 0 , 若C0≠0 , 得 r = -3 or -2
當n=1:
2
C1(r +7r+12) = C1(r+3)(r+4) = 0
因 r = -3 or -2
若要等式恆成立 則C1 = 0
可得Cn = 0 (n=odd)
當n≧2:
(i)r=-3代入:
1 1 1
C1n = --------------------- Cn-2 = ------- Cn-2 = -------- Cn-2
(n-3)(n-4)+6(n-3)+6 2 n(n-1)
n - n
1
= --- C10 (n=even)
n!
令 C10 = 1
∞ r+n -3 ∞ 1 n -3
y1 = ΣC1n x = x Σ --- x = x sinhx
n=0,2… n=0,2… n!
(ii)r=-2代入:
1 1 1
C2n = --------------------- Cn-2 = ------- Cn-2 = -------- Cn-2
(n-2)(n-3)+6(n-2)+6 2 n(n+1)
n + n
1
= -------- C20 (n=even)
(n+1)!
令C20 = 1
∞ r+n -2 ∞ 1 n -3 ∞ 1 n+1 -3
y2 = ΣC2n x = x Σ -------- x = x Σ -------- x = x coshx
n=0 n=0,2… (n+1)! n=0,2… (n+1)!
-3
y = a1 y1 + a2 y2 = x (a1 sinhx + a2 coshx)
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