※ 引述《zxnstu3104 (skykeeper)》之銘言:
: 一張照片是題目,一張照片是我的過程,懇請各位數學先進幫幫忙了
: https://i.imgur.com/Dy6zj2V.jpg
: https://i.imgur.com/VFvINy3.jpg
u = X(x)T(t)
X"/X = T"/T + 1 = -c_n^2
c_n > 0
k_n = √[1 + c_n^2]
inf
u = Sigma A_n sin(c_n x) cos(k_n t)
n = 1
接下來計算非常繁複,你要靠自己把結果拼湊出來
以下是結果
inf
u(x, 0) = Sigma - [(2/π) (-1)^k * 2 / (2k - 1)^2] * sin((2k - 1)x)
k = 1
=> A_(2k-1) = [(4/π) (-1)^(k+1) / (2k - 1)^2] k = 1, 2, 3...
c_(2k-1) = 2k - 1
inf
u = Sigma [(4/π) (-1)^(k+1) / (2k - 1)^2] sin((2k - 1)x)
k = 1
* cos(t√[1 + (2k - 1)^2])
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