這題是97台大土木第3題
(a)Slove the following ordinary differential equation
2
d ψ(x)
──── =1+x -1< x <1,which is subjected to the boundary condition
2 － －
dx
ψ(x = -1) = 0 and ψ(x = 1) = 0
And identity the homogeneous , particular solutions and resonant modes.
有問題的只有 resonant modes (共振項) 翻了兩份解答,發現不同老師對共振項的
定義也不同,當然答案也不一樣！
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(b)Express the solution of ψ(x) from Part(a) in terms of the
Chebyshev polynomisls Tn(x).
Hint:T (x) = 1 ; T (x) = x ; T (x) -2xTn(x)+T (x) = 0 ;
0 1 n+1 n-1
2 2
(1-x )Tn''(x)-xTn'(x)+n Tn(x) = 0 ;
1 Tm(x)Tn(x) 2 2 π
∫ ────── dx = 0,m≠n;∥To(x)∥ = π;∥Tn(x)∥ = ─;n =1,2,3...
-1 2 2
根號（1-x ）
(b)小題是要用Chebyshev polynomisls重解(a)小題！
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下列網址是原題的網頁
http://www.lib.ntu.edu.tw/exam/graduate/97/97212.pdf
謝謝！
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