作者s871526117 (安安)
看板NTU-Exam
標題[試題] 100下 徐治平 工程數學 第一次期中考
時間Fri Apr 13 14:46:16 2012
課程名稱︰工程數學
課程性質︰必修
課程教師︰徐治平
開課學院：工學院
開課系所︰化工系
考試日期（年月日）︰2012/4/13
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Solve the following Sturm-Liouville problems, where λ is a real value.(15%
each)
(a) y"+λy=0, y(-π)=y(π), y'(-π)=y'(π)
(b) y"+λy=0, y(0)=0, 3y(1)+y'(1)=0
(c) y"+λy=0, y(4)=0, y'(0)=0
2. Find the general solution of the following equations in terms of Bessel
functions. (10% each)
(a)9x^2y"-27xy'+(9x^2+35)y=0. Hint: let u=y/x^2
(b)4x^2y"+ 8xy'+(4x^2-35)y=0. Hint: let u=y√x
3. Show that the equation (15%)
d^2 y dy
sinθ─── + cos── + n(n+1)(sinθ)y=0
dθ^2 dθ
can be transformed into a Legendre's equttion by letting x=cosθ
4. Expand each of the following functions in a series of Legendre polynomials,
and evaluate the coefficients of the first three terms (10% each)
(a)cos(πx/2), -1<x<1
(b)1+2x-x^2
The first three Legendre polynomals are P0(x)=1, P1(x)=x, P2(x)=(1/2)(3x^2-1)
List of integrals
∫x sin(cx)dx = sin(cx)/c^2 - xcos(cx)/c
∫x^n cos(cx)dx = (x^n sin(cx)/c - (n/c)∫x^(n-1)sin(cx)dx
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1F：推 s93015a :第三題好像應該長這樣： 04/13 19:26
2F：→ s93015a : d^2 y dy 04/13 19:26
3F：→ s93015a :sinθ─── + 2cos── + n(n+1)(sinθ)y=0 04/13 19:26
4F：→ s93015a : dθ^2 dθ 04/13 19:26
5F：→ s93015a :才有可能推到Legendre equation耶 04/13 19:27
6F：→ s93015a :是題目出錯還是我算錯 04/13 19:28
7F：推 jsaon92 :題目沒錯 04/13 23:19
8F：推 princetonboy:安安你好安安 04/14 01:11
9F：→ s871526117 :題目上是原本的樣子ㄟ 04/16 00:43