[試題] 100下 徐治平 工程數學 第一次期中考

[s871526117 (安安)]

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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※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 140.112.22.252
推 s93015a     :第三題好像應該長這樣：
→ s93015a     :     d^2 y        dy
→ s93015a     :sinθ─── + 2cos── + n(n+1)(sinθ)y=0
→ s93015a     :     dθ^2        dθ
→ s93015a     :才有可能推到Legendre equation耶
→ s93015a     :是題目出錯還是我算錯
推 jsaon92     :題目沒錯
推 princetonboy:安安你好安安
→ s871526117  :題目上是原本的樣子ㄟ