[試題] 100下徐治平工程數學第一次期中考

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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 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.22.252

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