课程名称︰工程数学下 课程性质︰机械系大二下必修 课程教师︰施文彬 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2007.6.25 考试时限（分钟）：110 mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Rule: No calculators are allowed. You are allowed to bring an A4 size information sheet. Please provide the details of your calculation. Good luck! 1 1.(20%) Use convolution to find the inverse Fourier transform of ───── . (1+iω)^2 Then find the inverse Fourier transform of the function (e^6iω)sin(2ω) ──────── . (1+iω)^2 du d^2(u) du 2.(a)(10%) Transform ── = k( ──── + A── + Bu ) into a standard heat dt d(x^2) dx equation by letting u(x,t) = [e^(αx+βt)]v(x,t). du d^2(u) du (b)(10%) Solve ── = ( ──── + 6 ── ) for 0＜x＜4, t＞0; dt d(x^2) dx u(0,t)=u(4,t)=0 for t≧0; u(x,0)=1. 3.(a)(15%) Determine the eigenvalues (or at least the characteristic equation for them), eigenfunctions, and weighting function of the Sturm-Liouville problem (x^2)y"+ xy'+ (λx^2 - 4)y = 0, where y(0) isbounded and y(L)=0. (Hint: let t = (√λ)x ) (b)(10%) Prove that the eigen-functions are orthogonal to each other. d^2(y) d^2(y) 4.(20%) Consider the wave equation ──── = c^2 ──── on the line for d(t^2) d(x^2) dy c=2 and the given initial condition y(x,0)=f(x) and ─(x,0)=g(x). dt Here f(x)=┌ sin(x) for -2π≦x≦π ; g(x)= e^(-4|x|). └ 0 for x＜-2π and x＞π Solve the problom using the Fourier integral and then again using the Fourier transform. 5.(15%) Solve the boundary value problom using separation of variables. d^2(y) d^2(y) ──── = ──── - cos(x) for 0＜x＜2π, t＞0; d(t^2) d(x^2) y(0,t)= y(2π,t)= 0 for t≧0; dy y(x,0)=0, ─(x,0)= x for 0＜x＜2π. dt --

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