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課程名稱︰工程數學下
課程性質︰機械系大二下必修
課程教師︰施文彬
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2007.6.25
考試時限（分鐘）：110 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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