[試題] 100下 施文彬 工程數學下 期末考
課程名稱︰工程數學下
課程性質︰必修
課程教師︰施文彬
開課學院：工學院
開課系所︰機械工程學系
考試日期（年月日）︰2012/06/18
考試時限（分鐘）：110 min
是否需發放獎勵金：是, 感謝
（如未明確表示，則不予發放）
試題 :
Final Exam, Engineering Mathematics II, Spring 2012
Time: 10:20~23:10 noon, June 18, 2012.
Rule: No calculator and no information sheet is allowed. Points will not be
given without providing details of your calculation. Good luck!
6iz
e cos(z)
1. (20%) Given f(z)=──────────
z
(a) Find u and v so that f(z)=u(x,y)+i(x,y).
(b) Determine all points at which Cauchy-Riemann equations are satisfied, and
determine all points at which the f(z) is differentiable.
(c) Evaluate ∫f(z)dz ; Δ is any closed path enclosing z=-2i. (You may have to
discuss the solution for different paths.)
f(z)
(d) Evaluate ∫────── dz ; Δ is the circle │z+2i│= 4
2
(z+2i)
2. (20%) Consider the boundary value problem
2
c Yxx = Ytt + k for 0<x<L, t>0, and k is a constant;
Y(0,t)=Y(L,t)=0 for t≧0;
Y(x,0)=f(x), Yt(x,0)=g(x) for 0<x<L
(a) Solve the problem using separation of variables. (You may leave expansion
coefficients in integral forms.)
(b) What is the steady-state solution of this problem?
3. (20%) Consider the heat conduction
2
δu δ u
──= k ─── for -∞< x <∞, t>0 with
δt 2 2
δ x
u(x,0)=f(x) for -∞< x <∞.
(a) Determine the steady-state solution.
-x
(b) If f(x)=∕ e for -1≦x≦1
∣ , solve the problem by Fourier transform.
﹨ 0 for │x│>1 (Please carry out all integrals)
4. (20%) Consider the infinite string problem
2
c Yxx = Ytt, (-∞< x <∞, 0< t <∞)
y(x,0)=f(x), Yt(x,0)=g(x). (-∞< x <∞)
(a) Show that thewave equation becomes Yξη=0 by lettingξ=x-ct and η=x+ct.
(b) For g(x)＝0, derive the solution Y(x,t)=[f(x-ct)+f(x+ct)]/2.
f(x-ct)+f(x-ct) １ x+ct
(c) For g(x)≠0, derive the solution Y(x,t)=───────── + ─ ∫ g(s) ds.
2 2c x-ct
2 ,, , 2 2
5. (20%) Consider the differential equation x y +xy +(λx -n )y = 0 on the
interval (0,R). Here n is any given nonnegative integer. Let y(R)=0.
(a) Write the differential equation in Sturm-Liouville form and show that it is
a singular Sturm-Liouville problem with appropriate boundary condition at
x=0.
(b) Determine the eigenvalues and eigenfunctions of this Sturm-Liouville
problem.
(c) Write down the orthogonal condition of the eigenfunctions.
(d) Prove the orthogonal condition.
Some useful equations
-------------------------------------------------------------------------------
2 ,, , 2 2
Bessel's equation x y + xy +(x -v )y=0
A0 ∞ nπx nπx １ L １ L nπx
S(x)=─+Σ Ancos(───)+Bnsin(───), A0=─∫f(x)dx, An=─∫f(x)cos(───)dx
２ n=1 Ｌ Ｌ Ｌ-L Ｌ-L Ｌ
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