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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