課程名稱︰工程數學上 課程性質︰系定必修 課程教師︰楊馥菱 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰971210 考試時限（分鐘）：65 是否需發放獎勵金：是 試題 : Reference: A two sided A4 size summary. [ Series solution 40% ] 2 x 1. Find a series solution to y' + xy = e . If y(0) = -1, write out explicitly the first 4 terms of your solution. (15%) 2 2. Solve the ODE: 3x y" + 4xy' - (3x+2)y = 0 around x = 0. Explain why you choose Power series method or Frobenius method. Find two linearly independent series solutions for y(x) = C1y1(x) + C2y2(x). No need to determine C1, C2, but solve for the indicial equation and the recursion formula. (25%) [ Laplace Transform 35% ] Use Laplace transform to solve the ODE. Provide all the details. 0 for 0≦t＜4 3. y" + 4y = f(t) = { , where y(0) = 1 and y'(0) = 0. (20%) 3 for t≧4 4. y" + 4ty' - 4y = 0 subjected to y(0) = 0 and y'(0) = -7. (15%) The initial value theorem lim y(t) = lim Y(s) may be of use. t→0 s→∞ Please remember L[tf(t)] = -F'(s) [ Special function and Eigen function expansion 40% ] 5. Categorize the following Sturm-Liouville problem: 2 y" + λ y = 0 with y(0) - 2y'(0) = 0 and y'(1) = 0. Determine the eigenvalue. How many eigenvalues are there? ｄ┌ dy┐ ┌ ４┐ 6. Find the general solution to the ODE ─│x─│+│λx - ─│y = 0, where the ＿ dx└ dx┘ └ ｘ┘ transformation z = √λx will help. Now determine the solution to the ｄ┌ dy┐ ┌ ４┐ 1 １ following ODE: ─│x─│+│x - ─│y = 0, with y(0) = 0, y(-) = ─. dx└ dx┘ └ ｘ┘ 2 32 Please sketch your solution for x>0. The following information might be useful when interpreting the boundary conditions: At small x, 2 x x J0(x) ~ 1, J1(x) ~ -, J2(x) ~ ─, ... 2 ８ ２ ２ -1 ４ -2 Y0(x) ~ ─ ln x, Y1(x) ~ -─ x , Y2(x) ~ -─x , ... π π π -- As an Engineer, I shall participate in none but honest enterprises. When needed, my skill and knowledge shall be given without reservation for the public good. In the performance of duty and in fidelity to my profession, I shall give the utmost. excerpt from: The Obligation of an Engineer dn890221 --

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