課程名稱︰工程數學一 課程性質︰必修 課程教師︰李克強 開課學院：工學院 開課系所︰化工系 考試日期（年月日）︰2012/01/13 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : (1) Find the first three non-zero terms of each of the two linearly independent series solution to the following ODE: (20%) 2xy"+y'+xy=0, x>0 (2)(A) Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (15%) y"+λy=0, y(0)=0, y'(1)=0. (x=[0,1]) ∞ (B) Find the orthogonal function expansion of f(x)=x=Σ (AmΨm(x)), where m=1 Ψm(x) is the m-th eigenfunction corresponding to λm. Compute the first three coefficients, i.e., A1,A2 and A3. (15%) (3)(A) Find the first three eigenvalues and corresponding eigenfunctions of the following boundary value problem. Assume that all eigenvalues are real. (20%) x^2(y")+x'y+(λx^2-1)y=0, y(0) is finite, and y(1)=0. (x=[0,1]) ∞ (B) Find the orthogonal function expansion of f(x)=1=Σ (AmΨm(x)), where m=1 Ψm(x) is the m-th eigenfunction corresponding to λm. Compute the first two coefficients, i.e., A1,A2. (10%) (4) Does the following eigenvalues problem have real solutions? If it does, find its eigenvalues and eigenfunctions. If it doesn't, show your reasoning. (20%) x^2(y")+x'y-λx^2(y)=0, y(0) is finite, and y(1)=0. (x=[0,1]) Intergration Chart sin(ax) xcos(ax) ∫xsin(ax)dx = --------- - ---------- a^2 a cos(ax) xsin(ax) ∫xcos(ax)dx = --------- + ---------- a^2 a ∫xJ (x)dx = xJ (x) 0 1 ∫xJ (x)dx = -xJ (x) +∫J (x)dx 1 0 0 附表內含: J (x) , J (x) , Y (x) , Y (x) , K (x) , K (x) , I (x) , I (x) 0 1 0 1 0 1 0 1 在x=0~9整數時的值表 Jn(x) , Yn(x) , Jn'(x) , Yn'(x) 在n=0~6時 等於0的x值表 J , J , J 及 Y , Y , Y 的函數圖形 0 1 2 0 1 2 --

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