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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◆ From: 218.167.192.4
※ 編輯: tsf73 來自: 218.167.192.4 (01/13 19:44)