課程名稱︰工程數學上
課程性質︰系訂必修
課程教師︰蕭浩明
開課學院：工學院
開課系所︰機械工程學系
考試日期（年月日）︰2011/01/10
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 :
2
1.Solve (x + 2x + 1)y'' - 2(x + 1)y' + 2y = 0 y(0)=a0 y'(0)=a1
By Taylor Series Method (15%)
2.Use power series method to find the general solution near x=2 of
y'' - (x - 2)y' + 2y = 0 (15%)
Hint:Use a concept learned in ENG MATH 1 to transfer one variable to
another and SHIFT the expansion point to zero.
3.Determine the eigenvalues and eigenvectors of
┌ ┐
│cosθ -sinθ│
A = │sinθ cosθ│ (15%)
└ ┘
4.Find how many independent chemical reactions from the following list:
i. 4 NH + 5 O = 4 NO + 6 H O
3 2 2
ii. 4 NH + 3 O = 2 N + 6 H O
3 2 2 2
iii. 4 NH + 6 NO = 5 N + 6 H O
3 2 2
iv. 2 NO + O = 2 NO
2 2
v. 2 NO = N + O
2 2
vi. N + O = 2 NO
2 2 (20%)
Hint:Write the above six chemical reactions in the matrix form and
perform the necessary calculations.
-1 5
5.Use Cayley-Hamiton theorem to calculate A =？ and A =？
┌ ┐
│-4 5 5│
A = │-5 6 5│
│-5 5 6│
└ ┘ (20%)
6.Find the eigenvalues and eigenvectors of A and prove A's eigenvectors
are orthogonal.
┌ ┐
│2 4 -6│
A = │4 2 -6│
│6 -6 -15│ (20%)
└ ┘ T
Hint:Eigenvector Orthogonality: X X = 0 ,where X and X are eigenvectors.
j i i j
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