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1 Solve the equation xy   2 y  e x , y (1)  , 0 (a) the general solution

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國立勤益科技大學 100 學年度研究所碩士班招生筆試試題卷 所別：機械工程研究所 組別：

科目：工程數學

准考證號碼：□□□□□□□□（考生自填）

1 Solve the equation xy   2 ye x , y (1)  , 0 (a) the general solution

(b) the particular solution?

2. Solve the equation (using the separating variables method)

2 2

y 1 y xy

  x  , y (1) 1  (a) the general solution (b) the particular solution.

3. Find the solution of the following equation by the method of Laplace transform

2 t

y   y   yte , y (0) 1 , (0)  y   2

4. Find the solution of the following equation by the method of Laplace transform y    y t , y (0) 1 , (0) 1  y

5. If 3 1

6 2

A   

  

 

, 1 4

2 3

B  

  

  

find

(a) the inverse of A ? (b) the addition of A and B . 6. The matrix A find

(a) Eigen-values ?

(b) Eigen-vector of matrix A. ?

2 2 3

2 1 6

1 2 0

A

  

 

 

 

 

   

 

第 1 頁〈共 2 頁〉

考生注意事項：

1.考試時間 100 分鐘 3. 請依序做答於答案卷。

2.可用無程試之計算器 4.每題各 10%，共十題。

(2)

7. Evaluate the integral dr c   

, where ( , , ) x y z function is ( , , ) x y zxyyzxz , and the path c x :  cos( ), t y  sin( ), t zt , 0   t .

8. Evaluate the integral F dA s



, where the surface is

S : x 2  y 2  ( z  1) 2  9 , 1  z  4 , z  1 and [ , , 1]

F   x y z

9. Solve the Eigen value problems y "  y  0 , y ( 0 )  0 , y ( L )  0 (a) the Eigen values.

(b) the Eigen functions.

10. The time function is f t  , ( ) 1 0   t . (a) Expand in Fourier sine series.

(b) Sketch the function.

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