[試題] 98上 王昭男 工程數學一 2st期中考

[fong1014 ()]

課程名稱︰工程數學一
課程性質︰必修
課程教師︰王昭男
開課學院：工學院
開課系所︰工程科學及海洋工程學系
考試日期（年月日）︰2009.12.10
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：

1. Find the eigenvalues and eigenvectors of the problem,


    d    2x dy      2x
   ──(e  ──) + e  (λ+1)y = 0  ；  0 < x < π
    dx      dx

    y(0) =  0

    y(π) = 0


2. Use power series method (with center x= 0),find the solution of the

   following equation：

     2
   (x -x)y"-xy'+y = 0



3. Find the general soluion of the nonhomogenous equation：

                ┌        ┐    ┌  2       ┐
                │ 2   -4 │    │ t + 2t   │
  y'= Ay + g =  │        │y + │  2       │
                │ 1   -3 │    │ t +t+3   │
                └        ┘    └          ┘


4. If the general solution of the Bessel's equation

    2            2    2
   x y" + xy'+ (x - ν )y = 0  can be expressed by Bessel function


   y(x)=  AJ(x) +BY(x)
           ν     ν

  Find the general solution (in terms of the Bessel function) of the

  equatuon：

   2               2  2ν    2
  x y"+(1-2ν)xy'+ν(x   +1-ν)y = 0

             -ν         ν
  Hint: u = x   y   z = x

5. Find the critical points and its type of the following equation of

             3
   y"+y'-4y+y = 0


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