s 1 國立台灣海洋大學河海工程學系工程數學 二 班期末考 ( ) 2B

班級：_____________學號：______________姓名：______________

國立台灣海洋大學河海工程學系工程數學(二) 2B 班期末考

1. Find the Laplace transform of f(t). (20 %)

Table 1: Mapping for the Laplace transform (填入答案卷才給分)

f(t) p1(t) p2(t) p3(t) p4(t) p5(t) p6(t) p7(t) p8(t) p9(t) p10(t) F(s)

Time function, f(t) 1

)

1(t 

p p2⁽t⁾^cos(t⁾ p3⁽t⁾ ^sin(t⁾ p₄(t)tcos(t) p5⁽t⁾t^sin(t⁾ et

t

p₆( ) p7⁽t⁾^cosh(t⁾ p₈(t)sinh(t) p₉(t)tcosh(t) p₁₀(t)tsinh(t)

s function, F(s)

(1) s

1 (7) _{2 2}

2

) 1 (

1



 s

s

(2) s² 1

s (8) (s2 1)2

s

(3) s² 1

s (9) _{2 2}

) 1 (

2

 s

s

(4) 1 1

2 

s (10) _{2 2}

2

) 1 (

1



 s

s

(5) 1 1

2 

s (11) (s2 1)2

s

(6) 1 1



s (12) ( 2 1)2

2

 s

s

2. Find the inverse Laplace transform of F(s). (20 %) (1) ₂

1 se s

s



 (3) 1

(s3)(s5)

(2) ₂ 1

(s 0.5 )s (4) ^s2s16²

3. Using Laplace transform and inverse Laplace transform to solve the following ODE (1). y t'( )y t( ) 0, (0) 1 y  (10 %)

(2). y t'( )y t( )e y^t, (0) 0 (10%)

4. Find the Fourier transform of p function. (10%)

( ) 1 0

, -1 t p t 1

, otherwise

  

 

( ) g u

u

0 

5. Given

(a) Plot f(-u) versus u (5%)

(b) Plot f(t-u) versus u for (1) t0 (2) 0 t  (3)   t 2 (4) 2 t (5%) (c) Find

(1) t0 (2) 0 t  (3)   t 2 (4) 2 t (20%)

6 Find the Laplace transform of g function. (10%)

If the Laplace transform of g function is G(s), please find (5%) and plot (5%) the function whose Laplace transform is e G s^9s ( ). Please find (5%) and plot (5%) the function

whose Laplace transform is G s( ) /(1e^2s).

7. 學了一學年工數，又上完壹學期 有何心得? 建議學弟妹要如何學才會學好 ? (5%) 有何建議給 助教或老師作為下年度教學參考 ? (5%)(本題有寫才有分)

參考公式

transform ( ) 0 ( ) ^st , Fourier transform ( ) ( ) ^{i t}

Laplace F s ^ f t e dt^ F  ^ f t e^dt

 

1. 平移定理，s-平移L e f t( ^at ( ))F s a(  ) 平移定理，t-平移  L f t a u t a{ (  ) (  )}e F s^as ( ) 2. 微分的拉氏轉換: L f( ) sF s( ) f(0), ( )L f s F s² ( )sf(0) f(0)

3. 積分的拉氏轉換:

0

( ) 1 ( )

t

L f d F s

  s

 

 

 

4. 階梯函數的拉氏轉換: ( ( )) e as

L u t a s

   Dirac 衝擊函數的拉氏轉換: L( ( t a ))e^as

5. 摺積定義: ( * )( )f g t ^ f( ) ( g t  )d



   correlation 定義: (f g t)( ) ^ f( ) ( g t  )d



  



6. 摺積的拉氏轉換: ^{L f g t}^{( * )( )} ^{F s G s( ) ( )} 摺積的傅立葉轉換: ^F^{( * )( )f g t}  ^{F( ) ( ) G }

7. Parseval’s theorem ² 1 ²

( ) ( )

f t dt 2 F  d



 

  

 

8. correlation 的傅立葉轉換: ^F^{(f g t} ^{)( )} ^{F() ( )G  F( ) ( ) G }

9. 拉氏轉換的微分: L tf t( ( )) F s( )，拉氏轉換的積分 ^{( ( ))}  

s

L f t F s ds t



 ^{ }

10. 周期函數(t>0) 的 Laplace transform ₀ ( ) { ( )}

(1 )

T

st

sT

f t e dt L f t

e



 



 (T 為周期) 2012-exam-03.doc Chen J T

0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

( ) 0

, 0 u sin(u) f u , otherwise



  

 

( ) 0 , 0 u g u 1

, otherwise



  

 

( ) ( ) ( )

h t -^ g u f t-u du



( ) 1 0

, 0 t g t 1

, otherwise

  

 