  n  0,1,2,   

第二類習題：拉氏轉換

1. 試求下列階梯函數之拉普拉斯轉換(Laplace transform) :

 

0, 1, 1, f t



 

0 1 2

2 5

5 t t

 

 



【92 中山環工 20%】

2. Find the Laplace transform of the staircase function which is formed by the successive addition of unit step functions as 0, b, 2b, 3b, … , etc.【94 台大環 工15%】

3. 試求：(1)



^b



^

  

a

dt t f t t ₀

 ，at₀b；(2)^L



^



^tt₀

 

^，t₀ ^0；(3)L_^



^{a t}



^_{ ，}

0

a 。【94 台大環工 15%】

4. Find the integral ¹⁰

  

^



^



^

 

0

3 2

1 x x dx

x

e^x .【88 清大物理 8%】

5. 試證：(1)



x1



 x

 

x 或

 

 1



x1



x x ；(2)

 

1 1；(3)



n1



n!， 0, 1, 2,

n 。【91 元智工工 20%】

6. ^

 

^



^{  }

0 1 1t dt e

x ^x , then find



 







 





3 11

3 2

.【94 高科光電 10%】

7. If n is a positive integer and x n0,1,2,...., evaluate

 



x n



n x







 , where

 

^



^{  }



0

1e dt t

x ^{x t} is the Gamma function.【93 清大物理 10%】

8. Evaluate (1)^



e^ax dx

0

2 (2)



^xe^ax dx

0

2 (3)



^x e^ax dx

0

2 ² , a0.【92 交大應化 15%】

9. Solve y2y2y



t2



, y

 

0  y

 

0 0.【92 台科機械 10%】

10. Solve y ytu



t3



, y

 

0 3.【92 中央環工 30%】

11. Solve (1) I ^



^



x^



e^x dx

0

2 1

1 3 (2) dx

c

I x_x



c





0

2 , c0.【91 成大機械 8%】

12. We would like to evaluate an integral involving the derivative of the Dirac

 -function. Find the general formula for

   

0 -

- x t  t t dt







 .【91 中山電機 20%】

13.

 

2 t 1 0,

1 t 0 , {1 t

f  



  , f



t2

  

f t , find L

 

f

 

t .【92 台北化工 15%】

14. Show that !

4 1

0

4 



 









 dx

e ^x .【92 中山材料 20%】

15. If f

 

t can be expressed as

f

      

t  t1



u t1 u t2

 





u



t2

 

u t4

 





t5

  

u t4

 

u t5

 

. (1) Draw the figure of f

 

t versus t (use t as xaxis).

(2) Find the Laplace transform of f

 

t .【92 海洋電機 10%】

16. 若L

 

f

 

t F

 

s ，則

   





 



 

a F s at a

f

L 1

， af

 

at

a F s

L 



 



 



 





1 ，a0。【91 元

智工工10%】

17. Given that the Laplace transform ^{2 1 cos}

 

² ₂ ¹

1

L t ln s

s

 

    

    

  , please find the

value of

   







 t t  L 1t 1 cos 2

.【94 雲科電機 10%】

18. Solve ^{y6y9yL}

 

^xe3x , by Laplace transform.【94 高科電機 15%】

19. Solve y2y5y e ^xcosx, y

 

0  y

 

0 2 using the Laplace Transform.

【94 師大工數 15%】

20. Solve y 9 y f

 

t , y

 

0  y

 

0 1, f

 

t cost, t.【94 成大電機 18%】

21. If

 

















t 3 ,

3 t 0 2t,

0 t , 0

t2

t

f , find the Laplace transform of the given function f

 

t .【94

中原機械15%】

22. (1) Find the Laplace transform of the following function.

(2) Find the inverse Laplace transform of the following function:

35 10

12

2 

 s s

s

【94 中山電機 10%】

23. Solve the following ordinary differential equation by using the Laplace transform:



3



2

2   

 y y t

y  , y

 

0  y

 

0 0, where  is the Dirac delta function.

【93 淡江電機 20%，93 台大化工 10%】

24. Solve 0, ₃ 0

3 2

, 0

t

y y y t

e^ t

 

     .【94 高科通訊 20%】

25. Write the function f

 

t whose graph is shown in the following figure in terms of the Heaviside function, and find its Laplace transform.

【93 高科機械 15%】

26. Solve y3y2yr

 

t , r

 

t 4t, 0 t1, r

 

t 1, t1.【94 中山材料 16%】

27. Suppose f

 

t satisfies the difference-differential equation

 

 f

  

t  f tI



0 dt

t

df , t0, and the initial condition, f

 

t  f₀

 

t , 0

1 

 t where f₀

 

t is given. Show that the Laplace transform of f

 

t satisfies

    



 







   



  ⁰

1 0

0

1 1

0 e f t dt

e s e e

s s f

F _s ^u

s s

Find f

 

t , t0 when f₀

 

t 1.【93 交大電信】

28. Suppose Laplace transformation F

 

s L

 

f

 

t exists for s a0. Show that if a and b are constants with a0, then inverse Laplace transformation

 

 





 



 

 

a f t ae b as F

L ^a

1 bt

1 .

【91 中原物理 10%】

29. 一函數 f

 

t 如圖所示，且已知 f

 

t 之拉普拉斯(Laplace)轉換為

 

f

 

t F

 

s

L  ，求 



 



 



 



  b

a f t

L ，其中a,b0，a,b均為常數， f

 

t ^0 for

0

t 。【90 台科營建 15%】

30. 使用 Laplace 轉換方法計算下述系統之反應y

 

t ?



^{3 7 2 14 8}



2

2  yU t  t  t

dt y

d ，

   

0 ₀

0  

dt y dy

式中U 為單位步階函數(unit step function)。【90 嘉義土木 15%】

31. By Laplace transform, solve y4y4yt²e^2t, y

 

0 1, y

 

0 0.【92 中正 機械15%】

32. Find L

 

f

 

t , L

 

g

 

t , ^{f t}

 

^{sinatcoshbt,} g

 

t t²u

 

t1 .【92 海洋電機 15%】

33. Determine the current in the circuit:

20

R , L0.1, c1.562510^3, e

 

t 160t, 01

. 0

0 t , e

 

t 1.6, t0.01, i

   

0  i 0 0.

【92 大同電機 18%】

34. Solve y 2 yr

 

t , y

 

0  y

 

0 0, r

 

t 1, 0 t, r

 

t 0,   t2.

 

t t

r sin , t2 .【92 台大電機 15%】

35. Find

   

^_^











 

2 2 1^{2 2}

1

s s s L se

s

.【94 雲科電機 10%】

36. Given that

t s

L t 1

tan 1sin  _1



 , find 

 at L sin1t

.【94 大同電機 10%】

37. Find ^L



^tetsinh2t



.【94 中山電機 20%】

38. Solve y3y2y f

 

t , y

 

0  y

 

0 0, f

 

t 4t, 0 t1, f

 

t 8,

1

t .【94 海洋電機 10%】

39. Solve y4y21y2e^2tsin3t, y

 

0 1, y

 

0 0.【92 元智電機 20%】

40. Solve yy1



t2



, y

 

0 0, y

 

0 3.【93 台科化工 15%】

41. Solve mxcxkx

 

t , x

 

0  x

 

0 0, m,c,k,a are constants.【93 清

大物理20%】

42. Find ^L



^e3tf

 

^t



^, f

 

t 0, t6 and f

 

t  t²3, t6.【93 清大電機 10%】

43. Write the function whose graph is shown below in terms of the Heaviside function, and find its Laplace transform.

【92 高科機械 10%】

44. Consider the differential equation y

 

t 2 y2y f

 

t .

(1) Let f

 

t e^tsintcos2t. Solve the above differential equation.

(2) Let f

 

t be described as shown in figure. Solve the above differential equation.

【92 清大動機 20%】

45. Using Laplace transform to solve the following equation

 

^



^ty

 

^{x dxtH}



^t



dt t dy

0 2 ,y

 

0 1,

where H

 

t is the Heaviside step function.【92 暨南土木 15%】

46. Using Laplace transform to solve the following equations for y

 

t .

 

t f dt y

y

d²₂  4 

where f

 

t 1 for 0 t1 and f

 

t 0 everywhere else. The initial

condition for y are y

 

0 0 and

 

_{0 0}

dt

dy .【94 清大動機 15%】

47. Find the inverse transform of the function ln 1 ₂ s

  

 

 .【94 暨南電機 10%，94 高

科機械10%】

48. Find the inverse Laplace transform of the function

 

2 . tan ¹ 



 



 ^  s s

F 【92 雲科電

機10%】

49. Find L



tcos2t



.【94 清大電機 5%】

50. Find the Laplace transform of each of the following functions:

(1) ^cos

  

^{t u t1}



(2) te^3tsin2t【92 海洋電波 10%】

51. 求

   

 



²



²

2

4 2

4 2







  s s s

F 之反拉氏轉換

   

 

 

^















 ^  ₂

2 2 1

4 2

4 2 s L s s

F 。【94 高應電機

16%】

52. Solve the differential equation x16x f

 

x with the initial values x

 

0 0 and x

 

0 1, where

 

^{cos 4 , 0}

 

0, t

t t

f t 



  

 

  .【94 台科電機 15%，94 北科自

動化20%】

53. Find _^



^{eat ebt}



_^

L 1t

.【94 成大船舶 5%】

54. 求

   

 



²



²

2

4 2

4 2







  s s s

F 之反拉氏轉換

   

 

 

^















 ^  ₂

2 2 1

4 2

4 2 s L s t

f 。【94 高應電機

16%】

55. Solve the differential equation x16x f

 

t with the initial values x

 

0 0

and x

 

0 1, where

 

^{cos 4 , 0}

 

0, t

t t

f t 



  

 

  .【94 台科電機 15%，94 北科自

動化20%】

56. Find _^



^{eat ebt}



_^

L 1t

.【94 成大船舶 5%】

57. Using the Laplace transform to solve the given initial value problem.

 

t f y

y  , y

 

0 1, y

 

0 0, where

 









 

/2 t sint,

/2 t 0 , 1

 t 

f .

【94 中興機械 15%，93 台大電機 7%】

58. 若L

 

f

 

t F

 

s ，L

 

g

 

t G

 

s ，則試證：

           

0 0

t t

L^



f t g  d ^L^



f  g t d ^F s G s

【94 元智機械 10%，93 北科機電 15%】

59. Find



^{et e2t}



^*et.【94 清大電機 5%】

60. Apply the convolution of Laplace transform, find the solution of t

y

y 3cos2 ;y

 

0 0, y

 

0 0.

【93 中山物理 12%】

61. Solve the following differential equation by the method of Laplace transform.

2 2

2

1 2 1

3 x t

dt dx dt

x d

 



 , x0, 0

dt

dx for t0.

【93 中興機械 10%】

62. Show that y x

 

c1cosxsinx



0^x f s

  

sin x s ds



is a general solution to the differential equation yy f

 

x , where f

 

x is a continuous function on



 , 



.【92 交大電信 10%】

63. (1) Find the inverse Laplace transform of s cosh

 

as

1 where

 

cosh e^x 2e ^x x

 

 .

(Hint: cosh

 

z cos

 

iz where i 1)【93 台科化工 15%】

(2) Find









1 1

1 1 s s

L .【92 北科化工 10%】

64. Using the Laplace transform to solve Bessel’s equation of order zero.

0



y ty y

t , y

 

0 1.

【92 台科高分子 15%】

65. Find the Laplace transform of the given function:

 

0

sin

t d

t  







【93 交大機械 17%】

66. Solve the difference equation

 

t y

 

t y



t



t y 4 1 2 2 

3 ,

Using the Laplace transform if y

 

t 0 for t0.【93 清大動機 10%】

67. Find g

 

t

tlim0^ and g

 

t

t

lim if

    

²



²

2 3

5 2

128 216 72

16









 

s s

s s

t s g

L .【94 暨南電機

15%】

68. (1) Let y

 

t be the solution of w y



A m

  

t t

d y

d_{2 0}² / cos

2   , with

 

0 

 

0 0 dt

y dy . Assuming that  _b, find y

 

t

0

lim

 .

(2) How does this limit compare with the solution of y



A m

  

t t

d y d

0 2

2 0 2

cos

/ 

 

 ,

with

 

0 

 

0 0 dt

y dy .【94 交大物理 25%，94 海洋電機 15%】

69. (1) Solve y 4 y f

 

x , y

 

0  y

 

0 0.

(2) By convolution theorem, find h

 

t if

      _{ }

₂

2 1

 

Lh t s s s

H .【94 元智電

機10%】

70. By convolution theorem, find

 















 2 2 1

9 1

L s .【94 中興材料 10%】

71. _^1



^1et



_^?

L t 【94 清大材料】

72. Find

 

















2 2 2 1

a s

L s by using of convolution theorem.【94 交大環工 12%】

73. f

 

t t²2t1, t5, f

 

t 0, else. Find L

 

f

 

t .【93 北科電機 15%】

74. Find

 

_



 





 ^1 ₄ ³ ₄ 4a s L s t

f .【94 高應電子 15%】

75. 下圖函數 f

 

t ，在4 t6時為拋物線，其他時候為0，求其拉氏轉換 Laplace transform F

 

s 。

【94 北科光電 10%】

76. Find

 















 2 2 1

1 s

L s .【93 海洋電機 10%】

77. Solve y2y2y



t3



, y

 

0 0.【93 淡江電機 20%】

78. Solve y2y2yr

 

t , r

 

t 5sin2t if 0 t and 0 if t  ,  y

 

0 1,

 

0 5

y .【92 中山物理 15%】

79. Solve ^{y 9}



₀^{ty t dt}

 

^{cos 4t, y}

^{ }

^{0 0}.【92 中正機械 20%】

80. Find



1



^.

ln 1₂

2

1 



 







  s

L s 【92 高科電子 20%】

81. Find



^{28 2}



^{3 .}

3 1

















k s

s

L k 【92 中正電機 8%】

82. Let R1 ohm, L1 henry, and the input E

 

t 1 volt, when 0 t3sec, and E

 

t 0, when t3sec. Find the current I

 

t , assuming I

 

0 0.5 ampere.

【93 大同電機 18%】

83. Find the current I

 

t in the figure with R100 ohms, C0.1 farad and

 

t 100

v volts if 1 t2 and 0 otherwise, v_c

 

0 0.

【93 暨南電機 20%、93 師大電機 16%】

84. Find L

 

f

 

t , f

 

t 0, 0 t4, f

 

t e^3t, 4 t6, and f

 

t  t1, t6.

【93 雲科電機 15%】

85. Solve the following differential equation by Laplace Transformation:

0 2  

 y ty y

t , y

 

0 a.

【93 北科化工 20%】

86. Solve y16ty32y14, y

 

0  y

 

0 0.【94 雲科光電 15%】

87. Solve y2ty4y1, y

 

0  y

 

0 0.【94 成大製造 15%】

88. By applying the Laplace transformation technique to solve the following differential equation:



4 2



4 0

 t y y y

t , y

 

0 1, y

 

0 2. Please derive its solution. Does there exist a unique solution?

【92 海洋機械 20%、92 台科電子 12%】

89. Using the Laplace transform to solve the equation ₂

 

1 0

2   ny

dt t dy t

d y

td in

which n is any positive integer.

 

^{n n j}

j

y j x y n

x ¹

0





 

 



 

 

(Hint: The binomial expansion formula is useful.)

【95 清大電機 10%、94 海洋電機 20%】

90. Find the solution of a differential equation

2 0

2  y

dt tdy dt

y

td ; y

 

0 0,

 

0 5 dt

dy .

【94 台科化工 15%】

91. Given that t

 

1t y2y2y6t; y

 

0 0, y

 

2 0. Please use the Laplace transform to solve the problem.【91 成大電機 20%】

92. Let u

 

t denote the unit step function, find the Laplace transform of the following function

 

^{sin 3 4 4}

6 6

f t    t    u t .【94 台科電機 10%】

93. Using Laplace transform to solve the following system equations:

 

 





















2 1 2 2

1 2 1 1

ky y y k y

y y k ky y

with y1

 

0 1, y2

 

0 1, y1

 

0  3 ,k y2

 

0   3k .【93 成大醫工 20%】

94. Using Laplace transform to solve the following linear system:



















0 3 2

4 2 3 2

y x y

y y x x

with the initial conditions x

 

0 x

   

0 y 0 y

 

0 0.【93 海洋通訊導航 15%】

95. Using Laplace transform to solve the deflection u

 

x of a fixed-end beam of length l subjected to a concentrated loading P as shown in the following differential equation.



 

 

 3

4

4 l

x dx P

u

EI d  , 0xl,

with the boundary conditions u

   

0  lu 0 and

 

0 

 

₀ dx

l du dx

du , where

 

 is the Dirac delta function and the rigidity EI and P are constant.

【93 成大土木 20%】

96. The RLC in-series circuit with R 2

 

 , L 1

 

H , and C1/5

 

F . Using Laplace transform to solve the loop current, i

 

t , which the initial conditions are

   

A

i 0 2 , and ^i

 

^{0  4}

 

^{A .}【94 海洋電機 10%】

97. Given a mass-spring-damper system, with unknown values of K and C, an impulse function r

   

t  t generates an output response as y

 

t e^te^2t. Now if we are given another input function r

 

t sint, please find the

corresponding output response.【94 中正光機電 10%】

98. The initial value problem is given by

 

0

8

2x xy  , zxy, x

 

0 2, x

 

0 0,

 

t 2t² y  for

0 6



 t ,

 





 

 



 3 6

2 18

2  

 t

t

y for

6

 t ,

where x x

 

t , yy

 

t , zz

 

t and their first derivatives are continuous functions of t . Determine z

 

t for t0 and evaluate z



/6



, z



/6



,



/3



z and z



 /3



.【95 交大機械 17%】

99. Using Laplace transform to solve the boundary value problem:

 

x x y y

y 2   , y

 

0 0, y

 

1 2.

【95 清大工程科學 11%】

100. Solve y2ty4y1, y

 

0  y

 

0 0.【91 台科電機 10%、92 中原化工 14%】

101. Consider the following differential equation



1

 

3 2



0

2    

 x y x y

y x

where y

 

x is piecewise continuous on



0,



and of exponential order for T

t .

(1) Y

 

s is the Laplace transform of y

 

x . Please find Y

 

s

lims . Note that you have to present the calculation procedure to get the score.

(2) If y

 

0  and y 0

 

 , please find Y

 

s in terms of  and . (3) Find the inverse Laplace transform of Y

 

s , i.e., y

 

x .

(4) How many solutions do you get if  and  are given? Please explain why.

【91 台大電機 20%】

102. Solve y4ty4y0, y

 

0 0, y

 

0 10.【94 海洋機械 15%】

103. Find L

 

y

 

t ,

0^t sin 2

y  y e^t



t tdt.【94 成大機械 12%】

104. y^{ 4} 2yy1, y

 

0  y

 

0  y

 

0  y^{ 3}

 

0 0. Solve by Laplace transform.

【94 中央光電 10%】

105. Solve y^2y^6



0¹z

 

t dt^2u

 

t , 0yzz , y

 

0 5, z

 

0 6.【95 台

科機械20%】

106. Solve by Laplace transform:

 

1

2 1

1y  u t

y , y₂ y₁1u

 

t1 , y₁

 

0  y₂

 

0 0.

【93 中山材料 20%】

107. x4x2y2t, 1y8x4y  , x

 

0 3, y

 

0 5. Solve by Laplace transform.【93 中興化工 10%】

108. Solve y8ty16y3, y

 

0 0, y

 

0 0.【93 淡江機械 15%】

109. 0xy2yxy , y

 

0 0. Solve by Laplace transform.【93 台大電機 15%】

110. By Laplace transform, solve xxy2, 0yy2z , zxycost,

 

0 1

x , y

 

0 0, z

 

0 2.【93 元智通訊 20%】

111. Solve ty



t3



y2y0, y

 

0 0.【89 台科電子 10%】

112. Using Laplace transform to solve 0

3

2 _{2 3}

1 y  y 

y , y₁4y₂3y₃ t, y₁2y₂ 3y₃ 1.【92 成大電機 10%】

113. y₀²yBsint, y

 

0 0, y

 

0 0. (1) ₀ 

(2) ₀  【95 交大機械 20%】

114. Find the Laplace transform for the following periodic function.

(Note: In this problem, you should assign: y f

 

t , x and t t0.)

【94 中興化工 10%】

115. 單選題，每題恰有一解，答對一小題給 5 分，答錯或不答，不給分也不扣 分。

(1) Define a function g

 

t by

 

^{2 , if 0 and 3 3 3, 0,1, 2,}

0 , if 0

t n t n t n n

g t t

      

 

 



What is the Laplace transform of g

 

t ? (5%) (A) 2₂

s (B) _s

s

e s s

e









2 2

2 (C) _s

s

e s s

e s

2 2 2

2

2 2







 (D) _s

s s

e s s

e s e

3 2 2

3 2

2 3

2











 (E) none.

(2) Let

    















 ^  ^{ }

s s s

e e L s

t h

s s

8 2 8

2 3

2 2

1 . lim

 

?

1_ 

 h t

t

(A) 0 (B) 1 (C) 2 (D) 3 (E) none 【94 交大電機 10%】

116. 求sint 之拉氏轉換L



sint



。【93 北科電機 15%】

117. Given the periodic function

 

^{sin , 0}

0, 2

t t

f t t



 

  

    . Find the Laplace transform

of

 

f

 

t .【94 北科電機 15%，94 清大微電機 10%】

118. Solve the following differential equation with initial conditions given

 

t f y y

y2 10  , y

 

0 1, y

 

0 0, where f

 

t is given by the following figure.

【94 中原機械】

119. Consider the RLC circuit shown below. Initially there is no current in the circuit and no charge in the capacitor. At time t0, the switch is closed and left closed for 1 second. At time t1 second, the switch is opened and left open. Find the current in the circuit.



 150

R , L 1 H , C0.0002F, V 50V .

【93 交大電子 12%】

120. Find the current I

 

t in the figure with L1 Henry, c1 farad, zero initial current and charge on the capacitor, and v

 

t t if 0 t1 and v

 

t 1 if

1 t .

【94 師大電機 15%】

121. Solve yy f

 

t , t0, f

  

t  t 3



, y

 

0 0, f

 

t  tf



5



.【92 台大

生物環境15%】

122. Find steady state current of the following circuit.

250

R , L0.02, c210^6.【90 中興精密 20%】

123. (1) Find the Laplace transform of the function f

 

t as shown.

(2) What is the solution of the equation, if y₀ 1 ant if f

 

t is given as in figure with k1?

【90 中興精密 20%】

124. Find L

 

f

 

t , f t  f

 

t



 

 





2 , f

 

t 0,





0 t , f

 

t sint,







  t2 .【91 暨南電機 10%】

125. (1) Find a Laplace transform of the given infinite-duration pulses in Fig.1.

(2) For a first-order RL circuit in Fig.2 if v_t

 

t in part2 is used as an input, using the Laplace transform method, show that

  

^

      

^{ }

 









 











0 0

1 1

n n

n n t

nu t n e u t n

t i

where u

 

t is a unit step function.

【91 交大電子 12%】

126. Solve ^{f t}

 

^2t2



₀^{tsin 4}

  

^{ f t }



^d .【93 清大電子 7%、93 台大工程科 學20%】

127. Solve the following integral equation:

 

^{t cost} ₀^t

 

^{cos d sint} ₀^t

 

^{sin d sin 2t}

 



    



    

【94 暨南電機 15%】

128. Solve the following integral equation

 

^{sin 2}

 

₀^t

 

^{sin 2}

   

y t  t 



y  t d ^.

【94 中央電機 10%】

129. Prove that the Beta function:

       

 

1 1

, 0^{t m} 1 ⁿ m n

B m n x x dx

P m n

   

   



 ^{, m0, n0, and}  is Gamma

function.【91 北科機電 15%】

130. Find ^{2 2}

0^t cos3

t a

L e^_ ^ e  d ^_





by methods below.

(1) Convolution theorem. (10%)

(2) Using ^L



^eatf

 

^t



^F

^

^s

^

^{and L} ₀^{t f}

 

^{d 1F s}

 

  s

  

 





 ^{. (15%)}

【94 元智電機 25%】

131. Solve ^{y t}

 

^{ 2 3et }



₀^tety

 

^{ d} .【94 宜蘭電機 10%】

132. Solve ^{f t}

 

^6t2



₀^{t f t}



^



^{e d } .【94 淡江電機 10%、93 台科機械 20%】

133. Find y

 

t , ^{y t}

 

^{sint4et 2}



₀^ty

  

^{ cos t }



^d .【93 交大應化 10%】

134. Solve ^{y t}

 

^{ 6t}



₀^{ty t s}



^



^sinsds.【93 清大微機電 10%】

135. ^{y t}

 

^{ 1 sinh}



₀^t



^1

 

^{y 1 }



^{d ，求y}

^{ }

^{t 。}【93 台大生機 10%】

136. Find y

 

t , ^{y t}

 

^{cost e 2t}



₀^{t f}

 

^{ e d2 } .【92 中興物理 15%】

137. Solve y



ab



yaby f

 

t , y

 

0 c, y 0

 

d .【92 成大工程科學 15%】

138. Find y

 

t , ^{y te t 2et}



₀^{te y}

 

^{ d} .【92 成大製造 15%】

139. Find f

   

t g t , f

 

t t, g

 

t e^2t.【92 嘉義電機 20%】

140. Solve ^{y  y 4}



₀^ty

  

^{ sin t }



^{d e2t, y}

^{ }

^{0 1, y}

^{ }

^{0 1.【92 暨南電機}

10%】

141. (1) Derive ^L



^tsint



^



^{s222}



^{2 by using L}

^{ }

^tf

^{ }

^{t dsd L}

^{ }

^f

^{ }

^{t .}

(2) Solve ^{y 2t 4}



₀^ty

 

^{ t }



^d .【95 交大機械 10%】

142. Solve by Laplace transform xt t u x

x u 







 , u

 

x,0 0, u

 

0,t 0.【94 中山光 電15%】

143. A semi-infinite string at rest along the positive axis with the left end moving in a prescribed fashion. The displacement of the string can be described as following:

2 2 2 2 2

x a y t

y



 



 ,

 

^{sin 2}

 

^{, 0 1}

0, 0, 1

t t

y t

t



  

 

 

 

^{,0 y}

 

^{,0 0}



⁰



y x x x

t

  



Pease find y ,

 

x t using the Laplace transformation.【93 中興材料 20%】

144. Solve x

t x v x

v 2 2



 



 , v

 

x,0 1, v

 

0,t 1, by Laplace transform.【94 北科 化工20%】

145. By Laplace transform, solve

2 2

x u t u



 



 , x0, t0,

 

x,0 100

u , u

 

 t, 100, u

 

0,t 20, 0 t1, u

 

0,t 0, t1. Notice: The Laplace transform of complementary error function is

s

e a

t s erfc a

L  ^



 



 



 



 1

2 .【92 淡江化工 25%】

146. Solve ^{2 2}₂ ²₂ t

u x

a u



 



 , 0x, 0t, u

 

x,0 0, u_t

 

x,0 0, u ,

 

 t is finite.

【93 淡江航空 25%】

147. Solve ₂

2

x u t u







 , 0 x , t0, u

   

0,t u x,t 0, u

 

x,0 sinx.【92 中

原電機10%】

148. Solve the following boundary value problem by Laplace transform.



0, 0



2

2  



 



 x t

x u t u

  

x,0  A x0



u , u ,

 

 t is finite

 







 

0 0

, 0

0 , ,

0 for t t

t t for t B

u

Note: The Laplace transform of 



 



 t erfc a

2 is e ^{a s} s 1 

, erfc

 

x  1efc

 

x .【90

北科光電20%】

149. Solve a u² _xx  , u_tt x0, t0, a is a positive constant, u

 

x,0 0,

 

x k

u ,0  , u

 

0,t 0, u ,

 

 t is bounded, solve u.【94 淡江航空 15%】

150. Find the solution p ,

 

x t of the following partial differential equation:



x at



t p c x

p  



 



²₂ ¹₂ ²₂  , c a0, 0 x, 0 t.

 

x,0 0

p , p

 

x,0 /t0, p

 

0,t 0, p ,

 

x t  as x. Note:  denotes the Dirac delta function.【88 交大機械 20%】

151. Using Laplace Transformation (with respect to t ) to solve the partial differential equation.

 

x t a y

 

x t g

y_u ,  ² _xx ,  , x where a and g are constants.

And y ,

 

x t satisfies the boundary conditionsy

 

x,0  y_t

 

x,0 0, y

 

0,t 0,

 

, 0

lim 



 y_x x t

x .【89 逢甲土木 15%】

152. Solve ku

z a u t

u 



 



 , u

 

t,0 bsint, u

 

0,z 0.【92 台科化工 15%】

153. Calculate Laplace transforms of real-valued, square-wave function f

 

t with period 2c, where

 

t 0

f if t0

 

t 1

f if nct



n1



c, and

 

t 1

f if



n1



ct



n2



c, for n0, 2, 4...

【89 交大機械 16%】

154. 求 ^



at b x t dt

  

  



^。【88 台科電子 5%】

155. Find the integral



₀^10ex _^



^x1



^x2



^x3



^_^dx.【88 清大物理 8%】

156. 一函數 f

 

t 如圖所示，且已知 f

 

t 之拉普拉斯(Laplace)轉換為

 

f

 

t F

 

s

L  ，求 



 



 



 



  b

a f t

L ，其中a,b0，a, 均為常數。 b

【90 台科營建 15%】

157. Find L



sinatcoshatcosatsinhat



.【90 海洋光電 10%】

158. By Laplace transform, solve y8y16yt²e^4t，y

 

0 1，y

 

0 4.【90

台大電機10%】

159. Using the Laplace transform to solve the following differential equation.



 







 





 t t y t

y 3cos ,

0 ,

4 0 , y

 

0  y

 

0 1.

【90 雲科電機 10%】

160. Using the Laplace transform to solve the following initial value problem:

 

1 4

4   

 y y t

y  , y

 

0 0, y

 

0 1.

【90 中山電機 20%】

161. y5y6y8

 

t , y

 

0 3, y

 

0 0. (1) Find solution of ^{y t}

 

^.

(2) What are ^y

 

^{0 , y}

 

⁰ ? Using this information, what physical phenomena does delta function model?【90 海洋機械 14%】

162. (1) Solve y2y3u2u, y

 

0 2, where u

 

t is a unit step function.

(2) Check the solution of part (1) if y

 

0 2. If not, try to explain.【90 交大機械 25%】

163. (1) From the properties of Dirac delta function, expand ^



^14t2



as the sum of delta functions with simple argument; that is, find the parameters A and _n

a such that n 



14t²



A₁



ta₁



 A₂



ta₂



... holds.

(2) Solve the following equation by Laplace transform:



²



2 2

4 1 2

2 y t

dt dy dt

y

d     , y

 

0 0, y

 

0 0. (3) Is the solution y

 

t in (2) continuous at

2

1

t ? If not, how are ^y

 

^{t and}

 

^t

y related? Explain how you can figure out this relationship simply from the equation itself without actually solving for the solution.【88 清大電機 15%】

164. Show that the Laplace transform of ln

 

t is

    ^{ }

s t s

L 1 ln

ln   where the Gamma function is defined as ^

 

^{r }



₀^{u e dur1 u} .【87 清大動機 10%】

165. Solve the equation y4y4y f

 

t , y

 

0 1, y

 

0 2.

 

t 1

f , 0 t2

 

t 0

f , t2

【90 北科化工 20%】

166. Find the inverse Laplace transform of



3

 

^{2 4 13}



3 2

3



 







s s

e s

e ^{s s}

.【88 台科電子