Tarea Transoformada Directa

8/12/2019 Tarea Transoformada Directa

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30 THE LAPLACE TRANSFORM

Then differentiating with respect to r, we find

from which

r (r) f ' ur-1 e-u lnu du0

r (1) f ' e- u In u du0

Letting u = st 8 > 0, this becomesr (1) 8 f ' e s t (ln 8 + ln t) dt

0

Hence t: {In t} f ' e s t ln t dt0

T''(1) i - ln8 e s t dtAnother method. We have for k > -1 ,

ln 88

8 0

y + I n s8

f ' e s t tk dt0 r(k + 1)gk+ 1Then differentiating with respect to k

i ' e s t tk ln t dt0

Letting k =0 we have, as required,f ' e s t ln t dt

0t: {ln t}

r (k + 1) - r k + 1) In 8gk + 1

r (1) - ln 88

Supplementary roblems

y + Ins8

[CHAP.1

LAPLACE TRANSFORMS OF ELEMENTARY FUNCTIONS51 Find the Laplace transforms of each of the following functions. In each case specify the values of 8

for which the Laplace transform exists.(a) 2e4t Ans. (a) 2/ 8- 4), 8 > 4b) 3e-2t b) 3/ s + 2), s > - 2 .

(c) 5 t 3 (c) 5 - 3s)/s2, s > O(d) 2 t 2 - e t (d) (4 + 4 8 g3)fg3 8 + 1), 8> 0e) 3 cos 5t e) 38/ 82 + 25), 8 .> 0

(f) 10 sin 6t (f) 60/ 82 + 36), 8 > . 0(g) 6 sin 2t - 5 cos 2t (g) 12- 58)/ 82 + 4), 8 > 0h) t2_ + 1)2 h) s4 + 4s2 + 24)/s3, 8 > 0i) sin t - cos t)2 i) (82- 28 + 4)/8 82 + 4), 8 > 0J) 3 cosh 5t - 4 sinh 5t (j) 3 s - 20)/(s2- 25), 8 > 5


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CHAP: 1] THE LAPLACE TRANSFORM

52. Evaluate a) r: _ {(5e2t- 3)2}, b) r: _ {4 cos2 2t}.-25 30 9 2 2sAns. (a) 4 - 2 + s 8 > 4 b) 8 + s2 + 16' 8 > 0

53. Find r: _ {cosh2 4t}. s2 - 3 2Ans. s(82- 64)

54. Find r: _ {F(t)} if a) F t) = O< t < 2 (b) F(t) o;;;at;;;as=t > 2 t > 5Anso a) 4e-2s/8 b)

55. Prove that

2 - . . e-5s1 e-5s)g2 8n

8n+1 n = 1,2,3, 0 0 056. Investigate the existence of the Laplace transform of each of the following functions.

(a) 1/(t+ 1), b) t 2 te , c) cos t2 An8. a) exists, (b) does not exist, c) exists

LINEARITY, TRANSLATION AND CHANGE OF SCALE PROPERTIES57. Find r: _ {3t4 - 2t3 + 4e-3t - 2 sin 5t + 3 cos 2t}o

72 - 12 + _ 4 _ - ____ Q_ +A n8 8 5 84 8 + 3 s2 + 25 s2 + 458. Evaluate each of the following.

a) r: _ {t3e-3t} Ansb) e {e-t cos 2t}c) t: {2e3t sin 4t}d) t: { t + 2)2et}e) t: {e2t 3 sin 4t - 4 cos 4t)}f) r: _ {e-4t cosh 2t}g) r: _ {e-t 3 sinh 2t - 5 cosh 2t)}

59. Find a) r: _ {e- t sin2 t}, b) r: _ { 1 + te-t)3}o

a) 6/ 8 + 3)4b) (8 + 1)/ 82 + 28 + 5)c) 8/(s2- 68 + 25)d) 4s2 48 + 2)/ 8 -1)3e) (20- 48)/(s2- 48 + 20)/) (8 + 4)/ 82 + 88 + 12)g) (1 - 58)/ 82 + 28 - 3)

An8. a) 2 1 3 6 6b) 8 + (s + 1)2 + s + 2)3 + (s + 3)4s + 1) 82 + 2s + 5)60. Find r: _ {F t)} if F(t) = { ~ - 1 2 t > 1O< t < 1

31

61. If Ft(t), F 2 t), ooo Fn t) have Laplace transforms / 1 s), / 2 s), . oo, fn s) respectively and c1, c2 o oo, cn are any constants, prove that


3/6

32 THE LAPLACE TRANSFORM. , . ' 62. If t: {F(t)}

63. I f .e {F(t)} =

82 - 8+128 + 1)2 8- 1) , find t: {F(2t)}.

-8- find ;.e {e-t-F(3t)}.

64. I f / 8) = .e {F(t}}, prove that for r > 0,.e {rt F(at)}

LAPLACE TRANSFORMS OF DERIVATIVES65. a) If t: {F(t)} = / 8), prove that

Ans. 82- 2s + 4)/4 8 + 1)2 8 - 2)

Ans. e 3 s 1>8+1

1 I 8 ~ I n r)lnr at: {F '(t)} = 83 / 8) - 82 F(O) - 8 F'(O) - F O)

stating appropriate conditions on F(t).b) Generalize the result of a) and prove by use of mathematical induction.

CHAP.1

{ 2t 0 ~ t ~ 166. Given F(t) =. t t > 1 a) Find .e {F(t)}. b) Find t: {F'(t)}. . c) Does the resultt: {F (t}} = 8 .e {F(t}} - F(O) hold for this case?

2 e e 2 e sAns. a) 82 s 82 b) 8 8

67. (a) I f F(t) = { O < t ~t > 1 find .e {F (t)}.

Explain.

b) Does the result t: {F (t)} = 82 t: {F(t)} - 8 F(O) - F'(O} hold in this case? Explain.Ans. a) 2(1 - e-)/s

68. Prove: a) Theorem 1-7, Page 4; b) Theorem 1-8, Page 4.

LAPLACE TRANSFORMS OF INTEGRALS69. Verify directly that .e{t u2-u+e - u)du} = . . .e {t2- t + e-t}.870. I f / s) = .t:_{F t)}, show that .e{it F u)du} = 1:).

[The double integral is sometimes briefly written as iti t (t} dt2.J71. Generalize the result of Problem 70.

72. Show that t: {. 1 -:-uu} = In 1 +


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CHAP. 1)

MULTIPLICATION BY POWERS O t74. Prove that a) .t:.. {t cos at} =

b) .t:.. {t sin at} = 2as75. Find .t:.. { 3 sin 2t - 2 cos 2t)}. 8 +12s- 2s2An8. 82 + 4)276. Show that .t:.. {t2 sin t} 682 2s2 + 1)3 77. Evaluate a) .t:.. {t cosh 3t}, b) .t:.. {t sinh 2t}.

78. Find a) .t:.. {t2 cost}, b) .t:.. { t2- 3t+ 2) sin 3t}.Ans. a) s2 + 9)/ 82- 9)2, b) 4s/ s2- 4)2

Ans. a) 2s3 _ 6s)/ s2 + 1)3, b) 684 -1883 + ( ~ 2 s ; ; ; 162s + 432

79. Find .t:.. {t3 cost}. 6s4- 36s2 + 6Ans. . s2+ 1)480. Show that . ( ' t e -a t sin t dt

DIVISION BY t81. Show that ..,_ ={e. a t -t ebt.}

350.

I n ~ ) 8 +a. -

82. Show that .t:.. {cos at - cos bt} = . In s2 + b2 ) .t 2 82+a283. Find .t:.. { t} . 1 8 + 1ns. 2In 8 1

i ' St e 6 t dt84. Show that 0 t -[Hint Use P:r:oblem 81.]

85. Evaluate ( ' cos 6t - cos 4t dt.Jo t

' sin2 t rr86 . Show that0

r t = 2 .PERIODIC FUNCTIONS

In 2.

Ans. In 3/2)

87. Find .t:.. {F t)} where F t) is the periodic function shown graphically in Fig. 1-7 below.1 sAns. s anh 2

33


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34 THE LAPLACE TRANSFORM

F t) F t)2

1 2 31

Fig. 1-7 Fig.l-8

88. Find .J:. {F t)} -.vhere F t) is the periodic function shown graphically in Fig. 1-8 above.1 e- sA ns. s - -;-:;;--.,.,s 1 - e )

[CHAP.1

4

89. Let F t) = {:t O < t < 2 where F t) has period 4.2 oo where p > ~ l prove that / s ) - cr p+l ) / sP+l ass -> oo .


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CHAP.l] T H E LAPLACE TRANSFORM

THE GAMMA FUNCTION98. Evaluate (a) r(5), b) r < ~ } ~ t ) , (c) r(5/2), (d) r(3/2) r(4) .r(ll/2)

Ans. (a) 24, b) 1/60, c) 3 ~ / 4 . d) 32/315

99. Find (a) .e_ {tl/2 + t - 112}, (b) .e_ {t-113}, c) .e_ {(1 + vft )4}.Ans. (a) (2s + 1)y.;i/2s3l2, (b) r(2/3)/s213, . c) s2 + 2 ~ s 3 2 + 6s + 3y.;is 2 + 2)/sa

100. Find (a) .e_ {7.} (b) .e_ {t7 2 e3t}.Ans. (a) ..j1r (s + 2), (b) 105v;;:/16(s- 3)9/2

BESSEL FUNCTIONS101. Show that 1..j 2 - 2as + a2 + b2102. Show that .f._ {t J 0 (at)} = 8(s2 + a2)3/2 103. Find .(a) .f._ {e-at J 0 (4t)}, (b) .f._ {t J 0 (2t}}. 1 sAns. a) . . j s2+6s+25 (b) (s2+4)312

104. Prove that

105. f / 0 t) = J 0 (it), show that .e {10 at)} = 1 , a> 0..js2 a2106. Find .e {t J 0 (t) e-t}. A ns. s - 1)/(s2- 2s + 2)3/2107 . Show that (a) fo J0 t) dt = 1, (b) i e t J 0 (t) dt

d2108. Find the Laplace transform of dtZ {e2t J 0 (2t)}. s2Ans. - s - 2. .jsz 4s+ 8109. Show that

110. Prove that

1,e{tJdt)} = (s2+1)3/2e-a2/4s

111. Evaluate i t e 3 t J 0 (4t) dt. Ans. 3/125112. Prove that .f._ {J ( t)} (y 82+1 - s)nv s2 + 1 and thus obtain .f._ {Jn(at)}.

THE SINE, COSINE AND EXPONENTIAL INTEGRALS113. Evaluate (a) .e {e2t Si (t)}, (b) .e {t Si (t)}.

Ans. (a) t a n - 1 s ~ 2 ) / s - 2 ) , (b) t a n - l s - 1s2 s s2 + 1)

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