Deck 13: The Laplace Transform

Question 1

Find F ( s ) if

Answer 1

Consider the function.
Consider the formula for Laplace transform.
Substitute
for
.
Therefore, the Laplace transform of the function
is
.

Question 2

Find the Laplace transform of the function f ( t ) = te at sin ( t ) ( t 4).

Answer 2

Write the expression of the Laplace transform of
.
Substitute
for
.
Simplify further.
Thus, the Laplace transform of
is
.

Question 3

Given the following functions F ( s ), find the inverse Laplace transform of each function.
(a)

(b)

(c)

Answer 3

(a)
Consider the following function:
The function
is
Express
in a partial fraction expansion.
Calculate the value of
.
Calculate the value of
.
The partial fraction expansion of
is,
Now apply inverse Laplace transforms on both sides.
Therefore, the function
is,
(b)
Consider the following function:
The function
is
Express
in a partial fraction expansion.
Calculate the value of
.
Calculate the value of
.
The partial fraction expansion of
is,
Now apply inverse Laplace transforms on both sides.
Therefore, the function
is,
(c)
Consider the following function:
The function
is
Express
in a partial fraction expansion.
Calculate the value of
.
Calculate the value of
.
Calculate the value of
.
The partial fraction expansion of
is,
Now apply inverse Laplace transforms on both sides.
Therefore, the function
is,

Question 4

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 5

Find the inverse Laplace transform of the function F ( s ) using the convolution integral.

Question 6

The switch in the circuit in Fig. P13.63 has been closed for a long time and is opened at t = 0. Find i ( t ) for t 0, using Laplace transforms.

Figure P13.63

Question 7

The Laplace transform function for the output voltage of a network is expressed in the following form:

Determine the value of this voltage; that is, v o ( t ) at t
a. 6 V
b. 2 V
c. 12 V
d. 4 V

Question 8

Given
find f ( t ).

Question 9

In the circuit in Fig. E3.18, the switch opens at t = 0. Use Laplace transforms to find v o ( t ) for t 0.

Figure E3.18

Question 10

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 11

Find f ( t ) using the convolution integral if

Question 12

The switch in the circuit in Fig. P13.64 has been closed for a long time and is opened at t = 0. Find i ( t ) for t 0 using Laplace transforms.

Figure P13.64

Question 13

Use the time-shifting theorem to determine [ f ( t )], where f ( t ) = [e ( t 2) e 2( t 2) ] u ( t 2).

Question 14

Find F ( s ) if f ( t ) = e at sin t u(t 1).

Question 15

Given the following functions F ( s ), find f ( t ).
(a)

(b)

(c)

(d)

Question 16

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 17

Find f ( t ) using convolution if F ( s ) is

Question 18

The switch in the circuit in Fig. P13.65 has been closed for a long time and is opened at t = 0. Find i ( t ) for t 0 using Laplace transforms.

Figure P13.65

Question 19

If f ( t ) = te ( t 1) u ( t 1) e ( t 1) u ( t 1), determine F ( s ) using the time-shifting theorem.

Question 20

Determine f ( t ) if F ( s ) = s /( s + 1) 2.

Question 21

In the circuit in Fig. E3.19, the switch opens at t = 0. Use Laplace transforms to find i ( t ) for t 0.

Figure E3.19

Question 22

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 23

Find f ( t ) using convolution if F ( s ) is
(a)

(b)

Question 24

In the circuit shown in Fig. P13.66, switch action occurs at t = 0. Determine the voltage u 0 (t), t 0 using Laplace transforms.

Figure P13.66

Question 25

The output of a network is expressed as

Determine the output as a function of time.
a.

b.

c.

d.

Question 26

Find F ( s ) if f ( t ) = te at u ( t 4).

Question 27

Given the following functions F ( s ), find f ( t ).
(a)

(b)

(c)

(d)

Question 28

Find the inverse Laplace transform of the following functions.
(a)

(b)

(c)

Question 29

Find the initial and final values of f ( t ) if F ( s ) is given as
(a)

(b)

(c)

Question 30

Use property number 7 to find [ f ( t )] if f ( t ) = t e at u ( t 1).

Question 31

If F ( s ) = ( s + 2)/ s 2 ( s + 1), find f ( t ).

Question 32

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 33

Find f ( t ) if F ( s ) is given by the following functions:
(a)

(b)

(c)

Question 34

Determine the initial and final values of f ( t ) if F ( s ) is given by the expressions
(a)

(b)

(c)

Question 35

Find F ( s ) if f ( t ) = e 4 t ( t e t ).

Question 36

Use the results of property 3 and the fact that if f ( t ) = e t sin t , then F ( s ) = 1/( s + 1) 2 + 1 to find the Laplace transform of f ( t ) = e 2 t sin 2 t.

Question 37

Given the following functions F ( s ), find the inverse Laplace transform of each function.
(a)

(b)

Question 38

Find the inverse Laplace transform of the following functions.
(a)

(b)

(c)

(d)

Question 39

Find the final values of the time function f ( t ) given that
(a)

(b)

Question 40

Solve the following differential equation using Laplace transforms:

a. [ 2 e 2 t + e 4 t 3e 3t ] u ( t )
b. [ 3 e 2t + e 4 t + e 3 t ] u ( t )
c. [ e 2t + e 4 t 2 e 3 t ] u ( t )
d. [ 4 e 2t e 4 2 e 3 t ] u ( t )

Question 41

Given
find f ( t ).

Question 42

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 43

Find f ( t ) if F ( s ) is given by the following function:
.

Question 44

Find the final values of the time function f ( t ) given that
(a)

(b)

Question 45

Use property number 5 to find [ f ( t )] if f ( t ) = e at u ( t 1).

Question 46

Find [ f ( t )] if f ( t ) = t 2 e at ( t 2).

Question 47

Find f ( t ) if F ( s ) is given by the expression.
.

Question 48

Find the inverse Laplace transform of the function
.

Question 49

Find the initial and final values of the time function f ( t ) if F ( s ) is given as
(a)

(b)

(c)

Question 50

Find f ( t ) if F ( s ) = 10( s + 6)/ s ( s + 1)( s + 3).

Question 51

Find the initial and final values of the function f ( t ) if F ( s ) = is given by the expression

Question 52

Find the inverse Laplace transform of F ( s ) where
.

Question 53

Find f ( t ) if F ( s ) is given by the expression
.

Question 54

Find the initial and final values of f ( t ) if F ( s ) is given as
(a)

(b)

(c)

Question 55

If f ( t ) = e at , show that F ( s ) = 1/( s + a ).

Question 56

Find the Laplace transform of the function f ( t ) = e at ( t 1).

Question 57

If f ( t ) = t sin ( t ) u ( t 1), find F ( s ).

Question 58

Find the inverse Laplace transform of the function
.

Question 59

Use Laplace transforms to solve the following differential equations.
(a)

(b)

Question 60

In the network in Fig. P13.57, the switch opens at t = 0. Use Laplace transforms to find v o ( t ) for t 0.

Figure P13.57

Question 61

The output function of a network is expressed using Laplace transforms in the following form:

Find the output v o ( t ) as a function of time.
a. [ 12 + 3 e 2 t + 4 e t ] u ( t ) V
b. [ 2 + 4 e 2 t + 8 e t ] u ( t )V
c. [ 6 + 6 e 2 t 12 e t ] u ( t ) V
d. [ 3 + 2 e 2 t 6 e t ] u ( t )V

Question 62

If F ( s ) = 12( s + 2)/ s ( s + 1), find f ( t ).

Question 63

Find the initial and final values of the time function f ( t ) if

Question 64

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 65

Solve the following integrodifferential equation using Laplace transforms.
t 0

Question 66

In the circuit in Fig. P13.58, the switch moves from position 1 to position 2 at t = 0. Use Laplace transforms to find v ( t ) for t 0.

Figure P13.58

Question 67

Use the time-shifting theorem to determine [ f ( t )] where f ( t )=[ t 1 + e ( t 1) ] u ( t 1).

Question 68

If f ( t ) = e at sin t, show that

Question 69

If F ( s ) = ( s + 1) 2 ( s + 3) 2 /( s + 2)( s + 4), find f ( t ).

Question 70

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 71

Solve the following integrodifferential equation using Laplace transforms.
, y(0) = 0, t 0

Question 72

In the network in Fig. P13.59, the switch opens at t = 0. Use Laplace transforms to find i ( t) for t 0.

Figure P13.59

Question 73

If f ( t ) = sin t, show that F ( s ) = /( s 2 + 2 ).

Question 74

Given
, find f ( t ).

Question 75

Use the Laplace transform to find y ( t ) if

Question 76

Given the following functions F ( s ), find f ( t ).
(a)

(b)

Question 77

Solve the following differential equations using Laplace transforms.
(a)

(b)

Question 78

In the network in Fig. P13.60, the switch opens at t = 0. Use Laplace transforms to find i ( t ) for t 0.

Figure P13.60

Question 79

The Laplace transform function representing the output voltage of a network is expressed as

Determine the value of v o ( t ) at t = 100 ms.
a. 0.64 V
b. 0.45 V
c. 0.33 V
d. 0.24 V

Question 80

If f(t) = e at , show that F ( s ) =