The Laplace Transform HW
jav284
laplace_xfrm_questions.pdf
Laplace Transforms
1. Compute the Laplace Transforms of the following functions:
a) x t t u t( ) sin( ) ( )= 4 100 b) x t t u t( ) sin( ) ( . )= − −4 100 10 0 1 c) x t u t t t u t( ) ( ) ( ) cos( ) ( )= + − −2 4 5δ d) x t tu t t u t t u t( ) ( ) ( ) ( ) ( ) ( )= − − − + − −2 2 2 3 3
e) x t u t e tt( ) ( ) cos( )= − −2 10 u(t)
2. Compute the inverse Laplace Transforms of the following functions:
a) X s s
s s ( )
( ) =
+ + +
10 1
4 32
b) X s s
s s ( )
( ) =
+ + +
10 1
4 82
c) X s s
s s s ( )
( )( )( ) =
+ + + +
2 100
1 8 10
d) X s s
s s e s( )
( ) =
+ + +
−10 1
4 32 2
e) X s s s s
( ) ( )
= + +
20
10 162
f) X s s
s s s ( )
( )
( ) =
+
+ +
10 1
4 82
3. Find the limit as t→∞ of x(t) (if the limit exists)
a) X s s
s s s ( )
( )
( ) =
+
+ +
10 1
4 32
b) X s s
s s s ( )
( )
( ) =
+
+ +
10 1
4 82
c) X s s
s s s ( )
( )
( ) =
+
+ −
10 1
2 32
4. Give the general form of x(t) (do not solve for the coefficients explicitly).
a) X s s
s s s ( )
( )( )( ) =
+ + + +
2 100
2 6 10
b) X s s
s s s s ( )
( )( )( ) =
+ + + −
2 100
1 8 4
c) X s s
s s s ( )
( )( )( ) =
− + + +
40
1 8 10
d) X s s
s s s ( )
( )
( ) =
+
+ +
10 1
4 32
e) X s s
s s s ( )
( )
( ) =
+
+ +
10 1
4 82
f) X s s
s s s ( )
( )( ) =
+
+ +
1
4 82
g) X s s
s s s ( )
( )
( )(( ) )( ) =
+
+ + + +
20 1
16 4 25 12 2
