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8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 14
Worksheet 9 LaPlace Transforms
A LaPlace Transform is defined as intinfin
=
minus=
983088t
st dt t f et f L )()(
Use the definition to find the LaPlace Transform of each of the following1 983089=)(t f
2 t t f 983091=)(
3 t et f 983092
=)(
4 983090t t f =)(
5983089
983089
983090983090
983088
983090ge
lt
+minus=
t
t
t t t f
)(
6 )(t f L prime
7 Verify that the LaPlace Transform of )cos()( at t f = is 983088983090983090 gt
+s
ass
8122019 2320 Ws 9 Laplace
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8 Use the Table of Basic LaPlace Transforms shown above to find each of the following
(a) 983093 L
(b) t e L 983091minus
(c) t e L 983091983092
minus
(d) t te L 983091983092
minus
(e) 983090t L
(f) )cos( t L 983093
(g) ( ) t et L minus+minus983089983091
(h) y y y L 983090983091 +prime+primeprime
9 Find the inverse LaPlace Transforms of each of the following
(a)
+
minus
983089
983089983089
s L
(b)
+
minus
983089
983093983089
s L
(c)
minus
983090
983089 983094
s L
(d)
minus
983091
983089 983094
s L
(e)
++
+minus
))(( 983092983091
983090983089
ss
s L
(f)
+
minus
983094
983094
983090
983089
s L
(g)
++
+minus
983090983091
983091
983090
983089
ss
s L
10 Use LaPlace Transforms to solve 983088983088983089983088983088983090983091 =prime==+prime+primeprime )()( y y y y y
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 34
11 Find the inverse LaPlace Transform of each of the following
(a)983092983091
983090
983090minus+
=ss
sF )(
(b)
( )
2
1
53)(
minus
minus=
s
ssF
(c)( )( )983089983090
983090
983090
983090
minus+
+=
sss
ssF )(
12 Use LaPlace Transforms to solve the IVP t y y y 983090983090 =minusprime+primeprime where 983088983088 =)( y and
983089983088 =prime )( y
Theorem 76 First Translation Theorem
If )()( sF t f L = and a is any real number then )()( asF t f e L at minus=
13 Use the definition of the Laplace transform to show that )()( asF t f e L at
minus=
Find each of the following14 t te L 983095
15 ( ) t ee L t t
983092983091983089 983091983090 sin
minusminus+
16( )
minus
minus
983094
983089
983092
983089
s L
17
( )
minus
minusminus
983090
983089
983090
983089983090
s
s L
18 Solve the IVP 983088983088983088983088983092983092 983090983091
=prime==+primeminusprimeprime )()( y yet y y y t
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 44
Definition The Unit Step Function
The Unit Step Function ( )at
at at U
ge
ltle
=minus 983088
983089
983088
Write each of the following in terms of Unit Step Functions
19 ( )983089983088
983089983088983088
983088
983092
ge
ltle
=t
t t f
20 ( )
ge
ltle
ltle
=
983090983091
983090983091983090
983090983088
983088
983089
sin
π
π π
π
t
t
t
t t f
21
Theorem 77 Second Translation Theorem
If )()( t f LsF = and a gt 0 then )()()( sF eat U at f L asminus
=minusminus
Find each of the following
22 )( 983092983092
minusminus
t U e L t
23 )( 983089
983090
minust U t L
24
+
minus
minus
983091
983090
983089
s
e L
s
25( )
+minus
minus
minus
)( 983090983090
983089
ss
e L
s
26 Solve the IVP983088983088983088983088 =prime==+primeprime )()()( y yt f y y
8122019 2320 Ws 9 Laplace
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8 Use the Table of Basic LaPlace Transforms shown above to find each of the following
(a) 983093 L
(b) t e L 983091minus
(c) t e L 983091983092
minus
(d) t te L 983091983092
minus
(e) 983090t L
(f) )cos( t L 983093
(g) ( ) t et L minus+minus983089983091
(h) y y y L 983090983091 +prime+primeprime
9 Find the inverse LaPlace Transforms of each of the following
(a)
+
minus
983089
983089983089
s L
(b)
+
minus
983089
983093983089
s L
(c)
minus
983090
983089 983094
s L
(d)
minus
983091
983089 983094
s L
(e)
++
+minus
))(( 983092983091
983090983089
ss
s L
(f)
+
minus
983094
983094
983090
983089
s L
(g)
++
+minus
983090983091
983091
983090
983089
ss
s L
10 Use LaPlace Transforms to solve 983088983088983089983088983088983090983091 =prime==+prime+primeprime )()( y y y y y
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 34
11 Find the inverse LaPlace Transform of each of the following
(a)983092983091
983090
983090minus+
=ss
sF )(
(b)
( )
2
1
53)(
minus
minus=
s
ssF
(c)( )( )983089983090
983090
983090
983090
minus+
+=
sss
ssF )(
12 Use LaPlace Transforms to solve the IVP t y y y 983090983090 =minusprime+primeprime where 983088983088 =)( y and
983089983088 =prime )( y
Theorem 76 First Translation Theorem
If )()( sF t f L = and a is any real number then )()( asF t f e L at minus=
13 Use the definition of the Laplace transform to show that )()( asF t f e L at
minus=
Find each of the following14 t te L 983095
15 ( ) t ee L t t
983092983091983089 983091983090 sin
minusminus+
16( )
minus
minus
983094
983089
983092
983089
s L
17
( )
minus
minusminus
983090
983089
983090
983089983090
s
s L
18 Solve the IVP 983088983088983088983088983092983092 983090983091
=prime==+primeminusprimeprime )()( y yet y y y t
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 44
Definition The Unit Step Function
The Unit Step Function ( )at
at at U
ge
ltle
=minus 983088
983089
983088
Write each of the following in terms of Unit Step Functions
19 ( )983089983088
983089983088983088
983088
983092
ge
ltle
=t
t t f
20 ( )
ge
ltle
ltle
=
983090983091
983090983091983090
983090983088
983088
983089
sin
π
π π
π
t
t
t
t t f
21
Theorem 77 Second Translation Theorem
If )()( t f LsF = and a gt 0 then )()()( sF eat U at f L asminus
=minusminus
Find each of the following
22 )( 983092983092
minusminus
t U e L t
23 )( 983089
983090
minust U t L
24
+
minus
minus
983091
983090
983089
s
e L
s
25( )
+minus
minus
minus
)( 983090983090
983089
ss
e L
s
26 Solve the IVP983088983088983088983088 =prime==+primeprime )()()( y yt f y y
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 34
11 Find the inverse LaPlace Transform of each of the following
(a)983092983091
983090
983090minus+
=ss
sF )(
(b)
( )
2
1
53)(
minus
minus=
s
ssF
(c)( )( )983089983090
983090
983090
983090
minus+
+=
sss
ssF )(
12 Use LaPlace Transforms to solve the IVP t y y y 983090983090 =minusprime+primeprime where 983088983088 =)( y and
983089983088 =prime )( y
Theorem 76 First Translation Theorem
If )()( sF t f L = and a is any real number then )()( asF t f e L at minus=
13 Use the definition of the Laplace transform to show that )()( asF t f e L at
minus=
Find each of the following14 t te L 983095
15 ( ) t ee L t t
983092983091983089 983091983090 sin
minusminus+
16( )
minus
minus
983094
983089
983092
983089
s L
17
( )
minus
minusminus
983090
983089
983090
983089983090
s
s L
18 Solve the IVP 983088983088983088983088983092983092 983090983091
=prime==+primeminusprimeprime )()( y yet y y y t
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 44
Definition The Unit Step Function
The Unit Step Function ( )at
at at U
ge
ltle
=minus 983088
983089
983088
Write each of the following in terms of Unit Step Functions
19 ( )983089983088
983089983088983088
983088
983092
ge
ltle
=t
t t f
20 ( )
ge
ltle
ltle
=
983090983091
983090983091983090
983090983088
983088
983089
sin
π
π π
π
t
t
t
t t f
21
Theorem 77 Second Translation Theorem
If )()( t f LsF = and a gt 0 then )()()( sF eat U at f L asminus
=minusminus
Find each of the following
22 )( 983092983092
minusminus
t U e L t
23 )( 983089
983090
minust U t L
24
+
minus
minus
983091
983090
983089
s
e L
s
25( )
+minus
minus
minus
)( 983090983090
983089
ss
e L
s
26 Solve the IVP983088983088983088983088 =prime==+primeprime )()()( y yt f y y
8122019 2320 Ws 9 Laplace
httpslidepdfcomreaderfull2320-ws-9-laplace 44
Definition The Unit Step Function
The Unit Step Function ( )at
at at U
ge
ltle
=minus 983088
983089
983088
Write each of the following in terms of Unit Step Functions
19 ( )983089983088
983089983088983088
983088
983092
ge
ltle
=t
t t f
20 ( )
ge
ltle
ltle
=
983090983091
983090983091983090
983090983088
983088
983089
sin
π
π π
π
t
t
t
t t f
21
Theorem 77 Second Translation Theorem
If )()( t f LsF = and a gt 0 then )()()( sF eat U at f L asminus
=minusminus
Find each of the following
22 )( 983092983092
minusminus
t U e L t
23 )( 983089
983090
minust U t L
24
+
minus
minus
983091
983090
983089
s
e L
s
25( )
+minus
minus
minus
)( 983090983090
983089
ss
e L
s
26 Solve the IVP983088983088983088983088 =prime==+primeprime )()()( y yt f y y
