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MTH 256 Recitation Activity 8
1. Use the table of Laplace transforms from section 8.8 of the textbook, the linearity of the
Laplace transform, and the half angle formula
1 + cos(2 )
cos ! ( ) =
2
to compute the Laplace transform
ℒ {cos ! (5 )}.
Indicate the values of such that the Laplace transform is defined
1 + cos(10 )
ℒ{cos ! (5 )} = ℒ 2
4
2
1
1
= ℒ{1} + ℒ{cos(10 )}
2
2
11 1
=
+ !
,
>0
2 2 + 100
2. The general solution to
“” + 4 = 0
is
( ) = # cos(2 ) + ! sin(2 ).
Use the table of Laplace transforms from section 8.8 of the textbook to calculate the Laplace
transform ( ) given by
= ℒ { }.
Indicate the values of for which the Laplace transform is defined.
= ℒ { } = ℒ { # cos(2 ) + ! sin(2 )}
= # ℒ {cos(2 )} + ! ℒ {sin(2 )}
2
= # !
+ ! !
,
>0
+4
+4
3. The general solution to
“” + 2 ” + 2 = 0
is
( ) = # $% cos( ) + ! $% sin( ).
Use the table of Laplace transforms from section 8.8 of the textbook to calculate the Laplace
transform ( ) given by
= ℒ { }.
Indicate the values of for which the Laplace transform is defined.
= ℒ { } = ℒ{ # $% cos( ) + ! $% sin( )}
= # ℒ{ $% cos( )} + ! ℒ { $% sin( )}
+1
1
= #
+
,
>0
!
( + 1)! + 1
( + 1)! + 4
4. Determine the value of so that
is of the form
3
! + 9
! + !
.
Use this and the table of Laplace transforms from section 8.8 of the textbook to compute
3
ℒ $# A !
B
+9
=3
3
ℒ $# A !
B = sin(3 )
+9
5. Use the table of Laplace transforms from section 8.8 of the textbook and the linearity of the
inverse Laplace transform to compute
+3
ℒ $# A !
B.
+9
+3
3
ℒ $# A !
B = ℒ $# C !
D + ℒ $# A !
B
+9
+9
+9
= cos(3 ) + sin(3 )
6. Use the table of Laplace transforms from section 8.8 of the textbook and the linearity of the
inverse Laplace transform to determine
1
1
ℒ $# A +
B
( + 2)!
1
1
1
1
$#
$#
ℒ $# A +
B
=
ℒ
A
B
+
ℒ
A
B
( + 2)!
( + 2)!
= 1 + !%
7. Let > 0. Taking the Laplace transform of both side of
“” + 4 = sin(2 )
gives
! ( ) − ” (0) − (0) + 4 ( ) =
If we assume (0) = 0 and ” (0) = 0, then
( ! + 4) ( ) =
This gives
( ) =
2
.
! + 4
2
.
! + 4
2
( ! + 4)!
.
Use the table of Laplace transforms from section 8.8 of the textbook to determine
= ℒ $# { }.
2
( ! + 4)!
From the table
ℒ
So
ℒ $# A
$#
=
2
( ! + 2! )!
2 &
2 !
4 = sin( ) − cos( )
( + ! )!
2
1 2(2& )
=
( ! + 2! )! 8 ( ! − 2)!
2
1 $#
2(2& )
1
B
=
ℒ
2
4 = (sin(2 ) − 2 cos(2 ))
!
!
!
!
( + 4)
( − 2)
8
8
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