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Problem 3.04PP
Find the Laplace transform of the following time functions:
(a) f{t) = / sin /
(b) f{t) = t cos 3f
(c) f{f) = t e - t+ 2 tc o s t
(d) f[t) = ts\n 3 / -2 fc o s f
(e)/(0=1(fJ + 2/cos 2f
(e) /(0= 1(f)+ 2 /cos 2f
Step-by-step solution
step 1 of 9
(a)
Consider the time function,
Apply the Laplace transfomi on both the sides of the time function.
Since there are two variables in the above function we need to use the multiplication by time 
Laplace transform property. In order to do so. we solve the given time function by separating 
them.
First, consider ^ ^ /^ = s in r and find the Laplace transform of it.
Step 2 of 9
Now. find the Laplace transform for the whole function.
^ { « ( 0 } = — G(>)
f ( . » - . l ) ( 0 ) - ( l ) ( 2 . ) l
(.-.I)- J
2s
( s - . l f
2s
j ‘ + 2j ’ +1
Therefore, the Laplace transfomi of the time function, /s in / is
s* + 2ŝ + \
Step 3 of 9
(b)
Consider the time function, /]^;r^=toaB3f
Apply the Laplace transfomi on both the sides of the time function. £ [ f ( i ) ) = £ { tc o s 2 i\
Since there are two variables in the above function we need to use the multiplication by time 
Laplace transfomi property. In order to do so. we solve the given time function by separating 
them.
First, consider g ̂ /^ = cos3/ and find the Laplace transform of it.
Step 4 of 9
Now. find the Laplace transform for the whole function.
£ { , g i j ) } = - ± G ( s )
-s ‘ -9+2s‘
° j ' + 18s’ +81 
s ^ -9
s'+\Zs'*% \
Therefore, the Laplace transfomi of the time function, /cos3/ is 4 ^ -9
Step 5 of 9
(C)
Consider the time function, +2tcoB t
Appiy the Lapiace transfomi on both the sides of the time function.
/:{/(/)} = £ { le " +2f cost} 
F{s) = £ { le " } + 2r {/cos/}
1
( * + > r
__ i_
r - 2
- 2
( s ‘ + 1 ) - s ( 2 j ) 
s’ - 2 i ' + I
Step 6 of 9
Further simplification gives.
F (s )
1 -s ’
{s ^ iy
____1 , ^(^‘ - 0
~ { s ^ l f (s'+lf
(s ’ + 1 ) '+ 2 (s + 1)’ (s* - 1)
(s+l)’(s*+l)'
(s* + 2j’+1)+(ĵ +2s+l)(2s* - 2)
(s“+2s + l)(i* + 2s’ + l)
^ s *+ 2 s ^ + l + 2 s * -2 s *+ 4 s * -4 s + 2 s '- 2 
s‘ + 2s*+s’ + 2s*+4i’ + 2s+s*+ 2i“+ 1 
3s*+ 4 s’ +2j " - 4 s - I 
" j ‘ + 2s’ +3s*+4s’ +3s’ + 2j + 1
Therefore, the Laplace transform of the time function, te^ +2tcost '®
3s*+4s’ + 2 s ^ - 4 s - l
Step 7 of 9
(d)
Consider the time function,
Apply the Laplace transfomi on both the sides of the time function. 
/ ■ { / { / ) } = COS/ }
F ( j ) = ̂ { t s in y ) ~ 2 jC {/cos/}
, j (2 £ L + 2 [(£ i 1)z ! E ) 1
r+2 j ’ + l - 2 s '
* * c-^21
6s 2(»’ - l )
(s’ + 9 ) '" ( s '+ 1)'
Therefore, the Laplace transfomi of the time function, /s in 3 /-2 /c o s / is
6s 2 ( s ' - l )
( s '+ 9 y ( s ' t l ) ^
Step 8 of 9
(e)
Consider the time function, i ( /^ 4 '2 /oob2 /
Appiy the Lapiace transfomi on both the sides of the time function.
£ { f { i ) } = £{l(l)+2teos2l}
F ( s ) = / '{ l ( / ) } + 2 r { / c o s 2 / }
(s* + 4 ) - ( 2 j ) 5 
s '+ 4 - 2 s ’
= 1 - 2
= 1 - 2
( s ' . 4 f
Step 9 of 9
Further simplification yields, 
j ( j * + 4 ) *
^ s*+Bs^ + \ 6 + 2 s ^ - ^
s { s ^ + 4 f
s *+ 2 s^+ S s^ -S s+ l6
s(s^+ 4 )"
Therefore, the Laplace transfomi of the time function, l( / )+ 2 /c o s 2 / is
5*+2 j ’ +8s* - 8 s+16
s(s^ +A '^