Page 44 - LAPLACE TRANSFORM

Example 4

Find the Laplace Transforms using the Multiplication by of

the followings:-

2as
Prove that ( sin.L t at )=
2
(s + a 2 ) 2
a
We know that : {sin atL } = n = 1
(s + a 2 )
2

(−= ) 1 d { 2 a 2 }
ds (s + a )

*use Quotient Rule to solve this question

u = a v = s ( 2 + a )
2

du
Formula Quotient Rule
dv
Find the Laplace Transform using the Third Multiplication
0
=
=
s 2
ds
ds
n
of t of the following :- du − u dv
v
{ et .L 2 at } d ds ds
2
d } (s + a 2 )( ) 0 − (a )( 2s ) ds = v 2
at =
2
2
L { et . ds (s + a 2 ) 2
= ) 0 ( − 2as e ,
at
2
(s + a 2 ) 2 f(t) =
− 2as
=
f(t)
=
(s + a 2 ) 2 Prove e at
2

Therefore,
− 2as d 2  1 
=
(−
=n 1 )[ (s + a 2 ) 2 ] at } = ( − )1 2  
2, { et .L 22
ds 2  − as 
∴ 2as
2
(s + a 2 ) 2 2
= )1( d  1  

ds 2  − as 

37