Page 13 - LAPLACE TRANSFORM

s
−
A = e a st sin at + F s …….(4)
( )
a

From (4) we substitute into (2)

F s( ) = − e − st cos at − s e −st sin at + s F ( ) s 


a a  a a 

= − e − st cos at − se − 2 st sin at − s 2 2 F (s )

a a a

( ) sF + s 2 2 F ( ) s = − e − a st cos at − se − 2 st sin at
a
a

st
a
s


 2 a + s 2  = − e − a  cos at + a sin at
 ( ) sF


2




s
 a 2 e −st (cosat + a sin at )  ∞





( ) sF = − a (s 2 + a 2 ) 




 Ο 
 ae 0 (cos 0 + s sin 0   a

a
= −0 − s 2 + a 2   = s + a 2
2


  

6