Page 56 - LAPLACE TRANSFORM

Example 11

Find the Inverse Laplace Transform of the following.

 8s +13  Use Completing the Square
L −1   Method
s 2 + 4s + 5 
Formula

 s  + b  2 −   b  2 − c completing


 2   2  the Square

s
8 + 13 ( 8 + ) 2 − 3
s
L − 1 { } = { }
s 2 + s 5 ( + ) 2 2 − 9
4 −
s

s
8 + 13 ( 8 + ) 2 3
s
L 1 − { } = L 1 − { }− L 1 − { }
2
2
s
s 2 + s 5 ( + ) 2 − 9 ( + ) 2 − 9
s
4 −

s 3
= 8e − 2t L 1 − { }−e − 2t L 1 − { }
s 2 − 9 s 2 − 9

= e 8 − t 2 cosh t 3 − e − t 2 sinh t 3

Example 12

Find the Inverse Laplace Transform of the following.

L −1  25 


s 2 + 4s +13 
Use Completing the Square
25 25
L − 1 { } = { } Method
s
s 2 + s 13 ( + ) 2 2 + 9
4 +

25 25
L − 1 { } = L − 1 { }
s 2 + s 13 ( + ) 2 2 + 9
s
4 +

25  5 
= L 1 −  
5  ( + ) + 9 
s 2
2

= e 5 − t 2 sin t 3

49