Page 82 - LAPLACE TRANSFORM

If s= -1

24 = ) 0 ( A + B (− 1 )(− 1+ ) 2 + C ) 0 (
24 = B (− ) 1
B = − 24

24 12 24 12
Partial Fraction = = − +
s ( + 1 )( + ) 2 s ( + ) 1 ( + ) 2 Partial
s
s
s
s
s
10 + 30 = 20 − 10 fraction 2
s
(s )( 1 + + s ) 2 (s ) 1 + ( + ) 2

10s + 30 A B
(s + 1 )(s + ) 2 = (s + ) 1 + (s + ) 2
10s + 30 = A (s + ) 2 + B (s + 1 )....eq 1
If s=-1 if s=-2

10 (− ) 1 + 30 = A (− 1+ ) 2 + ) 0 ( B 10 (− ) 2 + 30 = ) 0 ( A + B (− 2 + ) 1

20 = ) 1 ( A 10 = B (− ) 1
A = 20 B = − 10 Partial fraction 1
+ Partial
fraction 2
20 − 10

s
( + ) 1 ( + ) 2
s
Partial Fraction =
12 24 12 20 10
= − + + −
Partial Fraction = s ( +s ) 1 ( +s ) 2 ( +s ) 1 ( +s ) 2
= 12 − 4 + 2
s
s ( + ) 1 ( + ) 2
s
Inverse Laplace Transform:

12 4 2
L 1 − { }− L 1 − { }+ L 1 − }
s ( +s ) 1 {( +s ) 2

t 2
12 −= 4 e + 2 e
t −

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