Need find a partial-fraction expansion of A=(1)/((s+1)^2 (s-1)(s+5))

Julia Chang 2022-09-26 Answered
Need find a partial-fraction expansion of A = 1 ( s + 1 ) 2 ( s 1 ) ( s + 5 ) .
I was solve equation differential using Laplace transform, but I need use partial fraction of A.
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Answers (1)

berzamauw
Answered 2022-09-27 Author has 9 answers
The partial fraction expansion is:
1 ( s + 1 ) 2 ( s 1 ) ( s + 5 ) = 1 24 ( s 1 ) 1 32 ( s + 1 ) 1 8 ( s + 1 ) 2 1 96 ( s + 5 )
The fraction expansion can be expressed as:
1 ( s + 1 ) 2 ( s 1 ) ( s + 5 ) = A s 1 + B s + 1 + C ( s + 1 ) 2 + D s + 5
A ( s + 1 ) 2 ( s + 5 ) + B ( s + 1 ) ( s 1 ) ( s + 5 ) + C ( s 1 ) ( s + 5 ) + D ( s 1 ) ( s + 1 ) 2 = 1
Using s = 1, we get:
24 A = 1 A = 1 24
Using s = 1, we get:
8 C = 1 C = 1 8
Using s = 5, we get:
96 D = 1 D = 1 96
For B, none of these will work. So we can choose something that makes life easier. Using s=0, we get
5 A 5 B 5 C D = 1
5 24 5 B + 5 8 + 1 96 = 1
B = 1 5 ( 1 20 96 60 96 1 96 ) = 1 32
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