Find L^(-1) [(9)/((s+3)^3)]

skauvzc

skauvzc

Answered question

2022-09-18

Find
L 1 [ 9 ( s + 3 ) 3 ] .
How do I go about with the fraction inside? There is no fixed formula for this expression. I did a partial fraction of the repeated linear factors as
9 = A s 2 + 6 A s + 9 A + B s + 3 B + C
This is just only going to tell me that 9 A + 3 B + C = 9

Answer & Explanation

geoforoiunpwd

geoforoiunpwd

Beginner2022-09-19Added 6 answers

Hint: If L ( f ( t ) ) = F ( s ) then
L ( f ( t ) e a t ) = F ( s a ) ,
and
L ( t n ) = n ! s n + 1 .
Zachariah Norris

Zachariah Norris

Beginner2022-09-20Added 2 answers

Note that L 1 { F ( s ) } = f ( t ), then L 1 { F ( s a ) } = e a t f ( t )
Therefore, we have
L 1 { 9 ( s + 3 ) 3 } = e 3 t L 1 { 9 s 3 }
Now find L 1 { 9 s 3 }
L 1 { 9 s 3 } = L 1 { 9 2 2 s 3 } = 9 2 L 1 { 2 s 3 } = 9 t 2 2
Therefore, L 1 { 9 ( s + 3 ) 3 } = 9 e 3 t t 2 2

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