Find inverse laplace transform of (1)/(s^4)

Evelyn Freeman

Evelyn Freeman

Answered question

2022-10-29

Find inverse laplace transform of 1 / s 4
I know that the result of the inverse laplace transform of the following is
L 1 { 1 s 4 } = 1 6 t 3
However I just can't seem to figure out where the fraction comes from.
1 6
Can anyone explain?

Answer & Explanation

pawia6g

pawia6g

Beginner2022-10-30Added 14 answers

L ( t 3 ) = 3 ! s 4 = 6 s 4 however we just have a 1 s 4 so we need to get rid of that 6, to do this we divide by 6. So therefore L 1 ( 1 s 4 ) = t 3 6
Nikolai Decker

Nikolai Decker

Beginner2022-10-31Added 3 answers

Hint. One may observe that by differentiating
0 e s t d t = 1 s , s > 0 ,
n times one has
( 0 e s t d t ) ( n ) = 0 ( 1 ) n t n e s t d t , s > 0 ,
and
( 1 s ) ( n ) = ( s 1 ) ( n ) = ( 1 ) n n ( n 1 ) 1 s ( n + 1 ) = ( 1 ) n n ! s ( n + 1 ) , s > 0 ,
giving
0 t n n ! e s t d t = 1 s n + 1 , s > 0 , n = 0 , 1 , 2 , .
The OP question corresponds to s = 1, n = 3

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