Find the Laplace transform of f(t)=t int_0^t tau e^(-tau) L(f)(s)= ??

Sara Solomon 2022-09-08 Answered
Find the Laplace transform of
f ( t ) = t 0 t τ e τ
L ( f ) ( s )=??
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Answers (2)

Peutiedw
Answered 2022-09-09 Author has 9 answers
First write the function as
f ( t ) = t 0 t τ e τ = t g ( t ) ,
then you can use the following properties of the Laplace transform :
1) L ( 0 t h ( τ ) d τ ) ( s ) = H ( s ) s ,
where H(s) is the Laplace transform of h.
2) L ( t n g ( t ) ) = ( 1 ) n G ( n ) ( s ) ,
where G(s) is the Laplace transform of g(t).
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Dana Russo
Answered 2022-09-10 Author has 2 answers
0 t d τ τ e τ = [ τ e τ ] 0 t + 0 t d τ e τ = 1 ( 1 + t ) e t
0 d t f ( t ) e s t = 0 d t t e s t 0 d t ( t + t 2 ) e ( s + 1 ) t = 1 s 2 1 ( s + 1 ) 2 2 ( s + 1 ) 3
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