Laplace transform of t^2e^(at)??

Noe Cowan 2022-11-20 Answered
Laplace transform of t 2 e a t ??
try to prove that
L { t 2 e a t } = 2 ( s a ) 3 .
I've gotten to the last integration by parts where
lim n 0 n 1 ( a s ) 2 2 e ( a s ) t d t = lim n 2 ( a s ) 3 e ( a s ) t | 0 n .
Now what do I do?
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Answers (1)

Prezrenjes0n
Answered 2022-11-21 Author has 19 answers
Evaluating the integral for a fixed n gives
2 ( a s ) 3 e ( a s ) n 2 ( a s ) 3 e 0
Assume that s>a and let n go to infinity. Then since ( a s ) n , the first term disappears and the limit is
2 ( a s ) 3 e 0 = 2 ( 1 ) 3 ( s a ) 3 = 2 ( s a ) 3
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