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?

Question

Laplace transform of       t    2        e          a      t      ??try to prove that      L    {      t    2        e          a      t        }  =      2          (      s      −      a              )        3              .I&#039;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?

Prezrenjes0n · Accepted Answer

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&amp;gt;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

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

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