Laplace Transform of t^2 by Cosine a t

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Theorem

Let $\sin$ denote the real sine function.

Let $\laptrans f$ denote the Laplace transform of a real function $f$.


Then:

$\laptrans {t^2 \cos a t} = \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}$


Proof

\(\displaystyle \laptrans {t^2 \cos a t}\) \(=\) \(\displaystyle -\map {\dfrac {\d^2} {\d s^2} } {\laptrans {\cos a t} }\) Higher Order Derivatives of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle -\map {\dfrac {\d^2} {\d s^2} } {\dfrac a {s^2 + a^2} }\) Laplace Transform of Sine
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}\) Quotient Rule for Derivatives

$\blacksquare$


Sources