Laplace Transform of Exponential times Function/Examples/Example 2

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Examples of Use of Laplace Transform of Exponential times Function

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


$\laptrans {t^2 e^{3 t} } = \dfrac 2 {\paren {s - 3}^3}$


Proof

\(\ds \laptrans {t^2}\) \(=\) \(\ds \dfrac {2!} {s^3}\) Laplace Transform of Positive Integer Power
\(\ds \leadsto \ \ \) \(\ds \laptrans {t^2 e^{3 t} }\) \(=\) \(\ds \dfrac {2!} {\paren {s - 3}^3}\) Laplace Transform of Exponential times Function
\(\ds \) \(=\) \(\ds \dfrac 2 {\paren {s - 3}^3}\) Definition of Factorial

$\blacksquare$


Sources