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
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(a)}$