Second Translation Property of Laplace Transforms/Examples/Example 1

Example of Use of Second Translation Property of Laplace Transforms

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

$\laptrans {\paren {t - 2}^3} = \dfrac {6 e^{-2 s} } {s^4}$

where $t > 2$.

Proof

\(\ds \laptrans {\paren {t - 2}^3}\)

\(=\)

\(\ds e^{-2 s} \laptrans {t^3}\)

Second Translation Property of Laplace Transforms

\(\ds \)

\(=\)

\(\ds e^{-2 s} \dfrac {3!} {s^4}\)

Laplace Transform of Positive Integer Power

\(\ds \)

\(=\)

\(\ds \dfrac {6 e^{-2 s} } {s^4}\)

simplification

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $3$. Second translation or shifting property