Laplace Transform of Derivative/Examples/Example 1

Examples of Use of Laplace Transform of Derivative

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


$\laptrans {-3 \sin 3 t} = \dfrac {-9} {s^2 + 9}$


Proof

\(\ds \laptrans {-3 \sin 3 t}\) \(=\) \(\ds s \laptrans {\cos 3 t} - \cos 0\) Laplace Transform of Derivative, Derivative of $\cos a x$
\(\ds \) \(=\) \(\ds s \dfrac s {s^2 + 9} - 1\) Laplace Transform of Cosine, Cosine of Zero is One
\(\ds \) \(=\) \(\ds \dfrac {-9} {s^2 + 9}\) 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: $5$. Laplace transform of derivatives
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