Laplace Transform of Exponential times Function/Examples/Example 5

Examples of Use of Laplace Transform of Exponential times Function

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

$\laptrans {e^{-3 t} \paren {3 \cos 6 t - 5 \sin 6 t} } = \dfrac {3 s - 24} {s^2 + 4 s + 40}$

$\ds \laptrans {3 \cos 6 t - 5 \sin 6 t}$

$=$

$\ds \dfrac {3 s} {s^2 + 6^2} - \dfrac {5 \times 6} {s^2 + 6^2}$

$\ds $

$=$

$\ds \dfrac {3 s - 30} {s^2 + 6^2}$

simplification

$\ds \leadsto \ \ $

$\ds \laptrans {e^{-3 t} \paren {3 \cos 6 t - 5 \sin 6 t} }$

$=$

$\ds \dfrac {3 \paren {s + 3} - 30} {\paren {s + 3}^2 + 6^2}$

$\ds $

$=$

$\ds \dfrac {3 s - 24} {s^2 + 4 s + 40}$

multiplying out

$\blacksquare$

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: $8 \ \text{(d)}$