Laplace Transform of Exponential times Function/Examples/Example 4

Example of Use of Laplace Transform of Exponential times Function

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

$\laptrans {e^{4 t} \cosh 5 t} = \dfrac {s - 4} {s^2 - 8 s - 9}$

$\ds \laptrans {\cosh 5 t}$

$=$

$\ds \dfrac 5 {s^2 - 5^2}$

$\ds \leadsto \ \ $

$\ds \laptrans {e^{4 t} \cosh 5 t}$

$=$

$\ds \dfrac {s - 4} {\paren {s - 4}^2 - 25}$

$\ds $

$=$

$\ds \dfrac {s - 4} {s^2 - 8 s - 9}$

multiplying out

$\blacksquare$

$\ds \laptrans {e^{4 t} \cosh 5 t}$

$=$

$\ds \laptrans {e^{4 t} \paren {\dfrac {e^{5 t} + e^{-5 t} } 2} }$

$\ds $

$=$

$\ds \dfrac 1 2 \laptrans {e^{9 t} + e^{-t} }$

$\ds $

$=$

$\ds \dfrac 1 2 \paren {\dfrac 1 {s - 9} + \dfrac 1 {s + 1} }$

$\ds $

$=$

$\ds \dfrac {s - 4} {s^2 - 8 s - 9}$

mulitplying out

$\blacksquare$