Laplace Transform of Exponential times Function/Examples/Example 4/Proof 1

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Example of Use of Laplace Transform of Exponential times Function

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


Proof

\(\ds \laptrans {\cosh 5 t}\) \(=\) \(\ds \dfrac 5 {s^2 - 5^2}\) Laplace Transform of Hyperbolic Cosine
\(\ds \leadsto \ \ \) \(\ds \laptrans {e^{4 t} \cosh 5 t}\) \(=\) \(\ds \dfrac {s - 4} {\paren {s - 4}^2 - 25}\) Laplace Transform of Exponential times Function
\(\ds \) \(=\) \(\ds \dfrac {s - 4} {s^2 - 8 s - 9}\) multiplying out

$\blacksquare$


Sources