Laplace Transform of Exponential times Function/Examples/Example 4/Proof 2
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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
| \(\ds \laptrans {e^{4 t} \cosh 5 t}\) | \(=\) | \(\ds \laptrans {e^{4 t} \paren {\dfrac {e^{5 t} + e^{-5 t} } 2} }\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \laptrans {e^{9 t} + e^{-t} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\dfrac 1 {s - 9} + \dfrac 1 {s + 1} }\) | Laplace Transform of Exponential | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {s - 4} {s^2 - 8 s - 9}\) | mulitplying out |
$\blacksquare$
Sources
- 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: $8 \ \text{(c)}$