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
- 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: $8 \ \text{(c)}$