MHB Dharshan's question via email about a Laplace Transform

AI Thread Summary
The Laplace Transform of the function 5sin(11t)sinh(11t) is evaluated using the identity for sinh and properties of Laplace Transforms. The process involves breaking down the function into its exponential components and applying the transform to each part. The final expression can be simplified to either a complex fraction or left in a more expanded form. The transforms used in the calculations are confirmed to be correct, although the final algebraic simplification may require further verification. The discussion emphasizes the importance of careful algebraic manipulation in obtaining the final result.
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Evaluate $\displaystyle \mathcal{L}\left\{ 5\sin{ \left( 11 \, t \right) } \sinh{ \left( 11\,t \right) } \right\} $.

$\displaystyle \begin{align*}
\mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\
&= \frac{5}{2} \,\mathcal{L} \left\{ \mathrm{e}^{11\,t} \sin{ \left( 11\,t \right) } - \mathrm{e}^{-11\,t} \sin{ \left( 11\,t \right) } \right\} \\
&= \frac{5}{2} \left[ \mathcal{L}\left\{ \mathrm{e}^{11\,t}\sin{ \left( 11\,t \right) } \right\} - \mathcal{L}\left\{ \mathrm{e}^{-11\,t} \sin{ \left( 11\,t \right) } \right\} \right] \\
&= \frac{5}{2} \left\{ \left[ \frac{11}{s^2 + 11^2} \right]_{s \to s - 11} - \left[ \frac{11}{s^2 + 11^2} \right]_{s \to s + 11} \right\} \\
&= \frac{55}{2} \left[ \frac{1}{\left( s - 11 \right) ^2 + 121} - \frac{1}{\left( s + 11 \right) ^2 + 121} \right]
\end{align*} $

It would be fine to leave your answer in this form, but if you get a common denominator and simplify, you could write the answer as $\displaystyle \frac{1210\,s}{s^4 + 58564}$.
 
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the transforms are correct, I didn’t check the algebra on your last simplification.
 
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