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Question

# Find F(s) if: (a) $f(t)=6 e^{-t} \cosh 2 t$ (b) $f(t)=3 t e^{-2 t} \sinh 4 t$ (c) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

Solution

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$\textbf{(a)}$

We can express $f(t)$ as

\begin{align*} f(t)&=6e^{-t}\left(\frac{e^{2t}+e^{-2t}}{2}\right)u(t)\\ &=3\left(e^{t}+e^{-3t}\right)u(t) \end{align*}

By the linearity property,

\begin{align*} F(s)&=3\mathcal{L}[e^{t}u(t)]+3\mathcal{L}[e^{-3t}u(t)]\\ &=3\left(\frac{1}{s-1}+\frac{1}{s+3}\right)\\ &=\boxed{\frac{6(s+1)}{s^2+2s-3}} \end{align*}

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