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Problem 474

Use the Laplace transform of $$ f(t)=e^{k t} $$ where \(\mathrm{k}\) is a complex constant of the form $\mathrm{k}=\operatorname{Re}\\{\mathrm{k}\\}+\mathrm{i} \operatorname{lm}\\{\mathrm{k}\\}$ with \(\operatorname{Re}\\{k\\}\) the real part of \(k, \operatorname{lm}\\{k\\}\) the imaginary part of \(k\), and $$ \mathrm{i} \equiv \sqrt{(-1)} $$ to find the Laplace transforms of $$ f(t)=\cosh k t, \sinh k t, \cos \mathrm{kt}, \text { and } \sin k t $$

Expert verified

In summary, the Laplace transforms for the given functions are:
1. \(L\\{\cosh k t}\\} = \frac{2s}{s^2 - k^2}\)
2. \(L\\{\sinh k t}\\} = \frac{2k}{s^2 - k^2}\)
3. \(L\\{\cos k t}\\} = \frac{2s}{s^2 + k^2}\)
4. \(L\\{\sin k t}\\} = \frac{2k}{s^2 + k^2}\)

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Chapter 16

Find the Laplace transform $$ \mathrm{L}\\{(\sin 3 \mathrm{t}) / \mathrm{t}\\} $$

Chapter 16

Prove that $$ L\\{f(t-a) \alpha(t-a)\\}=e^{-a s} L\\{f(t)\\} $$ for any function \(\mathrm{f}\) which has a Laplace transform, and where $$ \begin{array}{lr} a(t)=0, & t<0 \\ \text { and }=1, & t \geq 0 \end{array} $$ is the unit step function, and \(\mathrm{a}>0\).

Chapter 16

Find the Laplace transform \(L\\{h(t)\\}\), where
$$
\begin{array}{ll}
\mathrm{h}(\mathrm{t})=1, & 0<\mathrm{t}<\mathrm{c} \\
\text { and }=-1, & \mathrm{c}<\mathrm{t}<2 \mathrm{c}
\end{array}
$$
and \(\mathrm{h}(\mathrm{t}+2 \mathrm{c})=\mathrm{h}(\mathrm{t})\) for all
\(\mathrm{t}\), with c a constant in the following two ways. (See Figure 1.)
(a) Use the fact that
$$
L\\{g(t)\\}=\left[1 /\left\\{s\left(1+e^{-c s}\right)\right\\}\right]
$$
where
$$
\begin{array}{ll}
g(t)=1, & 0

Chapter 16

Find the Laplace transform of $$ f(t)=t^{n} $$ where \(\mathrm{n}\) is a positive integer.

Chapter 16

Find the Laplace transform of (a) \(g(t)=t e^{4 t}\) (b) \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{7 / 2}\)

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