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Problem 475

Find the Laplace transform, \(L\\{f(t)\\}=F(s)\), of (a) \(f(t)=2 \sin t+3 \cos 2 t\) (b) \(g(t)=\left[\left(1-e^{-t}\right) / t\right]\).

Expert verified

The Laplace transforms for the given functions are:
(a) \(F(s) = \frac{2}{s^2 + 1} + \frac{6s}{s^2 + 4}\)
(b) \(G(s) = - \int_0^{\infty} \ln(t)e^{-st} dt + \frac{1}{s}\int_0^{\infty} \frac{e^{-st}}{t} dt\)

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Chapter 16

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

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 transforms of (a) \(g(t)=e^{-2 t} \sin 5 t\) (b) $\mathrm{h}(\mathrm{t})=\mathrm{e}^{-\mathrm{t}} \mathrm{t} \cos 2 \mathrm{t}$

Chapter 16

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

Chapter 16

Solve the initial value problem $$ \begin{aligned} &y^{\prime \prime}(t)+2 y^{\prime}(t)+5 y(t)=H(t) \\ &y(0)=y^{\prime}(0)=0 \end{aligned} $$ where $$ \mathrm{H}(\mathrm{t})=1, \quad 0 \leq \mathrm{t}<\pi $$ $$ \text { and }=0, \quad t \geq \pi, $$ as shown in the accompanying graph.

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