Section 1
Laplace Transforn. Inverse Transform. Linearity. s-Shifting
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$r^{2}-2 t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\left(t^{2}-3\right)^{2}$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\cos 2 \pi t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\sin ^{2} 4 t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$e^{2 t} \cos 11 t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$e^{-t} \sinh 5 t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\cos (\omega t+\theta)$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\sin \left(3 t-\frac{1}{2}\right)$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$e^{2 a-2 b t}$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$-8 \sin 0.2 t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$\sin t \cos t$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)$$(t+1)^{3}$$
Find the Laplace transforms of the following functions. Show the details of your work. $(a, b, k, \omega, \theta$ are constants.)
Using $\mathscr{L}(f)$ in Prob. $13,$ find $\mathscr{L}\left(f_{1}\right)$, where $f_{1}(0)=0$ if $t \leq 2$ and $f_{1}(t)=1$ if $t>2$.
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Give simple examples of functions (defined for all $x \geq 0$ ) that have no Laplace transform.
Derive formula 6 from formulas 9 and 10.
Prove that $\mathscr{L}^{-1}$ is linear. Hink Use the fact that $\mathscr{C}$ is linear.
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{4 s-3 \pi}{s^{2}+\pi^{2}}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{2 s+16}{s^{2}-16}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{s^{4}-3 s^{2}+12}{s^{2}}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{10}{2 s+\sqrt{2}}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{n \pi L}{L^{2} s^{2}+n^{2} \pi^{2}}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{20}{(s-1)(s+4)}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{8}{s^{2}+4 t}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\sum_{k=1}^{4} \frac{(k+1)^{2}}{s+k^{2}}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{1}{(s-\sqrt{3})(s+\sqrt{5})}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{18 r-12}{9 s^{2}-1}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{1}{s^{2}+5}-\frac{1}{s+5}$$
Given $F(s)=\mathscr{L}(f)$Show the details. $\left(L, n, k, a,b\right.$ are constants.)$$\frac{1}{(s+a)(n+s)}$$
Find the transform.$$3.8 t e^{2.4 t}$$
Find the transform.$$-3 t^{4} e^{-0.51}$$
Find the transform.$$5 e^{-a x} \sin \omega t$$
Find the transform.$$e^{-3 t} \cos \pi t$$
Find the transform.$$e^{-k t}(a \cos t+b \sin t)$$
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Find the inverse transform. Show the details.$$\frac{7}{(x-1)^{3}}$$
Find the inverse transform. Show the details.$$\frac{\pi}{(s+\pi)^{2}}$$
Find the inverse transform. Show the details.$$(x+\sqrt{2})^{3}$$
Find the inverse transform. Show the details.$$\frac{s-6}{(a-1)^{2}+4}$$
Find the inverse transform. Show the details.$$\frac{15}{x^{2}+4 x+29}$$
Find the inverse transform. Show the details.$$\frac{4 s-2}{s^{2}-6 r+18}$$
Find the inverse transform. Show the details.$$\frac{\pi}{s^{2}+10 \pi s+24 \pi^{2}}$$