Question

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

58.$\mathcal{L}\mathbf{\left\{}\mathbf{sintsinht}\mathbf{\right\}}\mathbf{=}\frac{\mathbf{2}\mathbf{s}}{{\mathbf{s}}^{\mathbf{4}}\mathbf{+}\mathbf{4}}\mathbf{;}\mathbf{}\mathcal{L}\mathbf{\left\{}\mathbf{sinktsinhkt}\mathbf{\right\}}$

Short answer

$\mathcal{L}\left\{\mathrm{sinktsinhkt}\right\}=\frac{2{\mathrm{k}}^2\mathrm{s}}{{\mathrm{s}}^4+4{\mathrm{k}}^4}$.

Step-by-step solution

Definition of Laplace Transform

Let f be a function defined for $\mathbf{t}\mathbf{\ge }\mathbf{0}$. Then the integral

$\mathcal{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{xt}}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{dt}$

is said to be the Laplace transform of f, provided that the integral converges.

Use Laplace Transform

$\mathrm{F}\left(\mathrm{s}\right)=\mathcal{L}\left\{\mathrm{f}\right(\mathrm{t}\left)\right\}=\mathcal{L}\left\{\mathrm{sintsinht}\right\}=\frac{2\mathrm{s}}{{\mathrm{s}}^4+4}$

By definition of the change of scale theorem.

$\mathcal{L}\left\{\mathrm{f}\right(\mathrm{at}\left)\right\}=\frac{1}{\mathrm{a}}\mathrm{F}\left(\frac{\mathrm{s}}{\mathrm{a}}\right)$

Thus,$\mathcal{L}\left\{\mathrm{sinktsinhkt}\right\}\stackrel{\left(2\right)}{=}\frac{1}{\mathrm{k}}\mathrm{F}\left(\frac{\mathrm{s}}{\mathrm{k}}\right)$

$\stackrel{\left(1\right)}{=}\frac{1}{\mathrm{k}}\left(\frac{2\frac{\mathrm{s}}{\mathrm{k}}}{{\left(\frac{\mathrm{s}}{\mathrm{k}}\right)}^4+4}\right)$

$=\frac{1}{\mathrm{k}}\left(\frac{2{\mathrm{k}}^3\mathrm{s}}{{\mathrm{s}}^4+4{\mathrm{k}}^4}\right)$

Therefore,$\mathcal{L}\left\{\mathrm{sinktsinhkt}\right\}=\frac{2{\mathrm{k}}^2\mathrm{s}}{{\mathrm{s}}^4+4{\mathrm{k}}^4}$.