# determine a function f(t) that has the given Laplace transform F(s)=frac{4s+5}{s^{2}+9}

determine a function f(t) that has the given Laplace transform
$F\left(s\right)=\frac{4s+5}{{s}^{2}+9}$
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Step 1
Given Laplace transform is $F\left(s\right)=\frac{4s+5}{{s}^{2}+9}$
The function can be obtained as follows. $f\left(t\right)={L}^{-1}\left\{F\left(s\right)\right\}$
$={L}^{-1}\left\{\frac{4s+5}{{s}^{2}+9}\right\}$
$={L}^{-1}\left\{\frac{4s}{{s}^{2}+9}\right\}+{L}^{-1}\left\{\frac{5}{{s}^{2}+9}\right\}$
Step 2 On further simplifications, $f\left(t\right)=4{L}^{-1}\left\{\frac{s}{{s}^{2}+{3}^{2}}\right\}+\frac{5}{3}{L}^{-1}\left\{\frac{3}{{s}^{2}+{3}^{2}}\right\}$
$=4\mathrm{cos}\left(3t\right)+\frac{5}{3}\mathrm{sin}\left(3t\right)$