unecewelpGGi

Answered

2022-11-25

What's the correct way to go about computing the Inverse Laplace transform of this?
$\frac{-2s+1}{\left({s}^{2}+2s+5\right)}$
I Completed the square on the bottom but what do you do now?
$\frac{-2s+1}{\left(s+1{\right)}^{2}+4}$

Answer & Explanation

Dangelo Cain

Expert

2022-11-26Added 8 answers

Hints:
the inverse Laplace Transforms of the two forms are:
$\frac{a}{\left(s-b{\right)}^{2}+{a}^{2}}={e}^{bx}\mathrm{sin}ax$
$\frac{s-b}{\left(s-b{\right)}^{2}+{a}^{2}}={e}^{bx}\mathrm{cos}ax$
Can you use those two forms and get a result of:
${e}^{-t}\left(\frac{3}{2}\mathrm{sin}2t-2\mathrm{cos}2t\right)$

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