find the inverse of Laplace transform

$\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{({s}^{2}+4)}$

Josalynn
2021-01-08
Answered

find the inverse of Laplace transform

$\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{({s}^{2}+4)}$

You can still ask an expert for help

broliY

Answered 2021-01-09
Author has **97** answers

Step 1

Given Laplace transform is,

$\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{({s}^{2}+4)}$

Step 2

Taking inverse of Laplace transformation as,

${L}^{-1}\{\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{({s}^{2}+4)}\}={L}^{-1}\left\{\frac{3}{(s+2{)}^{2}}\right\}-{L}^{-1}\left\{\frac{2s+6}{({s}^{2}+4)}\right\}$

$=3{L}^{-1}\left\{\frac{1}{(s+2{)}^{2}}\right\}-2{L}^{-1}\left\{\frac{s}{{s}^{2}+4}\right\}-6{L}^{-1}\left\{\frac{1}{{s}^{2}+4}\right\}$

Step 3

Some formulas of inverse of Laplace transformation are

1)${L}^{-1}\{F(s)\}=f(t)\text{then}{L}^{-1}\{F(s-a)\}={e}^{at}f(t)$

$\Rightarrow {L}^{-1}\left\{\frac{1}{(s+2{)}^{2}}\right\}={e}^{-2t}{L}^{-1}\left\{\frac{1}{{s}^{2}}\right\}={e}^{-2t}t$

2)${L}^{-1}\left\{\frac{a}{{s}^{2}+{a}^{2}}\right\}=\mathrm{sin}at$

$\Rightarrow {L}^{-1}\left\{\frac{1}{{s}^{2}+4}\right\}=\frac{1}{2}{L}^{-1}\left\{\frac{2}{{s}^{2}+4}\right\}=\frac{(\mathrm{sin}2t)}{2}$

3)${L}^{-1}\{sF(s)\}={f}^{\prime}(t)+f(0)$

Then${L}^{-1}\left\{\frac{s}{{s}^{2}+4}\right\}=\frac{1}{2}{L}^{-1}\left\{s\frac{2}{{s}^{2}+4}\right\}=\frac{1}{2}((\mathrm{sin}2t{)}^{\prime}+\mathrm{sin}(0))=\mathrm{cos}2t$

Step 4

Then, consider equation (1),

$3{L}^{-1}\left\{\frac{1}{(s+2{)}^{2}}\right\}-2{L}^{-1}\left\{\frac{s}{{s}^{2}+4}\right\}-6{L}^{-1}\left\{\frac{1}{{s}^{2}+4}\right\}=3{L}^{-1}\left\{\frac{1}{(s+2{)}^{2}}\right\}-2{L}^{-1}\left\{\frac{s}{{s}^{2}+4}\right\}-\frac{6}{2}{L}^{-1}\left\{\frac{2}{{s}^{2}+4}\right\}=3{e}^{-2t}t-2\mathrm{cos}2t-3\mathrm{sin}2t$

Step 5

Therefore, the inverse of the given Laplace transformation is,

${L}^{-1}\{\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{{s}^{2}+4}\}=3{e}^{-2t}t-2\mathrm{cos}2t-3\mathrm{sin}2t$

Given Laplace transform is,

Step 2

Taking inverse of Laplace transformation as,

Step 3

Some formulas of inverse of Laplace transformation are

1)

2)

3)

Then

Step 4

Then, consider equation (1),

Step 5

Therefore, the inverse of the given Laplace transformation is,

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