Question

Please provide steps The inverse Laplace transform for displaystyle{F}{left({s}right)}=frac{8}{{{s}+{9}}}-frac{6}{{{s}^{2}-sqrt{{3}}}} is a) displayst

Laplace transform
ANSWERED
asked 2020-10-25
Please provide steps
The inverse Laplace transform for
\(\displaystyle{F}{\left({s}\right)}=\frac{8}{{{s}+{9}}}-\frac{6}{{{s}^{2}-\sqrt{{3}}}}\) is
a) \(\displaystyle{8}{e}^{{-{9}{t}}}-{6} \sin{{h}}{{\left({3}{t}\right)}}\)
b) \(\displaystyle{8}{e}^{{-{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}\)
c) \(\displaystyle{8}{e}^{{{9}{t}}}-{6} \sin{{h}}{\left({3}{t}\right)}\)
d) \(\displaystyle{8}{e}^{{{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}\)

Answers (1)

2020-10-26
Step 1
Consider the function:
\(\displaystyle{F}{\left({s}\right)}=\frac{8}{{{x}+{9}}}-\frac{6}{{{x}^{2}-\sqrt{{3}}}}\)
Find Inverse Laplace Transform.
Apply Partial fraction on \(\displaystyle\frac{6}{{{x}^{2}-\sqrt{{3}}}}\)
\(\displaystyle\frac{6}{{{x}^{2}-\sqrt{{3}}}}=\frac{A}{{{x}-{\sqrt[{{4}}]{{{3}}}}}}+\frac{B}{{{x}+{\sqrt[{{4}}]{{{3}}}}}}\)
\(\displaystyle{6}={A}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}+{B}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}\)
Let \(\displaystyle{x}={\sqrt[{{4}}]{{{3}}}}\ \text{ and }\ {x}=-{\sqrt[{{4}}]{{{3}}}}\) and find A and B
\(\displaystyle{A}=\frac{3}{{{\sqrt[{{4}}]{{{3}}}}}}\)
\(\displaystyle{B}=-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}}}\)
\(\displaystyle{F}{\left({s}\right)}=\frac{8}{{{x}+{9}}}-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}}}+\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}}}\)
Step 2
Inverse Laplace transform property
\(\displaystyle{L}^{ -{{1}}}{\left\lbrace{a} f{{\left({s}\right)}}+ g{{\left({s}\right)}}\right\rbrace}={a}{L}^{ -{{1}}}{\left\lbrace f{{\left({s}\right)}}\right\rbrace}+{L}^{ -{{1}}}{\left\lbrace g{{\left({s}\right)}}\right\rbrace}\ \text{ and }\ {L}^{ -{{1}}}{1}{s}+{a}={e}^{{-{a}{t}}}\)
\(\displaystyle{L}^{ -{{1}}}{\left\lbrace\frac{8}{{{x}+{9}}}-\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}-{\sqrt[{{4}}]{{{3}}}}\right)}}}+\frac{3}{{{\sqrt[{{4}}]{{{3}}}}{\left({x}+{\sqrt[{{4}}]{{{3}}}}\right)}}}\right\rbrace}={8}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}+{9}}}\right\rbrace}-{3}^{{\frac{3}{{4}}}}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}-{\sqrt[{{4}}]{{{3}}}}}}\right\rbrace}+{3}^{{\frac{3}{{4}}}}{L}^{ -{{1}}}{\left\lbrace\frac{1}{{{x}+{\sqrt[{{4}}]{{{3}}}}}}\right\rbrace}\)
\(\displaystyle={8}{e}^{{-{9}{t}}}-{3}^{{\frac{3}{{4}}}}{e}^{{{\sqrt[{{4}}]{{{3}}}}{t}}}+{3}^{{\frac{3}{{4}}}}{e}^{{-{\sqrt[{{4}}]{{{3}}}}{t}}}\)
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