Science, asked by thapaavinitika6765, 8 months ago

inverse\:laplace\:\frac{1}{x^{\frac{3}{2}}}

solve it

Answers

Answered by Anonymous
1

\mathrm{Laplace\:Inverse\:Transform\:of\:}\frac{1}{x^{\frac{3}{2}}}:\quad \frac{2t^{\frac{1}{2}}}{\pi ^{\frac{1}{2}}}

L^{-1}\left\{\frac{1}{x^{\frac{3}{2}}}\right\}

=L^{-1}\left\{\frac{2}{\pi ^{\frac{1}{2}}}\cdot \frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\right\}

\mathrm{Use\:the\:constant\:multiplication\:property\:of\:Inverse\:Laplace\:Transform:}

\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant\:}a:\quad L^{-1}\left\{a\cdot f\left(t\right)\right\}=a\cdot L^{-1}\left\{f\left(t\right)\right\}

=\frac{2}{\pi ^{\frac{1}{2}}}L^{-1}\left\{\frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\right\}

\mathrm{Use\:Inverse\:Laplace\:Transform\:table}:\quad \:L^{-1}\left\{\frac{\left(2n-1\right)!!\sqrt{\pi }}{2^{ns}^{n+\frac{1}{2}}}\right\}=t^{n-1/2}

L^{-1}\left\{\frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\right\}=t^{\frac{1}{2}}

=\frac{2}{\pi ^{\frac{1}{2}}}t^{\frac{1}{2}}

=\frac{2t^{\frac{1}{2}}}{\pi ^{\frac{1}{2}}}

Answered by mangalasingh00978
0

Answer:

\mathrm{Laplace\:Inverse\:Transform\:of\:}\frac{1}{x^{\frac{3}{2}}}:\quad \frac{2t^{\frac{1}{2}}}{\pi ^{\frac{1}{2}}}LaplaceInverseTransformof

x

2

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1

:

π

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2t

2

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L^{-1}\{\frac{1}{x^{\frac{3}{2}}}\}L

−1

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x

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}

=L^{-1}\{\frac{2}{\pi ^{\frac{1}{2}}}\cdot \frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\}=L

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π

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1

2

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x

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\mathrm{Use\:the\:constant\:multiplication\:property\:of\:Inverse\:Laplace\:Transform:}UsetheconstantmultiplicationpropertyofInverseLaplaceTransform:

\mathrm{For\:function\:}f(t)\mathrm{\:and\:constant\:}a:\quad L^{-1}\{a\cdot f(t)\}=a\cdot L^{-1}\{f(t)\}Forfunctionf(t)andconstanta:L

−1

{a⋅f(t)}=a⋅L

−1

{f(t)}

=\frac{2}{\pi ^{\frac{1}{2}}}L^{-1}\{\frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\}=

π

2

1

2

L

−1

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2

1

x

2

3

π

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\mathrm{Use\:Inverse\:Laplace\:Transform\:table}:\quad \:L^{-1}\{\frac{(2n-1)!!\sqrt{\pi }}{2^{ns}^{n+\frac{1}{2}}}\}=t^{n-1/2}

L^{-1}\{\frac{\pi ^{\frac{1}{2}}}{2^1x^{\frac{3}{2}}}\}=t^{\frac{1}{2}}L

−1

{

2

1

x

2

3

π

2

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}=t

2

1

=\frac{2}{\pi ^{\frac{1}{2}}}t^{\frac{1}{2}}=

π

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1

2

t

2

1

=\frac{2t^{\frac{1}{2}}}{\pi ^{\frac{1}{2}}}=

π

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2t

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