Question

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

prove it

Answer 1

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

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

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

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

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

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

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

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

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

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

Answer 2

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

3

1

:

π

2

1

2t

2

1

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

−1

{

x

2

3

1

}

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

−1

{

π

2

1

2

⋅

2

1

x

2

3

π

2

1

}

\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

{

2

1

x

2

3

π

2

1

}

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

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

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

π

2

1

2

t

2

1

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

π

2

1

2t

2

1